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 \(G(2^m, 2)\).