Let \(G\) be a connected graph. For \(x,y \in V(G)\) with \(d(x,y) = 2\), we define \(J(x,y) = \{u \in N(x) \cap N(y) | N[u] \cap N[x] \cup N[y]\}\) and \(J'(x,y) = \{u \in N(x) \cap N(y) |\) if \(v \in N(u) \setminus (N[x] \cup N[y])\) then \(N(x) \cup N(y) \cup N(u) \cap N[v]\}\). A graph \(G\) is quasi-claw-free if \(J(x,y) \neq \emptyset\) for each pair \((x,y)\) of vertices at distance \(2\) in \(G\). Broersma and Vumar introduced the class of \(P_3\)-dominated graphs defined as \(J(x,y) \cup J'(x,y) \neq \emptyset\) for each \(x,y \in V(G)\) with \(d(x,y) = 2\). Let \(\kappa(G)\) and \(\alpha_2(G)\) be the connectivity of \(G\) and the maximum number of vertices that are pairwise at distance at least \(2\) in \(G\), respectively. A cycle \(C\) is \(m\)-dominating if \(d(x,C) = \min\{d(x,u) | u \in V(C)\} \leq m\) for all \(x \in V(G)\). In this note, we prove that every \(2\)-connected \(\mathcal{P}_3\)-dominated graph \(G\) has an \(m\)-dominating cycle if \(\alpha_{2m+3}(G) \leq \kappa(G)\).

We initiate the study of signed edge majority total domination in graphs. The open neighborhood \(N_G(e)\) of an edge \(e\) in a graph \(G\) is the set consisting of all edges having a common vertex with \(e\). Let \(f\) be a function on \(E(G)\), the edge set of \(G\), into the set \(\{-1, 1\}\). If \(\sum_{x \in N_G(e)} f(x) \geq 1\) for at least half of the edges \(e \in E(G)\), then \(f\) is called a signed edge majority total dominating function of \(G\). The value \(\sum_{e\in E(G)}f(e)\), taking the minimum over all signed edge majority total dominating functions \(f\) of \(G\), is called the signed edge majority total domination number of \(G\) and denoted by \(\gamma’_{smt}(G)\). Obviously, \(\gamma’_{smt}(G)\) is defined only for graphs \(G\) which have no connected components isomorphic to \(K_2\). In this paper, we establish lower bounds on the signed edge majority total domination number of forests.

This article is a contribution to the study of the automorphism groups of \(2\)-\((v,k,1)\) designs. Let \(\mathcal{D}\) be a \(2\)-\((v,13,1)\) design, \(G \leq \mathrm{Aut}(\mathcal{D})\) be block transitive and point primitive. If \(G\) is unsolvable, then \(\mathrm{Soc}(G)\), the socle of \(G\), is not \(\mathrm{Sz}(q)\).

Using Cioaba’s inequality on the sum of the 3rd powers of the vertex degrees in connected graphs, we present an inequality on the Laplacian eigenvalues of connected graphs.

In this paper, the author studies the relation of vertices, edges, and cells of the quasi-cross-cut partition. Moreover, the three-term recurrence relations of \(\dim(S_d^0(\Delta))\) over the quasi-cross-cut partition and the triangulation are presented.

It has been known for at least \(2500\) years that mathematics and music are directly related. This article explains and extends ideas originating with Euler involving labeling parts of graphs with notes in such a way that other parts of the graphs correspond in a natural way to chords. The principal focus of this research is the notion of diatonic labelings of cubic graphs, that is, labeling the edges with pitch classes in such a way that vertices are incident with edges labeled with the pitch classes of a triad in a given diatonic scale. The pitch classes are represented in a natural way with elements of \(\mathbb{Z}_{12}\), the integers modulo twelve.

Several classes of cubic graphs are investigated and shown to be diatonic. Among the graphs considered are Platonic Solids, cylinders, and Generalized Petersen Graphs. It is shown that there are diatonic cubic graphs on \(n\) vertices for even \(n \geq 14\). Also, it is shown that there are cubic graphs on \(n\) vertices that do not have diatonic labelings for all even \(n \geq 4\). The question of forbidden subgraphs is investigated, and a forbidden subgraph for diatonic graphs, or “clash”, is demonstrated.

In this paper, we recall Konhauser polynomials. Approximation properties of these operators are obtained with the help of the Korovkin theorem. The order of convergence of these operators is computed by means of modulus of continuity, Peetre’s K-functional, and the elements of the Lipschitz class. Also, we introduce the \(r\)-th order generalization of these operators and we evaluate this generalization by the operators defined in this paper. Finally, we give an application to differential equations.

A graph \(X\) is said to be End-regular (resp., End-orthodox, End-inverse) if its endomorphism monoid \(\mathrm{End}(X)\) is a regular (resp., orthodox, inverse) semigroup. In this paper, End-regular (resp., End-orthodox, End-inverse) graphs which are the join of split graphs \(X\) and \(Y\) are characterized. It is also proved that \(X + Y\) is never End-inverse for any split graphs \(X\) and \(Y\).

Some new families of complete caps in Galois affine spaces \({AG}(N,q)\) of dimension \(N \equiv 0 \pmod{4}\) and odd order \(q \leq 127\) are constructed. No smaller complete caps appear to be known.

We give two Frankl-like results on set systems with restrictions on set difference sizes and set symmetric difference sizes modulo prime powers. Based on a similar method, we also give a bound on codes satisfying the properties of Hamming distance modulo prime powers.

In this note, a resolvable \((K_4 – e)\)-design of order \(296\) is constructed. Combining the results of \([2, 3, 4]\), the existence spectrum of resolvable \((K_4 – e)\)-designs of order \(v\) is the set \(\{v : v \equiv 16 \pmod{20}, v \geq 16\}\).

We study permutations of the set \([n] = \{1, 2, \ldots, n\}\) written in cycle notation, for which each cycle forms an increasing or decreasing interval of positive integers. More generally, permutations whose cycle elements form arithmetic progressions are considered. We also investigate the class of generalized interval permutations, where each cycle can be rearranged in increasing order to form an interval of consecutive positive integers.

In this paper, we study the symmetry for the generalized twisted Genocchi polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Genocchi polynomials using the symmetric properties for the \(p\)-adic invariant \(q\)-integral on \(\mathbb{Z}_p\).

In this paper, we use a simple method to derive different recurrence relations on the recursive sequence order-\(k\) and their sums, which are more general than that given in literature [J.Feng, More Identities on the Tribonacci Numbers, Ars Combinatoria, \(100(2011), 73-78]\). By using the generating matrices, we get more identities on the recursive sequence order-\(k\) and their sums, which are more general than that given in literature [E.Kihg, Tribonacci Sequences with Certain Indices and Their Sums, Ars Combinatoria, \(86(2008), 13-22]\) .

By applying discharging methods and properties of critical graphs, we proved that every simple planar graph \(G\) with \(\Delta(G) \geq 5\) is of class 1, if any 4-cycle is not adjacent to a 5-cycle in \(G\).

