Edge minimal Hamilton laceable bigraphs on \(2m\) vertices have at least \(\left\lfloor \frac{m+3}{6} \right\rfloor\) vertices of degree \(2\). If a bigraph is edge minimal with respect to Hamilton laceability, it is by definition edge critical, meaning the deletion of any edge will cause it to no longer be Hamilton laceable. The converse need not be true. The \(m\)-crossed prisms \([8]\) on \(4m\) vertices are edge critical for \(m \geq 2\) but not edge minimal since they are cubic. A simple modification of \(m\)-crossed prisms forms a family of “sausage” bigraphs on \(4m + 2\) vertices that are also cubic and edge critical. Both these families share the unusual property that they have exponentially many Hamilton paths between every pair of vertices in different parts. Even so, since the bigraphs are edge critical, deleting an arbitrary edge results in at least one pair having none.

In this paper, we investigate some new identities of symmetry for the Carlitz \(q\)-Bernoulli polynomials invariant under \(S_4\), which are derived from \(p\)-adic \(q\)-integrals on \(\mathbb{Z}_p\).

In this paper, we give the definition of acyclic total coloring and acyclic total chromatic number of a graph. It is proved that the acyclic total chromatic number of a planar graph \(G\) with maximum degree \(\Delta(G)\) and girth \(g\) is at most \(\Delta(G)+2\) if \(\Delta \geq 12\), or \(\Delta \geq 6\) and \(g \geq 4\), or \(\Delta = 5\) and \(g \geq 5\), or \(g \geq 6\). Moreover, if \(G\) is a series-parallel graph with \(\Delta \geq 3\) or a planar graph with \(\Delta \geq 3\) and \(g \geq 12\), then the acyclic total chromatic number of \(G\) is \(\Delta(G) + 1\).

Let \(G\) be a graph and \(\pi(G, x)\) its permanental polynomial. A vertex-deleted subgraph of \(G\) is a subgraph \(G – v\) obtained by deleting from \(G\) vertex \(v\) and all edges incident to it. In this paper, we show that the derivative of the permanental polynomial of \(G\) equals the sum of permanental polynomials of all vertex-deleted subgraphs of \(G\). Furthermore, we discuss the permanental polynomial version of Gutman’s problem [Research problem \(134\), Discrete Math. \(88 (1991) 105–106\)], and give a solution.

A semigraph G is edge complete if every pair of edges in G are adjacent. In this paper, we enumerate the non isomorphic semigraphs in one type of edge complete \((p,3)\) semigraphs without isolated vertices.

In this paper, the \(\lambda\)-number of the circular graph \(C(km, m)\) is shown to be at most \(9\) where \(m \geq 3\) and \(k \geq 2\), and the \(\lambda\)-number of the circular graph \(C(km + s, m)\) is shown to be at most \(15\) where \(m \geq 3\), \(k \geq 2\), and \(1 \leq s \leq m-1\). In particular, the \(\lambda\)-numbers of \(C(2m, m)\) and \(C(n, 2)\) are determined, which are at most \(8\). All our results indicate that Griggs and Yeh’s conjecture holds for circular graphs. The conjecture says that for any graph \(G\) with maximum degree \(\Delta \geq 2\), \(\lambda(G) \leq \Delta^2\). Also, we determine \(\lambda\)-numbers of \(C(n, 3)\), \(C(n, 4)\), and \(C(n, 5)\) if \(n \equiv 0 \pmod{7}\).

In this paper, we generalize the notion of solid bursts from classical codes equipped with Hamming metric \([14]\) to array codes endowed with RT-metric \([13]\) and obtain some bounds on the parameters of RT-metric array codes for the correction and detection of solid burst array errors.

Given a graph \(G = (V,E)\), a matching \(M\) of \(G\) is a subset of \(E\), such that every vertex of \(V\) is incident to at most one edge of \(M\). A \(k\)-matching is a matching with \(k\) edges. The total number of matchings in \(G\) is used in chemoinformatics as a structural descriptor of a molecular graph. Recently, Vesalian and Asgari (MATCH Commun. Math. Comput. Chem. \(69 (2013) 33–46\)) gave a formula for the number of \(5\)-matchings in triangular-free and \(4\)-cycle-free graphs based on the degrees of vertices and the number of vertices, edges, and \(5\)-cycles. But, many chemical graphs are not triangular-free or \(4\)-cycle-free, e.g., boron-nitrogen fullerene graphs (or BN-fullerene graphs). In this paper, we take BN-fullerene graphs into consideration and obtain formulas for the number of \(5\)-matchings based on the number of hexagons.

This paper aims to provide a systematic investigation of the family of polynomials generated by the Rodrigues’ formulas
\[K_{n_1,n_2}^{(\alpha_1, \alpha_2)}(x, k,p) = (-1)^{n_1+n_2} e^{px^k}[\prod\limits_{j=1}^2x^{-\alpha}\frac{d^nj}{dx^{n_j}} (x)^{\alpha_j+n_j}]e^{-px^k},\]
and
\[M_{n_1,n_2}^{(\alpha_0,p_1,p_2)}(x, k) = \frac{(-1)^{n_1+n_2}}{p_1^{n_1}p_2^{p_2}}x^{-\alpha_0}[\prod\limits_ {j=1}^{2}e^{p_jx^k}\frac{d^nj}{dx^{n_j}}{dx^{n_j}}e^{-p_jx^k}]x^{n_1+n_2+\alpha_0},\]
These polynomials include the multiple Laguerre and the multiple Laguerre-Hahn polynomials, respectively. The explicit forms, certain operational formulas involving these polynomials with some applications, and linear generating functions for \(K_{n_1,n_2}^{(\alpha_1, \alpha_2)}(x, k,p)\) and \(M_{n_1,n_2}^{(\alpha_0,p_1,p_2)}(x, k)\) are obtained.

For an \(n\)-connected graph \(G\), the \(n\)-wide diameter \(d_n(G)\), is the minimum integer \(m\) such that there are at least \(n\) internally disjoint \((di)\)paths of length at most \(m\) between any vertices \(x\) and \(y\). For a given integer \(l\), a subset \(S\) of \(V(G)\) is called an \((l, n)\)-dominating set of \(G\) if for any vertex \(x \in V(G) – S\) there are at least \(n\) internally disjoint \((di)\)paths of length at most \(l\) from \(S\) to \(z\). The minimum cardinality among all \((l, n)\)-dominating sets of \(G\) is called the \((l, n)\)-domination number. In this paper, we obtain that the \((l, n)\)-domination number of the \(d\)-ary cube network \(C(d, n)\) is \(2\) for \(1 \leq w \leq d\) and \(d_w(G) – f(d, n) \leq l \leq d_w(G) – 1 \) if \(d,n\geq 4\), where \(f(d, n) = \min\{e(\left\lfloor \frac{n}{2} \right\rceil + 1), \left\lfloor \frac{n}{2} \right\rfloor e\}\).

Let \(k\) be a positive integer, and let \(G\) be a simple graph with vertex set \(V(G)\). A function \(f: V(G) \rightarrow \{-1, 1\}\) is called a signed \(k\)-dominating function if \(\sum_{u \in N(v)} f(u) \geq k\) for each vertex \(v \in V(G)\). A set \(\{f_1, f_2, \ldots, f_d\}\) of signed \(k\)-dominating functions on \(G\) with the property that \(\sum_{i=1}^{d} f_i(v) \leq 1\) for each \(v \in V(G)\), is called a signed \(k\)-dominating family (of functions) on \(G\). The maximum number of functions in a signed \(k\)-dominating family on \(G\) is the signed \(k\)-domatic number of \(G\), denoted by \(d_{kS}(G)\). In this paper, we initiate the study of signed \(k\)-domatic numbers in graphs and we present some sharp upper bounds for \(d_{kS}(G)\). In addition, we determine the signed \(k\)-domatic number of complete graphs.

In this paper, \(E_2\)-cordiality of a graph \(G\) is considered. Suppose \(G\) contains no isolated vertex, it is shown that \(G\) is \(E_2\)-cordial if and only if \(G\) is not of order \(4n + 2\).

A Hamiltonian walk of a connected graph \(G\) is a closed spanning walk of minimum length in \(G\). The length of a Hamiltonian walk in \(G\) is called the Hamiltonian number, denoted by \(h(G)\). An Eulerian walk of a connected graph \(G\) is a closed walk of minimum length which contains all edges of \(G\). In this paper, we improve some results in [5] and give a necessary and sufficient condition for \(h(G) < e(G)\). Then we prove that if two nonadjacent vertices \(u\) and \(v\) satisfying that \(\deg(u) + \deg(v) \geq |V(G)|\), then \(h(G) = h(G + uv)\). This result generalizes a theorem of Bondy and Chvatal for the Hamiltonian property. Finally, we show that if \(0 \leq k \leq n-2\) and \(G\) is a 2-connected graph of order \(n\) satisfying \(\deg(u) + \deg(v) + \deg(w) \geq \frac{3n+k-2}{2}\) for every independent set \(\{u,v,w\}\) of three vertices in \(G\), then \(h(G) \leq n+k\). It is a generalization of Bondy's result.

Let \(G\) be a finite abelian group of order \(n\). The barycentric Ramsey number \(BR(H,G)\) is the minimum positive integer \(r\) such that any coloring of the edges of the complete graph \(K_r\) by elements of \(G\) contains a subgraph \(H\) whose assigned edge colors constitute a barycentric sequence, i.e., there exists one edge whose color is the “average” of the colors of all edges in \(H\). When the number of edges \(e(H) \equiv 0 \pmod{\exp(G)}\), \(BR(H,G)\) are the well-known zero-sum Ramsey numbers \(R(H,G)\). In this work, these Ramsey numbers are determined for some graphs, in particular, for graphs with five edges without isolated vertices using \(G = \mathbb{Z}_n\), where \(2 \leq n \leq 4\), and for some graphs \(H\) with \(e(H) \equiv 0 \pmod{2}\) using \(G = \mathbb{Z}_2^s\).

In this work, we consider the generalized Genocchi numbers and polynomials. However, we introduce an analytic interpolating function for the generalized Genocchi numbers attached to \(\chi\) at negative integers in the complex plane, and also we define the Genocchi \(p\)-adic \(L\)-function. As a result, we derive the value of the partial derivative of the Genocchi \(p\)-adic \(l\)-function at \(s = 0\).

Let \(G\) be a graph of order \(n\) and let \(\mu\) be an eigenvalue of multiplicity \(m\). A star complement for \(\mu\) in \(G\) is an induced subgraph of \(G\) of order \(n-m\) with no eigenvalue \(\mu\). Some general observations concerning graphs with the complete tripartite graph \(K_{r,s,t}\) as a star complement are made. We study the maximal regular graphs which have \(K_{r,s,t}\) as a star complement for eigenvalue \(\mu\). The results include a complete analysis of the regular graphs which have \(K_{n,n,n}\) as a star complement for \(\mu = 1\). It turns out that some well-known strongly regular graphs are uniquely determined by such a star complement.

In this paper, we first prove that if the edges of \(K_{2m}\) are properly colored by \(2m-1\) colors in such a way that any two colors induce a 2-factor of which each component is a 4-cycle, then \(K_{2m}\) can be decomposed into \(m\) isomorphic multicolored spanning trees. Consequently, we show that there exist three disjoint isomorphic multicolored spanning trees in any properly \((2m-1)\)-edge-colored \(K_{2m-1}\) for \(m \geq 14\).

The Merrifield-Simmons index, denoted by \(i(G)\), of a graph \(G\) is defined as the total number of its independent sets. A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than \(3\) in its unique cycle. Let \(\mathcal{U}_n^1\) be the set of fully loaded unicyclic graphs. In this paper, we determine graphs with the largest, second-largest, and third-largest Merrifield-Simmons index in \(\mathcal{U}_n^1\).

For a graph \(G = (V, E)\), the modified Schultz index of \(G\) is defined as \(S^0(G) = \sum\limits_{\{u,v\} \subset V(G)} (d_G(u) – d_G(v)) d_{G}(u, v)\), where \(d_G(u)\) (or \(d(u)\))is the degree of the vertex \(u\) in \(G\), and \(d_{G}(u, v)\) is the distance between \(u\) and \(v\). The first Zagreb index \(M_1\) is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index \(M_2\) is equal to the sum of the products of the degrees of pairs of adjacent vertices. In this paper, we present a unified approach to investigate the modified Schultz index and Zagreb indices of tricyclic graphs. The tricyclic graph with \(n\) vertices having minimum modified Schultz index and maximum Zagreb indices are determined.

Let \(T = (V, A)\) be a (finite) tournament and \(k\) be a non-negative integer. For every subset \(X\) of \(V\)\), the subtournament \(T[X] = (X, A \cap (X \times X))\) of \(T\), induced by \(X\), is associated. The dual tournament of \(T\), denoted by \(T^*\), is the tournament obtained from \(T\) by reversing all its arcs. The tournament \(T\) is self-dual if it is isomorphic to its dual. \(T\) is \((-k)\)-self-dual if for each set \(X\) of \(k\) vertices, \(T[V \setminus X]\) is self-dual. \(T\) is strongly self-dual if each of its induced subtournaments is self-dual. A subset \(I\) of \(V\) is an interval of \(T\) if for \(a,b \in I\) and for \(x \in V \setminus I\), \((a,x) \in A\) if and only if \((b,x) \in A\). For instance, \(\emptyset\), \(V\), and \(\{x\}\), where \(x \in V\), are intervals of \(T\) called trivial intervals. \(T\) is indecomposable if all its intervals are trivial; otherwise, it is decomposable. A tournament \(T’\), on the set \(V\), is \((-k)\)-hypomorphic to \(T\) if for each set \(X\) on \(k\) vertices, \(T[V \setminus X]\) and \(T'[V \setminus X]\) are isomorphic. The tournament \(T\) is \((-k)\)-reconstructible if each tournament \((-k)\)-hypomorphic to \(T\) is isomorphic to it.

Suppose that \(T\) is decomposable and \(|V| \geq 9\). In this paper, we begin by proving the equivalence between the \((-3)\)-self-duality and the strong self-duality of \(T\). Then we characterize each tournament \((-3)\)-hypomorphic to \(T\). As a consequence of this characterization, we prove that if there is no interval \(X\) of \(T\) such that \(T[X]\) is indecomposable and \(|V \setminus X| \leq 2\), then \(T\) is \((-3)\)-reconstructible. Finally, we conclude by reducing the \((-3)\)-reconstruction problem.

For a given graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(\pi\) containing \(H\) as a subgraph. In this paper, we characterize the potentially \(C_{2,6}\)-graphic sequences. This characterization partially answers Problem 6 in Lai and Hu [12].

We investigate two modifications of the well-known irregularity strength of graphs, namely the total edge irregularity strength and the total vertex irregularity strength.
In this paper, we determine the exact value of the total edge (vertex) irregularity strength for Halin graphs.

A signed \(k\)-dominating function of a graph \(G = (V, E)\) is a function \(f: V \rightarrow \{+1,-1\}\) such that \(\sum_{u \in N_G[v]} f(u) \geq k\) for each vertex \(v \in V\). A signed \(k\)-dominating function \(f\) of a graph \(G\) is minimal if no \(g \leq f\) is also a signed \(k\)-dominating function. The weight of a signed \(k\)-dominating function is \(w(f) = \sum_{v \in V} f(v)\). The upper signed \(k\)-domination number \(\Gamma_{s,k}(G)\) of \(G\) is the maximum weight of a minimal signed \(k\)-dominating function on \(G\). In this paper, we establish a sharp upper bound on \(\Gamma _{s,k}(G)\) for a general graph in terms of its minimum and maximum degree and order, and construct a class of extremal graphs which achieve the upper bound. As immediate consequences of our result, we present sharp upper bounds on \(\Gamma _{s,k}(G)\) for regular graphs and nearly regular graphs.

The paper contains enumerative combinatorics for positive braids, square free braids, and simple braids, emphasizing connections with classical Fibonacci sequence.

Suppose that \(D\) is an acyclic orientation of a graph \(G\). An arc of \(D\) is called dependent if its reversal creates a directed cycle. Let \(d_{\min}(G)\) (\(d_{\max}(G)\)) denote the minimum (maximum) of the number of dependent arcs over all acyclic orientations of \(G\). We call \(G\) fully orientable if \(G\) has an acyclic orientation with exactly \(d\) dependent arcs for every \(d\) satisfying \(d_{\min}(G) \leq d \leq d_{\max}(G)\). A graph \(G\) is called chordal if every cycle in \(G\) of length at least four has a chord. We show that all chordal graphs are fully orientable.