A graph \(G\) is pancyclic if it contains a cycle of every length from 3 to \(|V(G)|\) inclusive. A graph \(G\) is panconnected if there exists a path of length \(l\) joining any two different vertices \(x\) and \(y\) with \(d_G(x,y) \leq l \leq |V(G)| – 1\), where \(d_G(x,y)\) denotes the distance between \(x\) and \(y\) in \(G\). A hamiltonian graph \(G\) is panpositionable if for any two different vertices \(x\) and \(y\) of \(G\) and any integer \(k\) with \(d_G(x,y) \leq k \leq |V(G)|/2\), there exists a hamiltonian cycle \(C\) of \(G\) with \(d_C(x,y) = k\), where \(d_C(x,y)\) denotes the distance between \(x\) and \(y\) in a hamiltonian cycle \(C\) of \(G\). It is obvious that panconnected graphs are pancyclic, and panpositionable graphs are pancyclic.

The above properties can be studied in bipartite graphs after some modification. A graph \(H = (V_0 \cup V_1, E)\) is bipartite if \(V(H) = V_0 \cup V_1\) and \(E(H)\) is a subset of \(\{(u,v) | u \in V_0 \text{ and } v \in V_1\}\). A graph is bipancyclic if it contains a cycle of every even length from 4 to \(2\lfloor |V(H)|/2 \rfloor\) inclusive. A graph \(H\) is bipanconnected if there exists a path of length \(l\) joining any two different vertices \(x\) and \(y\) with \(d_H(x,y) \leq l \leq |V(H)| – 1\), where \(d_H(x,y)\) denotes the distance between \(x\) and \(y\) in \(H\) and \(l – d_H(x,y)\) is even. A hamiltonian graph \(H\) is bipanpositionable if for any two different vertices \(x\) and \(y\) of \(H\) and for any integer \(k\) with \(d_H(x,y) \leq k \leq |V(H)|/2\), there exists a hamiltonian cycle \(C\) of \(H\) with \(d_C(x,y) = k\), where \(d_C(x,y)\) denotes the distance between \(x\) and \(y\) in a hamiltonian cycle \(C\) of \(H\) and \(k – d_H(x,y)\) is even. It can be shown that bipanconnected graphs are bipancyclic, and bipanpositionable graphs are bipancyclic.

In this paper, we present some examples of pancyclic graphs that are neither panconnected nor panpositionable, some examples of panconnected graphs that are not panpositionable, and some examples of graphs that are panconnected and panpositionable, for nonbipartite graphs. Corresponding examples for bipartite graphs are discussed. The existence of panpositionable (or bipanpositionable, resp.) graphs that are not panconnected (or bipanconnected, resp.) is still an open problem.

In \([2]\) Stefano Innamorati and Mauro Zannetti gave a characterization of the planes secant to a non-singular quadric in \({P}G(4, q)\). Their result is based on a particular hypothesis (which we call “polynomial”) that, as the same authors wrote at the end of the paper, could not exclude possible sporadic cases. In this paper, we improve their result by giving a characterization without the “polynomial” hypothesis. So, possible sporadic cases are definitely excluded.

This paper generalizes the results of Guiduli [B. Guiduli, On incidence coloring and star arboricity of graphs. Discrete Math. \(163
(1997), 275-278]\) on the incidence coloring of graphs to the fractional incidence coloring. Tight asymptotic bounds analogous to Guiduli’s results are given for the fractional incidence chromatic number of graphs. The fractional incidence chromatic number of circulant graphs is studied. Relationships between the \(k\)-tuple incidence chromatic number and the incidence chromatic number of the direct products and lexicographic products of graphs are established. Finally, for planar graphs \(G\), it is shown that if \(\Delta(G) \neq 6\), then \(\chi_i(G) \leq \Delta(G) + 5\); if \(\Delta(G) = 6\), then \(\chi_i(G) \leq \Delta(G) + 6\); where \(\chi_i(G)\) denotes the incidence chromatic number of \(G\). This improves the bound \(\chi_i(G) \leq \Delta(G) + 7\) for planar graphs given in [M. Hosseini Dolama, E. Sopena, X. Zhu, Incidence coloring of k-degenerated graphs, Discrete Math. \(283 (2004)\), no. \(1-3, 121-128]\).

Let \(P(G, \lambda)\) be the chromatic polynomial of a graph \(G\). A graph \(G\) is chromatically unique if for any graph \(H\), \(P(H, \lambda) = P(G, \lambda)\) implies \(H \cong G\). Some sufficient conditions guaranteeing that certain complete tripartite graph \(K(l, n, r)\) is chromatically unique were obtained by many scholars. Especially, in 2003, H.W. Zou showed that if \(n > \frac{1}{3}(m^2+k^2+mk+2\sqrt{m^2 + k^2 + mk} + m – k)\), where \(n, k\), and \(m\) are non-negative integers, then \(K(n – m, n, n + k)\) is chromatically unique (or simply \(\lambda\)-unique). In this paper, we show that for any positive integers \(n, m\), and \(k\), let \(G = K(n – m, n, n + k)\), where \(m \geq 2\) and \(k \geq 1\), if \(n \geq \max\{\lceil \frac{1}{4}m^2 + m + k \rceil, \lceil \frac{1}{4}m^2 + \frac{3}{2}m + 2k – \frac{11}{4} \rceil, \lceil mk + m – k + 1 \rceil\}\), then \(G\) is \(\chi\)-unique. This improves upon H.W. Zou’s result in the case \(m \geq 2\) and \(k \geq 1\).

In this paper, it is proved that a toroidal graph without cycles of length \(k\) for each \(k \in \{4, 5, 7, 10\}\) is \(3\)-choosable.

In this paper, we investigate the transitive Cayley graphs of strong semilattices of rectangular groups, and of normal bands, respectively. We show under which conditions they enjoy the property of automorphism vertex transitivity in analogy to Cayley graphs of groups.

A family of connected graphs \(\mathcal{G}\) is said to be a family with constant metric dimension if its metric dimension is finite and does not depend upon the choice of \(G\) in \(\mathcal{G}\). In this paper, we study the metric dimension of the generalized Petersen graphs \(P(n,m)\) for \(n = 2m+1\) and \(m \geq 1\) and give a partial answer to the question raised in \([9]\): Is \(P(n, m)\) for \(n \geq 7\) and \(3 \leq m \leq \lfloor \frac{n-1}{2} \rfloor\) a family of graphs with constant metric dimension? We prove that the generalized Petersen graphs \(P(n,m)\) with \(n = 2m +1\) have metric dimension \(3\) for every \(m \geq 2\).

Let \(G\) be a graph on \(n\) vertices. \(\delta\) and \(\alpha\) be the minimum degree and independence number of \(G\), respectively. We prove that if \(G\) is a \(2\)-connected graph and \(|N(x) \cup N(y)| \geq n-\delta – 1\) for each pair of nonadjacent vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha – 1\), then \(G\) is hamiltonian or \(G \in \{G_1, G_2\}\) (see Figure 1.1 and Figure 1.2). As a corollary, if \(G\) is a 2-connected graph and \(|N(x) \cup N(y)| \geq n – \delta\) for each pair of nonadjacent vertices \(x,y\) with \(1 \leq |N(x) \cap N(y)| \leq \alpha – 1\), then \(G\) is hamiltonian. This result extends former results by Faudree et al. \(([5])\) and Yin \(([7])\).