A graph \(G\) with no isolated vertex is total restrained domination vertex critical if for any vertex \(v\) of \(G\) that is not adjacent to a vertex of degree one, the total restrained domination number of \(G – v\) is less than the total restrained domination number of \(G\). We call these graphs \(\gamma_{tr}\)-vertex critical. If such a graph \(G\) has total restrained domination number \(k\), then we call it \(k\)-\(\gamma_{tr}\)-vertex critical. In this paper, we study some properties in \(\gamma_{tr}\)-vertex critical graphs of minimum degree at least two.

In this paper, we give a necessary and sufficient condition for a function with the form \(tr(\sum_{i=1}^q a_ix^{i(q-1)})\) to be a generalized bent function. We indicate that these generalized bent functions are just those which could be constructed from partial spreads. We also introduce a method to calculate these generalized bent functions by means of interpolation.

Let \(G\) be a finite group and \(n\) a positive integer. The \(n\)-th commutativity degree \(P_n(G)\) of \(G\) is the probability that the \(n\)-th power of a random element of \(G\) commutes with another random element of \(G\). In 1968, P. Erdős and P. Turán investigated the case \(n = 1\), involving only methods of combinatorics. Later, several authors improved their studies and there is a growing literature on the topic in the last 10 years. We introduce the relative \(n\)-th commutativity degree \(P_n(H,G)\) of a subgroup \(H\) of \(G\). This is the probability that an \(n\)-th power of a random element in \(H\) commutes with an element in \(G\). The influence of \(P_n(G)\) and \(P_n(H,G)\) on the structure of \(G\) is the purpose of the present work.

It is known that determining the Lagrangian of a general \(r\)-uniform hypergraph is useful in practice and is non-trivial when \(r \geq 3\). In this paper, we explore the Lagrangians of \(3\)-uniform hypergraphs with edge sets having restricted structures. In particular, we establish a number of optimization problems for finding the largest Lagrangian of \(3\)-uniform hypergraphs with the number of edges \(m = \binom{k}{3} – a\), where \(a = 3\) or \(4\). We also verify that the largest Lagrangian has the colex ordering structure for \(3\)-uniform hypergraphs when the number of edges is small.

Let \(D\) be an acyclic orientation of a simple graph \(G\). An arc of \(D\) is called dependent if its reversal creates a directed cycle. Let \(d(D)\) denote the number of dependent arcs in \(D\). Define \(d_{\min}(G)\) (\(d_{\max}(G)\)) to be the minimum (maximum) number of \(d(D)\) over all acyclic orientations \(D\) of \(G\). We call \(G\) fully orientable if \(G\) has an acyclic orientation with exactly \(k\) dependent arcs for every \(k\) satisfying \(d_{\min}(G) \leq k \leq d_{\max}(G)\). In this paper, we prove that the square of a cycle \(C_n\) is fully orientable except for \(n = 6\).

Let \(G = (V, A)\) be a graph. For every subset \(X\) of \(V\), the sub-graph \(G(X) = (X, A \cap (X \times X))\) of \(G\) induced by \(X\) is associated. The dual of \(G\) is the graph \(G^* = (V, A^*)\)such that \(A^* = \{(x,y): (y,x) \in A\}\). A graph \(G’\) is hemimorphic to \(G\) if it is isomorphic to \(G\) or \(G^*\). Let \(k \geq 1\) be an integer. A graph \(G’\) defined on the same vertex set \(V\) of \(G\) is \((\leq k)\)-hypomorphic (resp. \((\leq k)\)-hemimorphic) to \(G\) if for all subsets \(X\) of \(V\) with at most \(k\) elements, the sub-graphs \(G(X)\) and \(G'(X)\) are isomorphic (resp. hemimorphic). \(G\) is called \((\leq k)\)-reconstructible (resp. \((\leq k)\)-half-reconstructible) provided that every graph \(G’\) which is \((\leq k)\)-hypomorphic (resp. \((\leq k)\)-hemimorphic) to \(G\) is hypomorphic (resp. hemimorphic) to \(G\). In 1972, G. Lopez {14,15] established that finite graphs are \((\leq 6)\)-reconstructible. For \(k \in \{3,4,5\}\), the \((<k)\)-reconstructibility problem for finite graphs was studied by Y. Boudabbous and G. Lopez [1,5]. In 2006, Y. Boudabbous and C. Delhommé [4] characterized, for each \(k \geq 4\), all \((\leq k)\)-reconstructible graphs. In 1993, J. G. Hagendorf and G. Lopez showed in [12] that finite graphs are \((\leq 12)\)-half-reconstructible. After that, in 2003, J. Dammak [8] characterized the \((\leq k)\)-half-reconstructible finite graphs for every \(7 \leq k \leq 11\). In this paper, we characterize for each integer \(7 \leq k \leq 12\), all \((\leq k)\)-half-reconstructible graphs.

In this paper, we study the relations between degree sum and extending paths in graphs. The following result is proved. Let \(G\) be a graph of order \(n\), if \(d(u)+d(v) \geq n+k\) for each pair of nonadjacent vertices \(u,v\) in \(V(G)\), then every path \(P\) of \(G\) with \(\frac{n}{k+2} \leq 2 < n\) is extendable. The bound \(\frac{n}{k+2}+2\) is sharp.

A median graph is a connected graph in which, for every three vertices, there exists a unique vertex \(m\) lying on the geodesic between any two of the given vertices. We show that the only median graphs of the direct product \(G \times H\) are formed when \(G = P_k\), for any integer \(k \geq 3\), and \(H = P_l\), for any integer \(l \geq 2\), with a loop at an end vertex, where the direct product is taken over all connected graphs \(G\) on at least three vertices or at least two vertices with at least one loop, and connected graphs \(H\) with at least one loop.

An urn contains \(m\) distinguishable balls with \(m\) distinguishable colors. Balls are drawn for \(n\) times successively at random
and with replacement from the urn. The mathematical expectation of the number of drawn colors is investigated. Some combinatorial identities on the Stirling number of the second kind \(S(n,m)\) are derived by using probabilistic method.

Let \(G\) be a finite group. The commutativity degree of \(G\), written \(d(G)\), is defined as the ratio \[\frac{|\{(x, y)x,y \in G, xy = yx\}|}{|G|^2}\]. In this paper, we examine the commutativity degree of groups of nilpotency class 2 and, by using numerical solutions of the equation \(xy \equiv zu \pmod{n}\), we give certain explicit formulas for some particular classes of finite groups. A lower bound for \(d(G)\) is obtained for \(2\)-generated groups of nilpotency class \(2\).

For a graph \(G\), the Hosoya index is defined as the total number of its matchings. A generalized \(\theta\)-graph \((r_1, r_2, \ldots, r_k)\) consists of a pair of end vertices joined by \(k\) internally disjoint paths of lengths \(r_1 + 1, r_2 + 1, \ldots, r_k + 1\). Let \(\Theta_k\) denote the set of generalized \(\theta\)-graphs with \(k \geq 4\). In this paper, we obtain the smallest and the largest Hosoya index of the generalized \(\theta\)-graph in \(\Theta_n^k\), respectively. At the same time, we characterize the corresponding extremal graphs.

The purpose of this paper is to solve the odd minimum \(S\)-cut, the odd minimum \(\bar{T}\)-cut, and the odd minimum \((S, T)\)-cut problems in directed graphs using triple families. We also provide here two properties of triple families.

Let \(G\) be a graph and let \(\delta(G)\) denote the minimum degree of \(G\). Let \(F\) be a given connected graph. Suppose that \(|V(G)|\) is a multiple of \(|V(F)|\). A spanning subgraph of \(G\) is called an \(F\)-factor if its components are all isomorphic to \(F\). In 2002, Kawarabayashi [5] conjectured that if \(G\) is a graph of order \(n\) (\(n \geq 3\)) with \(\delta(G) \geq \frac{\ell^2-3\ell+1}{\ell-2}\), then \(G\) has a \(K_\ell^-\)-factor, where \(K_\ell^-\) is the graph obtained from \(K_\ell\) by deleting just one edge. In this paper, we prove that this conjecture is true when \(\ell = 5\).

The \(b\)-chromatic number \(b(G)\) of a graph \(G\) is defined as the maximum number \(k\) of colors in a proper coloring of the vertices of \(G\) in such a way that each color class contains at least one vertex adjacent to a vertex of every other color class. Let \(\mu(G)\) denote the Mycielskian of \(G\). In this paper, it is shown that if \(G\) is a graph with \(b\)-chromatic number \(b\) and for which the number of vertices of degree at least \(b\) is at most \(2b – 2\), then \( b(\mu(G))\) lies in the interval \([b+1, 2b-1]\). As a consequence, it follows that \(b(G)+1 \leq b(\mu(G)) \leq 2b(G) -1\) for \(G\) in any of the following families: split graphs, \(K_{n,n} – \{a \ 1\text{-factor}\}\), the hypercubes \(Q_p\), where \(p \geq 3\), trees, and a special class of bipartite graphs. We show further that for any positive integer \(b\) and every integer \(k \in [b+1, 2b-1]\), there exists a graph \(G\) belonging to the family mentioned above, with \(b(G) = b\) and \(b(\mu(G)) = k\).

For a graph \(G = (V,E)\), the Schultz index of \(G\) is defined as \(S(G) = \sum\limits_{\{u,v \}\subseteq V(G)} (d_G(u) + d_G(v))d_G(u,v)\), where \(d_G(u)\) is the degree of the vertex \(u\) in \(G\), and \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\). In this paper, we investigate the Schultz index of tricyclic graphs. The \(n\)-tricyclic graphs with the minimum Schultz index are determined.

In this paper, we investigate the existence of perfect state transfer in integral circulant graphs between non-antipodal vertices—vertices that are not at the diameter of a graph. Perfect state transfer is considered on circulant quantum spin networks with nearest-neighbor couplings. The network is described by a circulant graph \(G\), which is characterized by its circulant adjacency matrix \(A\). Formally, we say that there exists perfect state transfer (PST) between vertices \(a, b \in V(G)\) if \(|F(\tau)_{ab}| = 1\) for some positive real number \(\tau\), where \(F(\tau) = \exp(itA)\). Saxena, Severini, and Shparlinski (International Journal of Quantum Information 5 (2007), 417-430) proved that \(|F(\tau)_{aa}| = 1\) for some \(a \in V(G)\) and \(t \in \mathbb{R}\) if and only if all the eigenvalues of \(G\) are integers (that is, the graph is integral). The integral circulant graph \(ICG_n(D)\) has the vertex set \(\mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}\) and vertices \(a\) and \(b\) are adjacent if \(\gcd(a-b, n) \in D\), where \(D \subseteq \{d: d|n, 1 \leq d \leq n\}\). We characterize completely the class of integral circulant graphs having PST between non-antipodal vertices for \(|D| = 2\). We have thus answered the question posed by Godsil on the existence of classes of graphs with PST between non-antipodal vertices. Moreover, for all values of \(n\) such that \(ICG_n(D)\) has PST (\(n \in 4\mathbb{N}\)), several classes of graphs \(ICG_n(D)\) are constructed such that PST exists between non-antipodal vertices.

Chemical indices are introduced to correlate chemical compounds’ physical properties with their structures. Among recently introduced such indices, the eccentric connectivity index of a graph \(G\) is defined as \(\xi^C(G) = \sum_{v \in V(G)} deg(v) ec(v)\), where \(deg(v)\) is the degree of a vertex \(v\) and \( ec(v)\) is its eccentricity. The extremal values of \(\xi^C(G)\) have been studied among graphs with various given parameters. In this note, we study trees with extremal values of the eccentric connectivity index with a given degree sequence. The extremal structures are identified; however, they are not unique.

A \(k\)-L\((d, 1)\)-labeling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to \(\{0, 1, \ldots, k\}\) such that \(|f(u) – f(v)| > 1\) if \(d(u,v) = 2\) and \(|f(u) – f(v)| \geq d\) if \(d(u,v) = 1\). The L\((d,1)\)-labeling number \(\lambda(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has a \(k\)-L\((d, 1)\)-labeling. In this paper, we show that \(2d+2 \leq \lambda(C_m \square C_n) \leq 2d+4\) if either \(m\) or \(n\) is odd. Furthermore, the following cases are determined: (a) \(\lambda_d(C_3 \square C_n)\) and \(\lambda_d(C_4 \square C_n)\) for \(d \geq 3\), (b) \(\lambda_d(C_m \square C_n)\) for some \(m\) and \(n\), (c) \(\lambda_d(C_{2m} \square C_{2n})\) for \(d \geq 5\) when \(m\) and \(n\) are even.

The purpose of this paper is to establish several identities involving \(q\)-harmonic numbers by the \(q\)-Chu-Vandermonde convolution formula and obtain some \(q\)-analogues of several known identities.

It will be proved that the problem of determining whether a set of vertices of a dually chordal graphs is the set of leaves of a tree compatible with it can be solved in polynomial time by establishing a connection with finding clique trees of chordal graphs with minimum number of leaves.

A vertex subset \(F\) is an \(R_k\)-vertex-cut of a connected graph \(G\) if \(G – F\) is disconnected and every vertex in \(G – F\) has at least \(k\) neighbors in \(G – F\). The cardinality of the minimum \(R_k\)-vertex-cut of \(G\) is the \(R_k\)-connectivity of \(G\), denoted by \(\kappa^k(G)\). This parameter measures a kind of conditional fault tolerance of networks. In this paper, we determine \(R_2\)-connectivity and \(R_3\)-connectivity of recursive circulant graphs \(G(2^m, 2)\).

In this paper, we introduce \(h(x)\)-Lucas quaternion polynomials that generalize \(k\)-Lucas quaternion numbers that generalize Lucas quaternion numbers. Also we derive the Binet formula and generating function of \(h(x)\)-Lucas quaternion polynomial sequence.

We determine the crossing numbers (i) of the complete graph \(K_n\) with an edge deleted for \(n \leq 12\) and (ii) of the complete bipartite graph \(K_{m,n}\) with an edge deleted for \(m \in \{3,4\}\) and for all natural numbers \(n$\), and also for the case \(m = n = 5\).

A \(G\)-design is called balanced if the degree of each vertex \(x\) is a constant. A \(G\)-design is called strongly balanced if for every \(i = 1, 2, \ldots, h\), there exists a constant \(C_i\) such that \(d_{A_i}(x) = C_i\) for every vertex \(x\), where \(A_i\) are the orbits of the automorphism group of \(G\) on its vertex-set and \(d_{A_i}(x)\) of a vertex is the number of blocks containing \(x\) as an element of \(A_i\). We say that a \(G\)-design is simply balanced if it is balanced, but not strongly balanced. In this paper, we determine the spectrum for simply balanced and strongly balanced House-systems. Further, we determine the spectrum for House-systems of all admissible indices nesting \(C_4\)-systems.

The Wiener index of a graph is the sum of the distances between all pairs of vertices. In this paper, we determine \(h\)-cacti and \(h\)-cactus chains with the extremal Wiener indices, respectively.

A cyclic coloring is a vertex coloring such that vertices incident with the same face receive different colors. Let \(G\) be a plane graph, and let \(\Delta^*\) be the maximum face degree of \(G\). In 1984, Borodin conjectured that every plane graph admits a cyclic coloring with at most \(\left\lfloor \frac{3\Delta^*}{2} \right\rfloor\) colors. In this note, we improve a result of Borodin et al. [On cyclic colorings and their generalizations, Discrete Mathematics 203 (1999), 23-40] by showing that every plane graph with \(\Delta^* = 6\) can be cyclically colored with 9 colors. This confirms the Cyclic Coloring Conjecture in the case \(\Delta^* = 6\).

In this paper, we derive some identities involving Genocchi polynomials and numbers. These identities follow by evaluating a certain integral in various ways. Also, we express the product of two Genocchi polynomials as a linear combination of Bernoulli polynomials.

Fuzzy graph theory is finding an increasing number of applications in modeling real-time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences, and the symbolic models used in expert systems. A bipolar fuzzy model is a generalized soft computing model of a fuzzy model that gives more precision, flexibility, and compatibility to a system when compared with systems designed using fuzzy models. In this research article, we introduce certain types of bipolar fuzzy competition graphs, including bipolar fuzzy \(k\)-competition, bipolar fuzzy \(p\)-competition, and bipolar fuzzy \(m\)-competition. We investigate some properties of these new concepts.