Arising from the VLSI design and network communication, the cutwidth problem for a graph \(G\) is to embed \(G\) into a path such that the maximum number of overlap edges (i.e., the congestion) is minimized. The characterization of forbidden subgraphs or critical graphs is meaningful in the study of a graph-theoretic parameter. This paper characterizes the set of \(4\)-cutwidth critical trees by twelve specified ones.

A path \(P\) in an edge-colored graph (not necessarily a proper edge-coloring) is a rainbow path if no two edges of \(P\) are assigned the same color. For a connected graph \(G\) with connectivity \(\kappa(G)\) and an integer \(k\) with \(1 \leq k \leq \kappa(G)\), the rainbow \(k\)-connectivity \(rc_k(G)\) of \(G\) is the minimum number of colors needed in an edge-coloring of \(G\) such that every two distinct vertices \(u\) and \(v\) of \(G\) are connected by at least \(k\) internally disjoint \(u-v\)rainbow paths. In this paper, the rainbow \(2\)-connectivity of the Petersen graph as well as the rainbow connectivities of all cubic graphs of order \(8\) or less are determined.

This paper investigates the number of rooted simple bipartite maps on the sphere and presents some formulae for such maps with the number of edges and the valency of the root-face as two parameters.

For a graph \(G = (V(G), E(G))\), the transformation graph \(G^{+-+}\) is the graph with vertex set \(V(G) \cup E(G)\) in which the vertices \(\alpha\) and \(\beta\) are joined by an edge if and only if \(\alpha\) and \(\beta\) are adjacent or incident in \(G\) while \(\{\alpha, \beta\} \not\subseteq E(G)\), or \(\alpha\) and \(\beta\) are not adjacent in \(G\) while \(\{\alpha, \beta\} \in E(G)\). In this note, we show that all but for a few exceptions, \(G^{+-+}\) is super-connected and super edge-connected.

In this paper, we give matrix representations of the \(k\)-generalized order-\(k\) Perrin numbers and we obtain relationships between these sequences and matrices. In addition, we calculate the determinant of this matrix.

A graph \(G\) is \(k\)-total domination edge critical, abbreviated to \(k\)-critical if confusion is unlikely, if the total domination number \(\gamma_t(G)\) satisfies \(\gamma_t(G) = k\) and \(\gamma_t(G + e) < \gamma_t(G)\) for any edge \(e \in E(\overline{G})\).Graphs that are \(4\)-critical have diameter either \(2\), \(3\), or \(4\). In previous papers, we characterized structurally the \(4\)-critical graphs with diameter four and found bounds on the order of \(4\)-critical graphs with diameter two. In this paper, we study a family \(\mathcal{H}\) of \(4\)-critical graphs with diameter three, in which every vertex is a diametrical vertex, and every diametrical pair dominates the graph. We also generalize the self-complementary graphs and show that these graphs provide a special case of the family \(\mathcal{H}\).

A finite planar set is \(k\)-isosceles for \(k \geq 3\) if every \(k\)-point subset of the set contains a point equidistant from two others. There exists no convex \(4\)-isosceles \(8\)-point set with \(8\) points on a circle.

In this note, motivated by the non-existence of a vertex-transitive strongly regular graph with parameters \((3250, 57, 0, 1)\), we obtain a feasibility condition concerning strongly regular graphs admitting an automorphism group with exactly two orbits on vertices. We also establish a result on the possible orbit sizes of a potential strongly regular graph with parameters \((3250, 57, 0, 1)\). We use our results to obtain a list of only 11 possible orbit size combinations for a potential strongly regular graph with parameters \((3250, 57, 0, 1)\) admitting an automorphism group with exactly two orbits.

In this note it is shown that the number of cycles of a linear hypergraph is bounded below by its cyclomatic number.

The Padmakar-Ivan \((PI)\) index is a Wiener-Szeged-like topological index. In this paper, we study the \(PI\) index of thorn graphs, and we present a generally useful method which can reduce the computational amount of \(PI\) index strikingly.

The concept of the sum graph and integral sum graph were introduced by F. Harary. In this paper, we gain some upper and lower bounds on the sum number and the integral sum number of a graph and these bounds are sharp, and some new properties on the integral sum graph. Using these results, we could directly investigate and determine the exclusive integral sum numbers, the exclusive sum numbers, the sum numbers and the integral sum numbers of the graphs \(K_n\backslash E(2P_3)\), \(K_n\backslash E(P_3)\) and any graph \(H\) with minimum degree \(\delta(H) = n-2\) respectively as \(2\) is more than a given number. Then they will be the beginning of a new thought of research on the (exclusive) sum graph and the (exclusive) integral sum graph.

Let \(G = (V, E)\) be a simple connected graph, where \(d_u\) is the degree of vertex \(u\), and \(d_G(u, v)\) is the distance between \(u\) and \(v\). The Schultz index of \(G\) is defined as \(\mathcal{W}_+(G) = \sum\limits_{u,v \subset V(G)} (d_u + d_v)d_G(u,v).\)In this paper, we investigate the Schultz index of a class of trees with diameter not more than \(4\).

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(g\) and \(f\) be two integer-valued functions defined on \(V(G)\) such that \(0 \leq g(x) \leq f(x)\) for each \(x \in V(G)\). A \((g, f)\)-factor of \(G\) is a spanning subgraph \(F\) of \(G\) such that \(g(x) \leq d_F(x) \leq f(x)\) for each \(x \in V(F)\). A \((g, f)\)-factorization of \(G\) is a partition of \(E(G)\) into edge-disjoint \((g, f)\)-factors. Let \({F} = \{F_1, F_2, \ldots, F_m\}\) be a factorization of \(G\) and \(H\) be a subgraph of \(G\) with \(m\) edges. If \(F_i\), \(1 \leq i \leq m\), has exactly one edge in common with \(H\), we say that \({F}\) is orthogonal to \(H\). In this paper, it is proved that every \((mg+k-1, mf-k+1)\)-graph contains a subgraph \( {R}\) such that \( {R}\) has a \((g, f)\)-factorization orthogonal to any given subgraph with \(k\) edges of \(G\) if \(f(x) > g(x) \geq 0\) for each \(x \in V(G)\) and \(1 \leq k \leq m\), where \(m\) and \(k\) are two positive integers.

Let \(G\) be a graph with a maximum matching of size \(q\), and let \(p \leq q\) be a positive integer. Then \(G\) is called \((p, q)\)-extendable if every set of \(p\) independent edges can be extended to a matching of size \(q\). If \(G\) is a graph of even order \(n\) and \(n = 2q\), then \((p,q)\)-extendable graphs are exactly the \(p\)-extendable graphs defined by Plummer \([11]\) in \(1980\).

Let \(d \geq 3\) be an integer, and let \(G\) be a \(d\)-regular graph of order \(n\) with a maximum matching of size \(q = \frac{n-t}{2}\geq 3\) for an integer \(t \geq 1\) such that \(n – t\) is even. In this work, we prove that if

(i) \(n \leq {(t+4)(d+1)-5}\) or

(ii) \(n \leq (t+4)(d+2) – 1\) when \(d\) is odd,

then \(G\) is \((2, q)\)-extendable.

A graph with vertex set \(V\) is said to have a prime cordial labeling if there is a bijection \(f\) from \(V\) to \(\{1,2,\ldots,|V|\}\) such that if each edge \(uv\) is assigned the label \(1\) for the greatest common divisor \(\gcd(f(u), f(v)) = 1\) and \(0\) for \(\gcd(f(u), f(v)) = 1\), then the number of edges labeled with \(0\) and the number of edges labeled with \(1\) differ by at most \(1\). In this paper, we show that the Flower Snark and its related graphs are prime cordial for all \(n \geq 3\).

In [2] it is proved that if \(X = Cay(G, S)\) is a connected tetravalent Cayley graph on a regular \(p\)-group \(G\) (for \(p \neq 2, 5\)), then the right regular representation of \(G\) is normal in the automorphism group of \(X\). In this paper, we prove that a similar result holds, for \(p = 5\), under a slightly stronger hypothesis. Some remarkable examples are presented.

In this paper, we define, for a graph invariant \(\psi\), the deck ratio of \(\psi\) by \(D_\psi(G) = \frac{\psi(G)}{\Sigma_{v\in V(G)}\psi(G-v)}\). We give generic upper and lower bounds on \(D_\psi\) for monotone increasing and monotone decreasing invariants \(\psi\), respectively.

Then, we proceed to consider the Wiener index \(W(G)\), showing that \(D_W(G) \leq \frac{1}{|V(G)|-2}\). We show that equality is attained for a graph \(G\) if and only if every induced \(P_3\) subgraph of \(G\) is contained in a \(C_4\) subgraph. Such graphs have been previously studied under the name of self-repairing graphs.

We show that a graph on \(n \geq 4\) vertices with at least \(\frac{n^2-3n+6}{2} – n + 3\) edges is necessarily a self-repairing graph and that this is the best possible result. We also show that a \(2\)-connected graph is self-repairing if and only if all factors in its Cartesian product decomposition are.

Finally, some open problems about the deck ratio and about self-repairing graphs are posed at the end of the paper.

For a graph \(X\) and a digraph \(D\), we define the \(\beta\) transformation of \(X\) and the \(\alpha\) transformation of \(D\) denoted by \(X^\beta\) and \(D^\alpha\), respectively.\(D^\alpha\) is defined as the bipartite graph with vertex set \(V(D) \times \{0,1\}\) and edge set \(\{((v_i,0), (v_j, 1)) \mid v_i v_j \in A(D)\}\).\(X^\beta\) is defined as the bipartite graph with vertex set \(V(X) \times \{0,1\}\) and edge set \(\{((v_i,0), (v_j, 1)) \mid v_i v_j \in A(X)\}\), where \(X\) is the associated digraph of \(X\).In this paper, we give the relation between the eigenvalues of the digraph \(D\) and the graph \(D^\alpha\) when the adjacency matrix of \(D\) is normal. Especially, we obtain the eigenvalues of \(D^\alpha\) when \(D\) is some special Cayley digraph.

To study orthogonal arrays and signed orthogonal arrays, Ray-Chaudhuri and Singhi (\(1988\) and \(1994\)) considered some module spaces. Here, using a linear algebraic approach, we define an inclusion matrix and find its rank. In the special case of Latin squares, we show that there is a straightforward algorithm for generating a basis for this matrix using the so-called intercalates. We also extend this last idea.

Integral circulant graphs have been proposed as potential candidates for modelling quantum spin networks with perfect state transfer between antipodal sites in the network. We show that the diameter of these graphs is at most \(O(\ln \ln n)\), and further improve the recent result of Saxena, Severini, and Shparlinski.

We present a bijective proof of the hook length formula for rooted trees based on the ideas of the bijective proof of the hook length formula for standard tableaux by Novelli, Pak, and Stoyanovskii \([10]\). In Section \(4\), we present another bijection for the formula.

In this paper, we give sufficient conditions for the existence of kernels by monochromatic directed paths (m.d.p.) in digraphs with quasi-transitive colorings. Let \(D\) be an \(m\)-colored digraph. We prove that if every chromatic class of \(D\) is quasi-transitive, every cycle is quasi-transitive in the rim and \(D\) does not contain polychromatic triangles, then \(D\) has a kernel by m.d.p. The same result is valid if we preserve the first two conditions before and replace the last one by: there exists \(k \geq 4\) such that every \(\overrightarrow{C}_k\) is quasi-monochromatic and every \(\overrightarrow{C}_{k-1}\) (\(3 \leq l \leq k-1\)) is not polychromatic. Finally, we also show that if every chromatic class of \(D\) is quasi-transitive, every cycle in \(D\) induces a quasi-transitive digraph and \(D\) does not contain polychromatic \(\overrightarrow{C}_3\), then \(D\) has a kernel by m.d.p. Some corollaries are obtained for the existence of kernels by m.d.p. in \(m\)-colored tournaments.

The determination of the zero-capacity of a noisy channel has inspired research on the independence number of the strong product of odd cycles. The independence number for two infinite families of the strong product of three odd cycles is considered in this paper. In particular, we present the independence number of \(C_7 \boxtimes C_9 \boxtimes C_{2k+1}\) and an upper bound on the independence number of \(C_{13} \boxtimes C_3 \boxtimes C_{2k+1}\). The results are partially obtained by a computer search.

The strong product \(G_1 \boxtimes G_2\) of graphs \(G_1\) and \(G_2\) is the graph with \(V(G_1) \times V(G_2)\) as the vertex set, and two distinct vertices \((x_1,x_2)\) and \((y_1,y_2)\) are adjacent whenever for each \(i \in \{1,2\}\) either \(x_i = y_i\) or \(x_iy_i \in E(G_i)\).

An edge irregular total \(k\)-labeling \(\varphi: V \cup E \to \{1,2,\ldots,k\}\) of a graph \(G = (V, E)\) is a labeling of vertices and edges of \(G\) in such a way that for any different edges \(xy\) and \(x’y’\) their weights \(\varphi(x) + \varphi(xy) + \varphi(y)\) and \(\varphi(x’) + \varphi(x’y’) + \varphi(y’)\) are distinct. The total edge irregularity strength, \(\text{tes}(G)\), is defined as the minimum \(k\) for which \(G\) has an edge irregular total \(k\)-labeling.

We have determined the exact value of the total edge irregularity strength of the strong product of two paths \(P_n\) and \(P_m\).

Given \(m, n\) and \(2 \leq l \leq mn\), we study the problem of separating \(l\) symbols on an \(m \times n\) array such that the minimum \(\ell_1\) distance between any two of the \(l\) symbols is as large as possible. This problem is similar in nature to the well-known Tammes’ problem where one tries to achieve the largest angular separation for a given number of points on a \(2-D\) or higher dimensional sphere. It is also closely related to the well-studied problem of constructing optimal interleaving schemes for correcting error bursts in multi-dimensional digital data where a burst can be an arbitrarily shaped connected region in the array. Moreover, the interest in studying this problem also arises from considerations of minimizing the risk of multiple nearby node failures in a distributed data storage system (or a similar industrial network) in the event of a relatively large scale random disruption. We derive bounds on the maximum possible distance of separation for general \(m,n\) and \(l\), and provide also optimal constructions in several special cases including small and large \(l\) values, small \(m\) (or \(n\)) values, and \(n-1 \geq (l-1)(m-1)\).