The \(\alpha\)-incidence energy of a graph is defined as the sum of \(a\)th powers of the signless Laplacian eigenvalues of the graph, where \(a\) is a real number such that \(\alpha \neq 0\) and \(\alpha \neq 1\). The \(\alpha\)-distance energy of a graph is defined as the sum of \(a\)th powers of the absolute values of the eigenvalues of the distance matrix of the graph, where \(\alpha\) is a real number such that \(\alpha \neq 0\). In this note, we present some bounds for the \(\alpha\)-incidence energy of a graph. We also present some bounds for the \(\alpha\)-distance energy of a tree.

Multi-sender authentication codes allow a group of senders to construct an authenticated message for a receiver such that the receiver can verify authenticity of the received message. In this paper, we construct one multi-sender authentication codes from
polynomials over finite fields. Some parameters and the probabilities of deceptions of this codes are also computed.

A graph \(G\) is called \((k, d)^*\)-choosable if for every list assignment \(L\) satisfying \(|L(v)| \geq k\) for all \(v \in V(G)\), there is an \(L\)-coloring of \(G\) such that each vertex of \(G\) has at most \(d\) neighbors colored with the same color as itself. In this paper, it is proved that every graph of nonnegative characteristic without \(4\)-cycles and intersecting triangles is \((3, 1)^*\)-choosable.

In this paper, we study \((2-d)\)-kernels in graphs. We shall show that the problem of the existence of \((2-d)\)-kernels is \(\mathcal{N}P\)-complete for a general graph. We also give some results related to the problem of counting \((2-d)\)-kernels in graphs. For special graphs, we show that the number of \((2-d)\)-kernels is equal to the Fibonacci numbers.

In 1989, Frankl and Füredi [1] conjectured that the \(r\)-uniform hypergraph with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of \(\mathbb{N}^{(r)}\) has the largest Lagrangian of all \(r\)-uniform hypergraphs of size \(m\). For \(2\)-graphs, the Motzkin-Straus theorem implies this conjecture is true. For \(3\)-uniform hypergraphs, it was proved by Talbot in 2002 that the conjecture is true while \(m\) is in a certain range. In this paper, we prove that the \(4\)-uniform hypergraphs with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of \(\mathbb{N}^{(r)}\) has the largest Lagrangian of all \(4\)-uniform hypergraphs with \(t\) vertices and \(m\) edges satisfying \(\binom{t-1}{4} \leq m \leq \binom{t-1}{4} + \binom{t-2}{3} – 17\binom{t-2}{2} + 1\).

A graph \(G\) on \(n \geq 3\) vertices is called claw-heavy if every induced claw of \(G\) has a pair of nonadjacent vertices such that their degree sum is at least \(n\). We say that a subgraph \(H\) of \(G\) is \(f\)-heavy if \(\max\{d(x), d(y)\} \geq \frac{n}{2}\) for every pair of vertices \(x, y \in V(H)\) at distance \(2\) in \(H\). For a given graph \(R\), \(G\) is called \(R\)-\(f\)-heavy if every induced subgraph of \(G\) isomorphic to \(R\) is \(f\)-heavy. For a family \(\mathcal{R}\) of graphs, \(G\) is called \(\mathcal{R}\)-\(f\)-heavy if \(G\) is \(R\)-\(f\)-heavy for every \(R \in \mathcal{R}\). In this paper, we show that every \(2\)-connected claw-heavy graph is hamiltonian if \(G\) is \(\{P_7, D\}\)-\(f\)-heavy, or \(\{P_7, H\}\)-\(f\)-heavy, where \(D\) is a deer and \(H\) is a hourglass. Our result is a common generalization of previous theorems of Broersma et al. and Fan on hamiltonicity of \(2\)-connected graphs.

An \(H_3\) graph is a multigraph on three vertices with double edges between two pairs of distinct vertices and a single edge between the third pair. In this paper, we decompose a complete multigraph \(2K_{10t}\) into \(H_3\) graphs.

In 1989, Zhu, Li, and Deng introduced the definition of implicit degree, denoted by \(\text{id}(v)\), of a vertex \(v\) in a graph \(G\). In this paper, we give a simple method to prove that: if \(G\) is a \(k\)-connected graph of order \(n\) such that the implicit degree sum of any \(k+1\) independent vertices is more than \((k+1)(n-1)/2\), then \(G\) is hamiltonian. Moreover, we provide an algorithm according to the proof.

Let \(D\) be a finite and simple digraph with vertex set \(V(D)\), and let \(f: V(D) \to \{-1, 1\}\) be a two-valued function. If \(\sum_{x \in N_D^-[v]} f(x) \geq 1\) for each \(v \in V(D)\), where \(N_D^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\), then \(f\) is a signed dominating function on \(D\). The sum \(\sum_{v \in V(D)} f(v)\) is called the weight of \(f\). The signed domination number, denoted by \(\gamma_S(D)\), of \(D\) is the minimum weight of a signed dominating function on \(D\). In this work, we present different lower bounds on \(\gamma_S(D)\) for general digraphs, show that these bounds are sharp, and give an improvement of a known lower bound obtained by Karami in 2009 [H. Karami, S.M. Sheikholeslami, A. Khodkar, Lower bounds on the signed domination numbers of directed graphs, Discrete Math. 309 (2009), 2567-2570]. Some of our results are extensions of well-known properties of the signed domination number of graphs.

Let \(G\) be a graph of order at least \(2k\) and \(s_1, s_2, \ldots, s_k, t_1, t_2, \ldots, t_k\) be any \(2k\) distinct vertices of \(G\). If there exist \(k\) disjoint paths \(P_1, P_2, \ldots, P_k\) such that \(P_i\) is an \(s_i – t_i\) path for \(1 \leq i \leq k\), we call \(G\) \(k\)-linked. K. Kawarabayashi et al. showed that if \(n \geq 4k – 1\) (\(k \geq 2\)) with \(\sigma_2(G) \geq n + 2k – 3\), then \(G\) is \(k\)-linked. Li et al. showed that if \(G\) is a graph of order \(n \geq 232k\) with \(\sigma_2^*(G) \geq n + 2k – 3\), then \(G\) is \(k\)-linked. For sufficiently large \(n\), it implied the result of K. Kawarabayashi et al. The main purpose of this paper is to lower the bound of \(n\) in the result of Li et al. We show that if \(G\) is a graph of order \(n \geq 111k + 9\) with \(\sigma_2^*(G) \geq n + 2k – 3\), then \(G\) is \(k\)-linked. Thus, we improve the order bound to \(111k + 9\), and when \(n \geq 111k + 9\), it implies the result of \(K\). Kawarabayashi \(et al\).

The classification of all dihedral f-tilings of the Riemannian sphere \(S^2\) ,whose prototiles are two right triangles with at least one isosceles, is given.The combinatorial structure and the symmetry group of each tiling is also achieved.

In [4], the author introduced a new metric on the space \(\text{Mat}_{m \times s}(\mathbb{Z}_q)\), which is the module space of all \(m \times s\) matrices with entries from the finite ring \(\mathbb{Z}_q\) (\(q \geq 2\)), generalizing the classical Lee metric [5] and the array RT-metric [8], and named this metric as GLRTP-metric, which is further renamed as LRTJ-metric (Lee-Rosenbloom-Tsfasman-Jain Metric) in [1]. In this paper, we introduce a complete weight enumerator for codes over \(\text{Mat}_{m \times s}(\mathbb{Z}_q)\) endowed with the LRTJ-metric and obtain a MacWilliams-type identity with respect to this new metric for the complete weight enumerator.

The Zagreb indices and the modified Zagreb indices are important topological indices in mathematical chemistry. In this paper we study the relationship between the modified Zagreb indices and the reformulated modified Zagreb indices with respect to trees.

In this work, we first prove that every prime number \(p \equiv 1 \pmod{4}\) can be written in the form \(p = P^2 – 4Q\)with two positive integers \(P\) and \(Q\). Then, we define the sequence \(B_n(P, Q)\) to be \(B_0 = 2\), \(B_1 = P\), and \(B_n = PB_{n-1} – QB_{n-2}\) for \(n \geq 2\), and derive some algebraic identities on it. Also, we formulate the limit of the cross-ratio for four consecutive numbers \(B_n\), \(B_{n+1}\), \(B_{n+2}\), and \(B_{n+3}\).