In this paper, we apply the concepts of intuitionistic fuzzy sets to coalgebras. We give the definition of intuitionistic fuzzy subcoalgebras and investigate some properties of intuitionistic fuzzy subcoalgebras. Considering the applications of intuitionistic fuzzy subcoalgebras, we discuss their properties under homomorphisms of coalgebras.

In this paper, the joint tree method of graph embeddings, which was introduced by Liu, is generalized to digraph embeddings. The genus distributions of a new type of digraphs in orientable surfaces are determined.

In this paper, the \(m\)-hull sets in the join and composition of two connected graphs are characterized and their \(m\)-hull numbers are shown to be direct consequences of these characterizations.

The generalized de Bruijn digraph denoted by \(G_B(n,m)\) is the digraph \((V, A)\) where \(V = \{0,1,\ldots,m-1\}\) and \((i,j) \in A\) if and only if \(j \equiv ni + \alpha \pmod{m}\) for some \(\alpha \in \{0,1,\ldots,n-1\}\). By replacing each arc of \(G_B(n,m)\) with an undirected edge and eliminating loops and multi-edges, we obtain a generalized undirected de Bruijn graph \(UG_B(n,m)\). In this paper, we prove that the diameter of \(UG_B(n,m)\) is equal to 3 whenever \(n \geq 2\) and \(n^2 + (\frac{\sqrt{5}+1}{2})\leq m \leq 2n^2.\)

The zeroth-order general Randić index of a graph \(G\) is defined as \({}^{0}{}{R}_\alpha = \sum\limits_{v\in V(G)} d(v)^\alpha\)
where \(d(v)\) is the degree of the vertex \(v\) in \(G\) and \(\alpha\) is an arbitrary real number. In the paper, we give sharp lower and upper bounds on the zeroth-order general Randić index of cacti.

Let \(G\) be a connected \(k\)-colourable graph of order \(n \geq k\). A subgraph \(H\) of \(G\) is \(k\)-colourfully panconnected in \(G\) if there is a \(k\)-colouring of \(G\) such that the colours are close together in \(H\), in two different senses (called variegated and panconnected) to be made precise. Let \(s_k(G)\) denote the smallest number of edges in a spanning \(k\)-colourfully panconnected subgraph \(H\) of \(G\). It is conjectured that \(s_k(G) = n-1\) if \(k \geq 4\) and \(G\) is not a circuit (a connected \(2\)-regular graph) with length \(\equiv 1 \pmod{k}\). It is proved that \(s_k(G) = n-1\) if \(G\) contains no circuit with length \(\equiv 1 \pmod{k}\), and \(s_k(G) \leq 2n-k-1\) whenever \(k \geq 4\).

Multisender 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 give the model of multisender authentication codes and the calculation formulas on probability of success in attacks by malicious groups of senders. A construction of multisender authentication codes from symplectic geometry over finite fields is given, and the parameters and the probabilities of deceptions are also calculated.

Let \((X,{B})\) be an \(\alpha\)-fold block design with block size \(4\). If a star is removed from each block of \({B}\), the resulting collection of triangles \({T}\) is a partial \(\lambda\)-fold triple system \((X,{T})\). If the edges belonging to the deleted stars can be arranged into a collection of triangles \({S}^*\), then \((X,{T} \cup {S}^*)\) is an \(\lambda\)-fold triple system, called a metamorphosis of the \(\lambda\)-fold block design \((X, {B})\) into a \(4\)-fold triple system.

Label the elements of each block \(b\) with \(b_1, b_2, b_3\) and \(b_4\) (in any manner). For each \(i = 1,2,3,4\), define a set of triangles \({T}_i\) and a set of stars \({S}_i\) as follows: for each block \(b = (b_1, b_2, b_3, b_4)\) belonging to \({B}\), partition \(b\) into a triangle and a star centered at \(b_i\), and place the triangle in \({T}_i\) and the star in \({S}_i\). Then \((X,\mathcal{T}_i)\) is a partial \(\alpha\)-fold triple system.

Now if the edges belonging to the stars in \({S}_i\) can be arranged into a collection of triangles \({S}_i^*\), then \((X,{T}_i \cup {S}_i^*)\) is an \(\lambda\)-fold triple system and we say that \(M_i = (X,{T}_i \cup {S}_i^*)\) is the \(i\)th metamorphosis of \((X,{B})\).

The full metamorphosis of \((X,{B})\) is the set of four metamorphoses \(\{M_1, M_2, M_3, M_4\}\). The purpose of this work is to give a complete solution of the following problem: For which \(n\) and \(\lambda\) does there exist an \(\lambda\)-fold block design with block size \(4\) having a full metamorphosis into \(\lambda\)-fold triple systems?

A labeling of a graph is any map that carries some set of graph elements to numbers (usually to the positive integers). An \((a, d)\)-edge-antimagic total labeling on a graph with \(p\) vertices and \(q\) edges is defined as a one-to-one map taking the vertices and edges onto the integers \(1,2,…,p+q\) with the property that the sums of the labels on the edges and the labels of their endpoints form an arithmetic sequence starting from \(a\) and having a common difference \(d\). Such a labeling is called super if the smallest possible labels appear on the vertices.

We use the connection between \(a\)-labelings and edge-antimagic labelings for determining a super \((a,d)\)-edge-antimagic total labelings of disconnected graphs.

Let \(P\) be an \(n \times n\) array of symbols. \(P\) is called avoidable if for every set of \(z\) symbols, there is an \(n \times n\) Latin square \(L\) on these symbols so that corresponding cells in \(P\) and \(L\) differ. Due to recent work of Cavenagh and Ohman, we now know that all \(n \times n\) partial Latin squares are avoidable for \(n \geq 4\). Cavenagh and Ohman have shown that partial Latin squares of order \(4m + 1\) for \(m \geq 1\) [1] and \(4m – 1\) for \(m \geq 2\) [2] are avoidable. We give a short argument that includes all partial Latin squares of these orders of at least \(9\). We then ask the following question: given an \(n \times n\) partial Latin square \(P\) with some specified structure, is there an \(n \times n\) Latin square \(L\) of the same structure for which \(L\) avoids \(P\)? We answer this question in the context of generalized sudoku squares.

For a graph \(G\) with vertices labeled \(1,2,\ldots,n\) and a permutation \(\alpha\) in \(S_n\), the symmetric group on \(\{1,2,\ldots,n\}\), the \(\alpha\)-generalized prism over \(G\), \(\alpha(G)\), consists of two copies of \(G\), say \(G_x\) and \(G_y\), along with the edges \((x_i, y_{\alpha(i)})\), for \(1 \leq i \leq n\). In [10], the importance of building large graphs by using generalized prisms is indicated. A graph \(G\) is supereulerian if it has a spanning eulerian subgraph. In this note, we consider results of the form that if \(G\) has property \(P\), then for any \(\alpha \in S_{|V(G)|}\), \(\alpha(G)\) is supereulerian. As a result, we obtain a few properties of \(G\) which implies that for any \(\alpha \in S_{|V(G)|}\), \(\alpha(G)\) is supereulerian. Also, while the permutations are restricted, the related result is discussed.