An edge set \(F\)is called a restricted edge-cut if \(G – F\) is disconnected and contains no isolated vertices. The minimum cardinality over all restricted edge-cuts is called the restricted edge-connectivity of \(G\), and denoted by \(\lambda'(G)\). A graph \(G\) is called \(\lambda’\)-optimal if \(\lambda'(G) = \xi(G)\), where \(\xi(G) = \min\{d_G(u) + d_G(v) – 2: uv \in E(G)\}\). In this note, we obtain a sufficient condition for a \(k( \geq 3)\)-regular connected graph with two orbits to be \(\lambda’\)-optimal.

Let \(G = (V, E)\) be a connected graph. An edge set \(S \subset E\) is a \(k\)-restricted edge cut if \(G – S\) is disconnected and every component of \(G – S\) has at least \(k\) vertices. The \(k\)-restricted edge connectivity \(\lambda_k(G)\) of \(G\) is the cardinality of a minimum \(k\)-restricted edge cut of \(G\). A graph \(G\) is \(\lambda_k\)-connected if \(k\)-restricted edge cuts exist. A graph \(G\) is called \(\lambda_k\)-optimal if \(\lambda_k(G) = \xi_k(G)\), where \[\xi_k(G) = \min\{|[X, Y]|: X \subseteq V, |X| = k \text{ and } G[X] \text{ is connected}\};\] Here, \(G[X]\) is the subgraph of \(G$\) induced by the vertex subset \(X \subseteq V\), and \(Y = V \setminus X\) is the complement of \(X\); \([X, Y]\) is the set of edges with one end in \(X\) and the other in \(Y\). \(G\) is said to be super-\(\lambda_k\) if each minimum \(k\)-restricted edge cut isolates a connected subgraph of order \(k\). In this paper, we give some sufficient conditions for triangle-free graphs to be super-\(\lambda_3\).

The \(k\)-path-connectivity \(\pi_k(G)\) of a graph \(G\) was introduced by Hager in \(1986\). Recently, Mao investigated the \(3\)-path-connectivity of lexicographic product graphs. Denote by \(G \circ H\) the lexicographic product of two graphs \(G\) and \(H\). In this paper, we prove that \(\pi_4(G \circ H) \geq \lfloor\frac{|V(H)|-2}{3}\rfloor\) for any two connected graphs \(G\) and \(H\). Moreover, the bound is sharp. We also derive an upper bound of \(\pi_4(G \circ H)\), that is, \(\pi_4(G \circ H) \leq 2\pi_4(G)|V(H)|\).

This paper devotes to the investigation of \(3\)-designs admitting the special projective linear group \(\mathrm{PSL}(2, 2^n)\) as an automorphism, and we determine all the possible values of \(n\) in the simple \(3-(2^n + 1, 7, \lambda)\) designs admitting \(\mathrm{PSL}(2,2^n)\) as an automorphism group.

In this note, we characterize graphs with a given small matching number. Specifically, we characterize graphs with minimum degree at least \(2\) and matching number at most \(3\). The characterization when the matching number is at most \(2\) strengthens the result of Lai and Yan’s that characterized the non-supereulerian \(2\)-edge connected graphs with matching at most \(2\). Furthermore, the characterization of graphs with matching number at most \(3\) addresses a conjecture of Lai and Yan in [SuperEulerian graphs and matchings, Applied Mathematics Letters 24 (2011) 1867-1869].

Let \(D(G)\) be the distance matrix of a connected graph \(G\). The distance spectral radius of \(G\) is the largest eigenvalue of \(D(G)\) and has been proposed as a molecular structure descriptor. In this paper, we study the distance spectral radius of graphs with a given independence number. Special attention is paid to graphs with a given independence number and maximal distance spectral radius.

Key distribution is paramount for Wireless Sensor Networks (WSNs). The design of key management schemes is the most important aspect and basic research field in WSNs. A key distribution scheme based on symplectic geometry over fields is proposed, where a 2-dimensional subspace in symplectic geometry represents a node, and all \(2s\)-dimensional non-isotropic subspaces represent the key pool, guaranteeing that every pair of nodes has a shared key, thus improving network connectivity. The performance analysis shows that the scheme has good connectivity and higher resilience to node compromise compared to other key pre-distribution schemes.

For a given graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(\pi\) containing \(H\) as a subgraph. Let \(K_{ r+1} – C_k\) be the graph obtained from \(K_{ r+1}\) by removing the \(k\) edges of a \(k\)-cycle. In this paper, we first characterize potentially \(A_{ r+1} – C_k\)-graphic sequences (\(3 \leq k \leq r+1\)), analogous to Yin et al.’s characterization [19], using a system of inequalities. Then, we obtain a sufficient and necessary condition for a graphic sequence \(\pi\) to have a realization containing \(K_{r+1} – C_k\) as an induced subgraph.

A graph \(G\) with \(1 \leq n \leq |V(G)| – 2\) is said to be \(n\)-factor-critical if any \(n\) vertices of \(G\) are deleted, then the resultant graph has a perfect matching. An odd graph \(G\) with \(2k \leq |V(G)| – 3\) is said to be near \(k\)-extendable if \(G\) has a \(k\)-matching and any \(k\)-matching of \(G\) can be extended to a near perfect matching of \(G\). Lou and Yu [Australas. J. Combin. 29 (2004) 127-133] showed that any \(5\)-connected planar odd graph is \(3\)-factor-critical. In this paper, as an improvement of Lou and Yu’s result, we prove that any \(4\)-connected planar odd graph is \(3\)-factor-critical and also near \(2\)-extendable. Furthermore, we prove that all \(5\)-connected planar odd graphs are near \(3\)-extendable.

Determining the biplanar crossing number of the graph \(C_n \times C_n \times C_n \times P_n\) was a problem proposed in a paper by Czabarka, Sykora, Székely, and Vito [2]. We find as a corollary to the main theorem of this paper that the biplanar crossing number of the aforementioned graph is zero. This result follows from the decomposition of \(C_n \times C_n \times C_n \times P_m\) into one copy of \(C_{n^2} \times P_{lm},l-2\) copies of \(C_{n^2} \times P_m\), and a copy of \(C_{n^2} \times P_{2m}\).

Let \(A_n\) be the alternating group of degree \(n\) with \(n \geq 5\). Set \(S = \{(1ij), (1ji) \mid 2 \leq i, j \leq n, i \neq j\}\). In this paper, it is shown that the full automorphism group of the Cayley graph \(\mathrm{Cay}(A_n, S)\) is the semi-product \(R(A_n) \rtimes \mathrm{Aut}(A_n, S)\), where \(R(A_n)\) is the right regular representation of \(A_n\) and \(\mathrm{Aut}(A_n, S) = \{\phi \in \mathrm{Aut}(A_n) \mid S^\phi = S\} \cong \mathrm{S_{n-1}}\).

Topological indices of graphs, and trees in particular, have been vigorously studied in the past decade due to their many applications in different fields. Among such indices, the number of subtrees (BC-subtrees), along with their variations, have received much attention. In this paper, we provide some new evaluation results related to these two indices on specific structures, such as generalized Bethe trees, Bethe trees, and dendrimers, which are of practical interest. Using generating functions, we also examine the asymptotic behavior of subtree (resp. BC-subtree) density of dendrimers.

For an integer \(k \geq 0\), a graphical property \(P\) is said to be \(k\)-stable if whenever \(G + uv\) has property \(P\) and \(d_G(u) + d_G(v) \geq k\), where \(uv \notin E(G)\), then \(G\) itself has property \(P\). In this note, we present spectral sufficient conditions for several stable properties of a graph.

Let \(G\) be a connected graph. The degree resistance distance of \(G\) is defined as \(D_R(G) = \sum\limits_{\{u,v\} \in V(G)} (d(u) + d(v))r(u,v)\), where \(d(u)\) (and \(d(v)\)) is the degree of the vertex \(u\) (and \(v\)), and \(r(u,v)\) is the resistance distance between vertices \(u\) and \(v\). A fully loaded unicyclic graph is a unicyclic graph with the property that there is no vertex with degree less than \(3\) in its unique cycle. In this paper, we determine the minimum and maximum degree resistance distance among all fully loaded unicyclic graphs with \(n\) vertices, and characterize the extremal graphs.

The cyclic edge-connectivity of a cyclically separable graph \(G\), denoted by \(c\lambda(G)\), is the minimum cardinality of all edge subsets \(F\) such that \(G – F\) is disconnected and at least two of its components contain cycles. Since \(c\lambda(G) \leq \zeta(G)\), where \(\zeta(G) = \min\{w(A) \mid A \text{ induces a shortest cycle in } G\}\), for any cyclically separable graph \(G\), a cyclically separable graph \(G\) is said to be cyclically optimal if \(c\lambda(G) = \zeta(G)\). The mixed Cayley graph is a kind of semi-regular graph. The cyclic edge-connectivity is a widely studied parameter, which can be used to measure the reliability of a network. Because previous work studied cyclically optimal mixed Cayley graphs with girth \(g \geq 5\), this paper focuses on mixed Cayley graphs with girth \(g < 5\) and gives some sufficient and necessary conditions for these graphs to be cyclically optimal.

Let \(p\) be an odd prime, \(q\) be a prime power coprime to \(p\), and \(n\) be a positive integer. For any positive integer \(d \leq n\), let \(g_1(x) = {x^{p^{n-d}} – 1}\),\(g_2(x)=1+{x^{p^{n – d+1}}}+x^{2p^{n-d+1}}+ \ldots +x^{(p^{d-1}-1)p^{n-d+1}}\),and , \(g_3(x) =1+x^{p^{n-d}}+x^{2p^{n-d}}+ \ldots +x^{(p-1)p^{n-d}} \). In this paper, we determine the weight distributions of \(q\)-ary cyclic codes of length \(pn\) generated by the polynomials \(g_1(x)\), \(g_2(x)\), \(g_3(x)\), \(g_4(x)\), and \(g_5(x)\), by employing the techniques developed in Sharma \& Bakshi [11]. Keywords: cyclic codes, Hamming weight, weight spectrum.

A construction of authentication codes with arbitration from singular symplectic geometry over finite fields is given, and the parameters of the codes are computed. Assuming that the encoding rules of the transmitter and the receiver are chosen according to a uniform probability distribution, the probabilities of success for different types of deceptions are also computed.

Let \(M\) be a simple connected binary matroid with corank at least two such that \(M\) has no connected hyperplane. Seymour proved that \(M\) has a non-trivial series class. We improve this result by proving that \(M\) has at least two disjoint non-trivial series classes \(L_1\) and \(L_2\) such that both \(M \backslash L_1\) and \(M \backslash L_2\) are connected. Our result extends the corresponding result of Kriesell regarding critically \(2\)-connected graphs.

For a non-complete graph \(\Gamma\), a vertex triple \((u,v,w)\) with \(v\) adjacent to both \(u\) and \(w\) is called a \(2\)-geodesic if \(u \neq w\) and \(u,w\) are not adjacent. Then \(\Gamma\) is said to be \(2\)-geodesic transitive if its automorphism group is transitive on both arcs and \(2\)-geodesics. In this paper, we classify the family of connected \(2\)-geodesic transitive graphs of valency \(3p\), where \(p\) is an odd prime.

We generalize the well known congruence Lucas\(^1\) Theorem for binomial coefficient to the bi\(^s\)nomial coefficients.

The linear arboricity \(la(G)\) of a graph \(G\) is the minimum number of linear forests that partition the edges of \(G\). In this paper, it is proved that if \(G\) is a planar graph with maximum degree \(\Delta \geq 7\) and every \(7\)-cycle of \(G\) contains at most two chords, then \(la(G) = \left\lceil \frac{\Delta(G)}{2} \right\rceil\).

In this paper, we study the generalized Pell \(p\)-sequences modulo \(m\). Additionally, we define the generalized Pell \(p\)-sequences and the basic generalized Pell sequences in groups, and then examine these sequences in finite groups. Furthermore, we obtain the periods of the generalized Pell \(p\)-sequences and the basic periods of the basic generalized Pell sequences in the binary polyhedral groups \(\langle n,2,2\rangle\), \(\langle2,n,2\rangle\), and \(\langle2,2,n\rangle\).

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those incident to a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those incident to a single vertex. It is defined as the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find this number and classify all optimal sets for the star graphs, one of the most popular interconnection networks.

The terminal Wiener index of a tree is the sum of distances for all pairs of pendent vertices, which recently arose in the study of phylogenetic tree reconstruction and the neighborhood of trees. This paper presents sharp upper and lower bounds for the terminal Wiener index in terms of its order and diameter and characterizes all extremal trees that attain these bounds. Additionally, we investigate the properties of extremal trees that attain the maximum terminal Wiener index among all trees of order \(n\) with fixed maximum degree.

Based on some results of Shult and Yanushka [7], Brouwer [1] proved that there exists a unique regular near hexagon with parameters \((s,t,t_2) = (2,11,1)\), namely the one related to the extended ternary Golay code. His proof relies on the uniqueness of the Witt design \(S(5,6,12)\), Pless’s characterization of the extended ternary Golay code \(G_{12}\), and some properties of \(S(5,6,12)\) and \(G_{12}\). It is possible to avoid all this machinery and provide an alternative, more elementary and self-contained proof for the uniqueness. The author recently observed that such an alternative proof is implicit in the literature, obtainable by combining results from [1], [4], and [7]. This survey paper aims to bring this fact to the attention of the mathematical community. We describe the relevant parts of the above papers for this alternative proof of classification. Additionally, we prove several extra facts not explicitly contained in [1], [4], or [7]. This paper can also be seen as an addendum to Section 6.5 of [3], where the uniqueness of the near hexagon was not proved.

Recently, Belbachir and Belkhir gave some recurrence relations for the \(r\)-Lah numbers. In this paper, we give other properties for the \(r\)-Lah numbers, we introduce and study a restricted class of these numbers.

An \(H\)-polygon is a simple polygon whose vertices are \(H\)-points, which are points of the set of vertices of a tiling of \(\mathbb{R}^2\) by regular hexagons of unit edge. Let \(G(v)\) denote the least possible number of \(H\)-points in the interior of a convex \(H\)-polygon \(K\) with \(v\) vertices. In this paper, we prove that \(G(8) = 2\), \(G(9) = 4\), \(G(10) = 6\), and \(G(v) \geq \lceil \frac{v^2}{16\pi^2}-\frac{v}{4}+\frac{1}{2}\rceil – 1\) for all \(v \geq 11\), where \(\lceil x \rceil\) denotes the minimal integer more than or equal to \(x\).

Row-cyclic array codes have already been introduced by the author \([9]\). In this paper, we give some special classes of row-cyclic array codes as an extension of classical BCH and Reed-Solomon codes.

The harmonic weight of an edge is defined as reciprocal of the average degree of its end-vertices. The harmonic index of a graph \(G\) is defined as the sum of all harmonic weights of its edges. In this work, we give the minimum value of the harmonic index for any \(n\)-vertex connected graphs with minimum degree \(\delta\) at least \(k(\geq n/2)\) and show the corresponding extremal graphs have only two degrees,i.e., degree \(k\)and degree \(n – 1\), and the number of vertices of degree \(k\) is as close to \(n/2\) as possible.

In this note, we consider one type of \(k\)-tridiagonal matrix family whose permanents and determinants are specified to the balancing and Lucas-balancing numbers. Moreover, we provide some properties between Chebyshev polynomial properties and the given number
sequences,

Let \(G = (V, E)\) be a graph. A subset \(D \subseteq V\) is a dominating set if every vertex not in \(D\) is adjacent to a vertex in \(D\). The domination number of \(G\) is the smallest cardinality of a dominating set of \(G\). The bondage number of a nonempty graph \(G\) is the smallest number of edges whose removal from \(G\) results in a graph with larger domination number than \(G\). In this paper, we determine that the exact value of the bondage number of an \((n-3)\)-regular graph \(G\) of order \(n\) is \(n-3\).

A graph is closed when its vertices have a labeling by \([n]\) with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal, claw-free, and has a property we call narrow, which holds when every vertex is distance at most one from all longest shortest paths of the graph.