Using the model of words, we give bijective proofs of Gould-Mohanty’s and Raney-Mohanty’s identities, which are respectively multivariable generalizations of Gould’s identity

\[\sum\limits_{k=0}^{n} \left(
\begin{array}{c}
x-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y+kz \\
n-k \\
\end{array}
\right)
= \sum\limits_{k=0}^{n}
\left(
\begin{array}{c}
x+\epsilon-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y-\epsilon+kz \\
n-k \\
\end{array}
\right)
\]

and Rothe’s identity
\[\sum\limits_{k=0}^{n}\frac{x}{x-kz}
\left(
\begin{array}{c}
x-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y+kz \\
n-k \\
\end{array}
\right)
=
\left(
\begin{array}{c}
x+y \\
n \\
\end{array}
\right)\]

Ryjáček introduced a closure concept in claw-free graphs based on local completion at a locally connected vertex. He showed that the closure of a graph is the line graph of a triangle-free graph. Broušek and Holub gave an analogous closure concept of claw-free graphs, called the edge-closure, based on local completion at a locally connected edge. In this paper, it is shown that the edge-closure is the line graph of a multigraph.

For a graph \(G = (V,E)\), a function \(f : V \rightarrow \{0,1,2\}\) is called a Roman dominating function (RDF) if for any vertex \(v\) with \(f(v) = 0\), there is at least one vertex \(w\) in its neighborhood with \(f(w) = 2\).

The weight of an RDF \(f\) of \(G\) is the value \(f(V) = \sum_{v\in V} f(v)\). The minimum weight of an RDF of \(G\) is its Roman domination number, denoted by \(\gamma_R(G)\). In this paper, we show that \(\gamma_R(G) + 1 \leq \gamma_R(\mu(G)) \leq \gamma_R(G) + 2\), where \(\mu(G)\) is the Mycielekian graph of \(G\), and then characterize the graphs achieving equality in these bounds.

A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. A fan \(F_n\) is the graph obtained from the join of the path \(P_n\) and the null graph \(N_1\). In this paper, we investigate the cordiality of the join and the union of pairs of fans and graphs consisting of a fan with a path, and a cycle.

We consider the problem of covering a unit cube with smaller cubes. The size of a cube is given by its side length and the size of a covering is the total size of the cubes used to cover the unit cube. We denote by \(g_3(n)\) the smallest size of a minimal covering using \(n\) cubes. We present tight results for the upper and lower bounds of \(g_3(n)\).

Let \(G\) be a graph. The cardinality of any largest independent set of vertices in \(G\) is called the independence number of \(G\) and is denoted by \(\alpha(G)\). Let \(a\) and \(b\) be integers with \(0 \leq a \leq b\). If \(a = b\), it is assumed that \(G\) be a connected graph, furthermore, \(a \geq \alpha(G)\), \(a/|V(G)| = 0 \pmod{2}\) if \(a\) is odd. We prove that every graph \(G\) has an \([a, b]\)-factor if its minimum degree is at least \((\frac{b+\alpha(G)a-\alpha(G)}{b})\lfloor \frac{b+\alpha(G)a}{2\alpha(G)} \rfloor -\frac{\alpha(G)}{b}(\lfloor \frac{b+\alpha(G)a}{2\alpha(G)}\rfloor )^2+ \theta\frac{\alpha(G)^2}{b}+\frac{a}{b}\alpha(G)\), where \(\theta = 0\) if \(a < b\), and \(\theta = 1\) if \(a = b\). This degree condition is sharp.

Suppose that graphs \(H\) and \(G\) are graceful, and that at least one of \(H\) and \(G\) has an \(\alpha\)-labeling. Four graph operations on \(H\) and \(G\) are provided. By utilizing repeatedly or in turn the four graph operations, we can construct a large number of graceful graphs. In particular, if both \(H\) and \(G\) have \(\alpha\)-labelings, then each of the graphs obtained by the four graph operations on \(H\) and \(G\) has an \(\alpha\)-labeling.

In this paper, we present three algebraic constructions of authentication codes from power functions over finite fields with secrecy and realize an application of some properties about authentication codes in [1]. The first and the third class are optimal. Some of the codes in the second class are optimal, and others in the second class are asymptotically optimal. All authentication codes in the three classes provide perfect secrecy.

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by \(q\)-series and compositions exhibiting particular patterns are specified by generating functions for these patterns. Here we view compositions as alternating sequences of partitions (i.e., alternating blocks) and obtain results for the asymptotic expectations of the number of such blocks (or parts per block) for different ways of defining the blocks.

For any integer \(k \geq 1\), a signed (total) \(k\)-dominating function is a function \(f : V(G) \rightarrow \{-1, 1\}\) satisfying \(\sum_{u \in N(v)} f(u) > k\) (\(\sum_{w \in N[v]} f(w) \geq k\)) for every \(v \in V(G)\), where \(N(v) = \{u \in V(G) | uv \in E(G)\}\) and \(N[v] = N(v) \cup \{v\}\). The minimum of the values of \(\sum_{v \in V(G)} f(v)\) , taken over all signed (total) \(k\)-dominating functions \(f\), is called the signed (total) \(k\)-domination number and is denoted by \(\gamma_{kS}(G)\) (\(\gamma’_{kS}(G)\), resp.). In this paper, several sharp lower bounds of these numbers for general graphs are presented.

Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices, where any two distinct vertices \(x\) and \(y\) are joined by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-packing design (\(G\)-covering design) of \(K_v\), denoted by \((v,G,\lambda)\)-PD (\((v,G,\lambda)\)-CD) is a pair \((X,B)\), where \(X\) is the vertex set of \(\lambda K_v\) and \(B\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in at most (at least) \(\lambda\) blocks of \(B\). A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. There are four graphs with 7 points, 7 edges and a 5-circle, denoted by \(G_i\), \(i = 1,2,3,4\). In this paper, we have solved the existence problem of the maximum \((v, G_i,\lambda)\)-PD and the minimum \((v, G_i, \lambda)\)-CD.

It was shown by Gaborit et al. [10] that a Euclidean self-dual code over \({GF}(4)\) with the property that there is a codeword whose Lee weight \(\equiv 2 \pmod{4}\) is of interest because of its connection to a binary singly-even self-dual code. Such a self-dual code over \({GF}_4\) is called Type I. The purpose of this paper is to classify all Type I codes of lengths up to 10 and extremal Type I codes of length 12, and to construct many new extremal Type I codes over \({GF}(4)\) of lengths from 14 to 22 and 34. As a byproduct, we construct a new extremal singly-even self-dual binary [36, 18, 8] code, and a new extremal singly-even self-dual binary [68, 34, 12] code with a previously unknown weight enumerator \(W_2\) for \(\beta = 95\) and \(\gamma = 1\).