Let \(\Sigma = (X, \mathcal{B})\) be a \(4\)-cycle system of order \(v = 1 + 8k\). A \(c\)-colouring of type \(s\) is a map \(\phi: \mathcal{B} \to C\), where \(C\) is a set of colours, such that exactly \(c\) colours are used and for every vertex \(x\), all the blocks containing \(x\) are coloured exactly with \(s\) colours. Let \(4k = qs + r\), with \(r \geq 0\). \(\phi\) is equitable if for every vertex \(x\), the set of the \(4k\) blocks containing \(x\) is partitioned into \(r\) colour classes of cardinality \(q + 1\) and \(s – r\) colour classes of cardinality \(q\). In this paper, we study colourings for which \(s | k\), providing a description of equitable block colourings for \(c \in \{s, s+1, \ldots, \lfloor \frac{2s^2+s}{3} \rfloor\}\).

In this paper, we first introduce a linear program on graphical invariants of a graph \(G\). As an application, we attain the extremal graphs with lower bounds on the first Zagreb index \(M_1(G)\), the second Zagreb index \(M_2(G)\), their multiplicative versions \(\Pi_1^*(G)\), \(\Pi_2(G)\), and the atom-bond connectivity index \(ABC(G)\), respectively.

Let \(\Gamma\) be an oriented graph. We denote the in-neighborhood and out-neighborhood of a vertex \(v\) in \(\Gamma\) by \(\Gamma^-(v)\) and \(\Gamma^+(v)\), respectively. We say \(\Gamma\) has Property \(A\) if, for each arc \((u,v)\) in \(\Gamma\), each of the graphs induced by \(\Gamma^+(u) \cap \Gamma^+(v)\), \(\Gamma^-(u) \cap \Gamma^-(v)\), \(\Gamma^-(u) \cap \Gamma^+(v)\), and \(\Gamma^+(u) \cap \Gamma^-(v)\) contains a directed cycle. Moreover, \(\Gamma\) has Property B if each arc \((u,v)\) in \(\Gamma\) extends to a \(3\)-path \((x,u), (u,v), (v,w)\), such that \(|\Gamma^+(x) \cap \Gamma^+(u)| \geq 5\) and \(|\Gamma^-(v) \cap \Gamma^-(w)| \geq 5\). We show that the only oriented graphs of order at most \(17\), which have both properties \(A\) and \(B\), are the Tromp graph \(T_{16}\) and the graph \(T^+_{16}\), obtained by duplicating a vertex of \(T_{16}\). We apply this result to prove the existence of an oriented planar graph with oriented chromatic number at least \(18\).

By the partial fraction decomposition method, we establish a \(q\)-harmonic sum identity with multi-binomial coefficient, from which we can derive a fair number of harmonic number identities.

A fall \(k\)-coloring of a graph \(G\) is a proper \(k\)-coloring of \(G\) such that each vertex of \(G\) sees all \(k\) colors on its closed neighborhood. We denote \(\text{Fall}(G)\) the set of all positive integers \(k\) for which \(G\) has a fall \(k\)-coloring. In this paper, we study fall colorings of the lexicographic product of graphs and the categorical product of graphs. Additionally, we show that for each graph \(G\), \(\text{Fall}(M(G)) = \emptyset\), where \(M(G)\) is the Mycielskian of the graph \(G\). Finally, we prove that for each bipartite graph \(G\), \(\text{Fall}(G^c) \subseteq \{\chi(G^c)\}\) and it is polynomial time to decide whether \(\text{Fall}(G^c) = \{\chi(G^c)\}\) or not.

In this paper, we first provide two necessary conditions for a graph \(G \) to be \(E_k\)-cordial, then we prove that every \(P_n(n \geq 3)\) is \(E_p\)-cordial if \(p\) is odd. In the end, we discuss the \(E_2\)-cordiality of a graph \)G\) under the condition that some subgraph of \(G\) has a \(1\)-factor.

In this paper, we consider the problem of determining the structure of a minimal critical set in a latin square \(L\) representing the elementary abelian \(2\)-group of order \(8\).

In this paper, the first two (resp. four) largest signless Laplacian spectral radii together with the corresponding graphs in the class of bicyclic (resp. tricyclic) graphs of order n are determined, and the first two (resp. four) largest signless Laplacian spreads together with the corresponding graphs in the class of bicyclic (resp. tricyclic) graphs of order \(n\) are identified.

An edge-magic total labeling of a graph \(G\) is a one-to-one map \(\lambda\) from \(V(G) \cup E(G)\) onto the integers \(\{1, 2, \ldots, |V(G) \cup E(G)|\}\) with the property that there exists an integer constant \(c\) such that \(\lambda(x) + \lambda(x,y) + \lambda(y) = c\) for any \((x, y) \in E(G)\). If \(\lambda(V(G)) = \{1, 2, \ldots, |V(G)|\}\), then the edge-magic total labeling is called super edge-magic total labeling. In this paper, we formulate super edge-magic total labeling on subdivisions of stars \(K_{1,p}\), for \(p \geq 5\).

In this paper, we briefly survey Euler’s works on identities connected with his famous Pentagonal Number Theorem. We state a partial generalization of his theorem for partitions with no part exceeding an identified value \(k\), along with some identities linking total partitions to partitions with distinct parts under the above constraint. We derive both recurrence formulas and explicit forms for \(\Delta_n(m)\), where \(\Delta_n(m)\) denotes the number of partitions of \(m\) into an even number of distinct parts not exceeding \(n\), minus the number of partitions of \(m\) into an odd number of distinct parts not exceeding \(n\). In fact, Euler’s Pentagonal Number Theorem asserts that for \(m \leq n\), \(\Delta_n(m) = \pm 1\) if \(m\) is a Pentagonal Number and \(0\) otherwise. Finally, we establish two identities concerning the sum of bounded partitions and their connection to prime factors of the bound integer.

Consider the following one-person game: let \(S = {F_1, F_2,\ldots, F_r}\) be a family of forbidden graphs. The edges of a complete graph are randomly shown to the Painter one by one, and he must color each edge with one of \(r\) colors when it is presented, without creating some fixed monochromatic forbidden graph \(F\); in the \(i\)-th color. The case of all graphs \(F\); being cycles is studied in this paper. We give a lower bound on the threshold function for online \(S\)-avoidance game,which generalizes the results of Marciniszyn, Spdhel and Steger for the symmetric case. [Combinatorics, Probability and Computing, Vol. \(18, 2009: 271-300.\)]

Given positive integers \(n\), \(k\), and \(m\), the \((n,k)\)-th \(m\)-restrained Stirling number of the first kind is the number of permutations of an \(n\)-set with \(k\) disjoint cycles of length \(\leq m\). By inverting the matrix consisting of the \((n,k)\)-th \(m\)-restrained Stirling number of the first kind as the \((n+1,k+1)\)-th entry, the \((n,k)\)-th \(m\)-restrained Stirling number of the second kind is defined. In this paper, we study the multi-restrained Stirling numbers of the first and second kinds to derive their explicit formulae, recurrence relations, and generating functions. Additionally, we introduce a unique expansion of multi-restrained Stirling numbers for all integers \(n\) and \(k\), and a new generating function for the Stirling numbers of the first kind.

Employing \(q\)-commutive structures, we develop binomial analysis and combinatorial applications induced by an important operator in
analogue Fourier analysis associated with well-known \(q\)-series of L.J. Rogers.

In [H. Ngo, D. Du, New constructions of non-adaptive and error-tolerance
pooling designs, Discrete Math. \(243 (2002) 167-170\)], by using subspaces
in a vector space Ngo and Du constructed a family of well-known pooling
designs. In this paper, we construct a family of pooling designs by using
bilinear forms on subspaces in a vector space, and show that our design and
Ngo-Du’s design have the same error-tolerance capability but our design is
more economical than Ngo-Du’s design under some conditions.

A transverse Steiner quadruple system \((TSQS)\) is a triple \((X, \mathcal{H}, \mathcal{B})\) where \(X\) is a \(v\)-element set of points, \(\mathcal{H} = \{H_1, H_2, \ldots, H_r\}\) is a partition of \(X\) into holes, and \(\mathcal{B}\) is a collection of transverse \(4\)-element subsets with respect to \(\mathcal{H}\), called blocks, such that every transverse \(3\)-element subset is in exactly one block. In this article, we study transverse Steiner quadruple systems with \(r\) holes of size \(g\) and \(1\) hole of size \(u\). Constructions based on the use of \(s\)-fans are given, including a construction for quadrupling the number of holes of size \(g\). New results on systems with \(6\) and \(11\) holes are obtained, and constructions for \(\text{TSQS}(x^n(2n)^1)\) and \(\text{TSQS}(4^n2^1)\) are provided.