Let \(j\) and \(k\) be two positive integers. An \(L(j,k)\)-labeling of a graph \(G\) is an assignment of nonnegative integers to the vertices of \(G\) such that the difference between labels of any two adjacent vertices is at least \(j\), and the difference between labels of any two vertices that are at distance two apart is at least \(k\). The minimum range of labels over all \(L(j,k)\)-labelings of a graph \(G\) is called the \(\lambda_{j,k}\)-number of \(G\), denoted by \(\lambda_{j,k}(G)\). Similarly, we can define \(L(j,k)\)-edge-labeling and \(L(j,k)\)-edge-labeling number, \(\lambda’_{j,k}(G)\), of a graph \(G\). In this paper, we show that if \(G\) is \(K_{1,3}\)-free with maximum degree \(\Delta\) then \(\lambda_{j,k}(G) \leq k\lfloor\Delta^2/2\rceil + j\Delta – 1\) except that \(G\) is a 5-cycle and \(j = k\). Consequently, we obtain an upper bound for \(\lambda’_{j,k}(G)\) in terms of the maximum degree of \(L(G)\), where \(L(G)\) is the line graph of \(G\). This improves the upper bounds for \(\lambda’_{2,1}(G)\) and \(\lambda’_{1,1}(G)\) given by Georges and Mauro [Ars Combinatoria \(70 (2004), 109-128]\). As a corollary, we show that Griggs and Yeh’s conjecture that \(\lambda_{2,1}(G) \leq \Delta^2\) holds for all \(K_{1,3}\)-free graphs and hence holds for all line graphs. We also investigate the upper bound for \(\lambda’_{j,k}(G)\) for \(K_{1,3}\)-free graphs \(G\).

Let \(G = (V, E)\) be a hamiltonian graph. A hamiltonian cycle \(C\) of \(G\) is described as \((v_1, v_2, \ldots, v_{n(G)}, v_1)\) to emphasize the order of vertices in \(C\). Thus, \(v_1\) is the beginning vertex and \(v_i\) is the \(i\)-th vertex in \(C\). Two hamiltonian cycles of \(G\) beginning at \(u\), \(C_1 = (u_1, u_2, \ldots, u_{n(G),u_1})\) and \(C_2 = (v_1, v_2, \ldots, v_{n(G)},v_1)\) of \(G\) are independent if \(u_1 = v_1 = u_1\) and \(u_i \neq v_i\) for every \(2 \leq i \leq n(G)\). A set of hamiltonian cycles \(\{C_1, C_2, \ldots, C_k\}\) of \(G\) are mutually independent if they are pairwise independent. The mutually independent hamiltonianicity of graph \(G\), \(\text{IHC}(G)\), is the maximum integer \(k\) such that for any vertex \(u\) there are \(k\)-mutually independent hamiltonian cycles of \(G\) beginning at \(u\). In this paper, we prove that \(\text{IHC}(G) \geq \delta(G)\) for any hamiltonian graph and \(\text{IHC}(G) \geq 2\delta(G) – n(G) + 1\) if \(\delta(G) \geq \frac{n(G)}{2}\). Moreover, we present some graphs that meet the bound mentioned above.

Using connectivity and planarity constraints we characterise all \(5\)-regular planar graphs with diameter \(3\).

In this paper, we investigate how the Wiener index of unicyclic graphs varies with graph operations. These results are used to present a sharp lower bound for the Wiener index of unicyclic graphs of order \(n\) with girth \(g\) and matching number \(\beta \geq \frac{3g}{2}\), Moreover, we characterize all extremal graphs which attain the lower bound.

The main aim of this paper is to present the idea of \(L\)-presheaves on a topological space \(X\). Categorical properties of \(L\)-presheaves are studied. The nature of \(L\)-presheaves locally in the neighbourhood of some point is summarized. This aim required constructing the notions of category of \(L\)-sets, \(L\)-direct systems and their \(L\)-limits and \(L\)-functors with their \(L\)-natural transformations. We prove that the ”\(L\)-stalk” is an \(L\)-functor from the category of \(L\)-presheaves to the category of \(L\)-sets.

A \( t \)-strong biclique covering of a graph \( G \) is an edge covering \(
E(G) = \bigcup_{i=1}^{t} E(H_i)\) where each \( H_i \) is a set of disjoint bicliques; say \( H_{i,1}, …, H_{i,r_i} \), such that the graph \( G \) has no edge between \( H_{i,k} \) and \( H_{i,j} \) for any \( 1 \leq j < k \leq r_i \). The strong biclique covering index \( S(G) \) is the minimum number \( t \) for which there exists a \( t \)-strong biclique covering of \( G \). In this paper, we study the strong biclique covering index of graphs. The strong biclique covering index of graphs was introduced in [H. Hajiabolhassan, A. Cheraghi, Bounds for Visual Cryptography Scheme, Discrete Applied Mathematics, 158 (2010), 659-665] to study the pixel expansion of visual cryptology. We present a lower bound for the strong biclique covering index of graphs and also we introduce upper bounds for different products of graphs.

We introduce vertex-transitive graphs \(\Gamma_n\), that are also embeddings of the strong product of triangular graphs \(L(K_n)\) and the complete graph \(K_2\). For any prime \(p\), linear codes obtained from the row span of incidence matrices of the graphs over \(\mathbb{F}_p\), are considered; their main parameters (length, dimension and minimum distance) and automorphism groups are determined. Unlike most codes that have been obtained from incidence and adjacency matrices of regular graphs by others, binary codes from the row span of incidence matrices of \(\Gamma_n\) have other minimum words apart from the rows of the matrices. Using a specific information set, PD-sets for full permutation decoding of the codes are exhibited.

Let \(G\) be a connected graph of order \(p \geq 2\). The closed interval \(I[x,y]\) consists of all vertices lying on some \(x-y\) geodesic of \(G\). If \(S\) is a set of vertices of \(G\), then \(I[S]\) is the union of all sets \(I\{x, y\}\) for \(x, y \in S\). The geodetic number \(g(G)\) is the minimum cardinality among the subsets \(S\) of \(V(G)\) with \(I[S] = V\). A geodetic set of cardinality \(g(G)\) is called a \(g\)-set of \(G\). For any vertex \(z\) in \(G\), a set \(S_x \subseteq V\) is an \(x\)-geodominating set of \(G\) if each vertex \(v \in V\) lies on an \(z-y\) geodesic for some element \(y\) in \(S_z\). The minimum cardinality of an \(x\)-geodominating set of \(G\) is defined as the \(x\)-geodomination number of \(G\), denoted by \(g_x(G)\) or simply \(g_x\). An \(x\)-geodominating set \(S_x\) of cardinality \(g_x(G)\) is called a \(g_x\)-set of \(G\). If \(S_x \cup \{x\}\) is a \(g\)-set of \(G\), then \(x\) is called a geo-vertex of \(G\). The set of all geo-vertices of \(G\) is called the geo-set of \(G\) and the number of geo-vertices of \(G\) is called the geo-number of \(G\) and it is denoted by \(gn(G)\). For positive integers \(r, d\) and \(n \geq 2\) with \(r < d \leq 2r\), there exists a connected graph \(G\) of radius \(r\), diameter \(d\) and \(gn(G) = n\). Also, for each triple \(p, d\) and \(n\) with \(3 \leq d \leq p – 1, 2 \leq n \leq p – 2\) and \(p – d – n + 1 \geq 0\), there exists a graph \(G\) of order \(p\), diameter \(d\) and \(gn(G) = n\). If the \(x\)-geodomination number \(g_x(G)\) is same for every vertex \(x\) in \(G\), then \(G\) is called a vertex geodomination regular graph or for short VGR-graph. If \(S \cup \{x\}\) is same for every vertex \(x\) in \(G\), then \(G\) is called a perfect vertex geodomination graph or for short PVG-graph. We characterize a PVG-graph.

The Wiener index, one of the oldest molecular topological descriptors used in mathematical chemistry, was well-studied during the past decades. For a graph \(G\), its Wiener index is defined as \(W(G) = \sum\limits_{\{u, v\} \subseteq V(G)} d_G(u, v)\), where \(d_G(u, v)\) is the distance between two vertices \(u\) and \(v\) in \(G\). In this paper, we study the Wiener index of a class of composite graph, namely, double graph. We reveal the relation between the Wiener index of a given graph and the one of its double graph as well as the relation between Wiener index of a given graph and the one of its \(k\)-iterated double graph. As a consequence, we determine the graphs with the maximum and minimum Wiener index among all double graphs and \(k\)-iterated double graphs of connected graphs of the same order, respectively.

The set of unicyclic graphs with \(n\) vertices and diameter \(d\) is denoted by \(\mathcal{U}_{n,d}\). For \(3 \leq i \leq d\), let \(P_{n-d-1}(i)\) be the graph obtained from path \(P_{d+1}: v_1 v_2 \ldots v_{d+1}\) by adding \(n-d-1\) pendant edges at \(v_i\), and \(U_{n-d-2}(i)\) be the graph obtained from \(P_{n-d-1}(i)\) by joining \(v_{i-2}\) and a pendant neighbor of \(v_{i}\). In this paper, we determine all unicyclic graphs in \(\mathcal{U}_{n,d}\) whose largest Laplacian eigenvalue is greater than \(n-d+2\). For \(n-d \geq 6\) and \(G \in \mathcal{U}_{n,d}\), we prove further that the largest Laplacian eigenvalue \(\mu(G) \leq \max\{\lambda(U_{n,d-2}(i)) \mid 3 \leq i \leq d\}\), and conjecture that \(\mathcal{U}_{n,d}.\) is the unique graph which has the greatest value of the greatest Laplacian eigenvalue in \(\mathcal{U}_{n,d}\). We also prove that the conjecture is true for \(3 \leq d \leq 6\).

The Padmakar-Ivan \((PI)\) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the PI index with respect to the extremal simple pericondensed hexagonal systems and we solve it completely.

Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices. Let \(G\) be a finite simple graph. A \(G\)-design (\(G-GD_\lambda)(v)\) (\(G\)-packing (\(G-PD_\lambda)(v)\), \(G\)-covering (\(G-CD_\lambda)(v)\)) of \(K_v\) is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined exactly (at most, at least) in \(\lambda\) blocks. In this paper, we will discuss the maximum packing designs and the minimum covering designs for four particular graphs each with six vertices and nine edges.

Let \(a\) and \(b\) be integers such that \(1 \leq a < b\), and let \(G\) be a graph of order \(n\) with \(n \geq \frac{(a+b)(2a+2b-3)}{a+1}\) and the minimum degree \(\delta(G) \geq \frac{(b-1)^2-(a+1)(b-a-2)}{a+1} \). Let \(g(x)\) and \(f(x)\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) \leq f(x) \leq b\) for each \(x \in V(G)\). We prove that if \(|N_G(x) \cup N_G(y)| \geq \frac{(b-1)n}{a+b} \) for any two nonadjacent vertices \(x\) and \(y\) in \(G\), then \(G\) has a \((g, f)\)-factor. Furthermore, it is shown that the result in this paper is best possible in some sense.

Let \(D\) be a digraph with order at least two. The transformation digraph \(D^{++-}\) is the digraph with vertex set \(V(D) \cup A(D)\) in which \((x, y)\) is an arc of \(D^{++-}\) if one of the following conditions holds:(i) \(x, y \in V(D)\), and \((x, y)\) is an arc of \(D\);(ii) \(x, y \in A(D)\), and the head of \(x\) is the tail of \(y\);(iii) \(x \in V(D), y \in A(D)\), and \(x\) is not the tail of \(y\);(iv) \(x \in A(D), y \in V(D)\), and \(y\) is not the head of \(x\).In this paper, we determine the regularity and diameter of \(D^{++-}\). Furthermore, we characterize maximally-arc-connected or super-arc-connected \(D^{++-}\). We also give sufficient conditions for this kind of transformation digraph to be maximally-connected or super-connected.

For a graph \(G\) and any two vertices \(u\) and \(v\) in \(G\), let \(d_G(u,v)\) denote the distance between them and let \(diam(G)\) be the diameter of \(G\). A multi-level distance labeling (or radio labeling) for \(G\) is a function \(f\) that assigns to each vertex of \(G\) a positive integer such that for any two distinct vertices \(u\) and \(v\), \(d_G(u,v) + |f(u) – f(v)| = diam(G) + 1\). The largest positive integer in the range of \(f\) is called the span of \(f\). The radio number of \(G\), denoted \(rn(G)\), is the minimum span of a multi-level distance labeling for \(G\).

A helm graph \(H_n\) is obtained from the wheel \(W_n\) by attaching a vertex of degree one to each of the \(n\) vertices of the cycle of the wheel. In this paper, the radio number of the helm graph is determined for every \(n \geq 3\): \(rn(H_3) = 13\), \(rn(H_4) = 21\), and \(rn(H_n) = 4n + 2\) for every \(n \geq 5\). Also, a lower bound of \(rn(G)\) related to the length of a maximum Hamiltonian path in the graph of distances of \(G\) is proposed.

In this paper, firstly, we define the generalized \(k\)-Horadam sequence and investigate some of its properties. In addition, by also defining the circulant matrix \(C_n(H)\) whose entries are the generalized \(k\)-Horadam numbers, we compute the spectral norm, eigenvalues, and the determinant of this matrix.

The generating function for \(p\)-regular partitions is given by \(\frac{{(q^p;q^p)}_\infty}{{(q;q)}_\infty}\) .In this paper, we will investigate the reciprocal of this generating function. Several interesting results will be presented, and as a corollary of one of these, we will get a parity result due to Sellers for \(p\)-regular partitions with distinct parts.

Motivated by the results from [J. Li, W. Shiu, W. Chan, The Laplacian spectral radius of some graphs, Linear Algebra Appl. \(431 (2009) 99-103]\), we determine the extremal graphs with the second largest Laplacian spectral radius among all bipartite graphs with vertex connectivity \(k\).

Let \(\omega(K_{1,1,t,}{n})\) be the smallest even integer such that every \(n\)-term graphic sequence \(\pi = (d_1,d_2,\ldots,d_n)\) with \(\sigma(\pi) = d_1+d_2+\cdots+d_n \geq \sigma(K_{1,1,t,}{n})\) has a realization \(G\) containing \(K_{1,1,t,}{n}\) as a subgraph, where \(K_{1,1,t,}{n}\) is the \(1 \times 1 \times t\) complete \(3\)-partite graph. Recently, Lai (Discrete Mathematics and Theoretical Computer Science, \(7(2005), 75-81)\) conjectured that for \(n \geq 2t+4\),

\[\sigma(K_{1,1,t,}{n}) = \begin{cases}
(t+1)(n-1)+2 & \text{if \(n\) is odd or \(t\) is odd,}\\
(t+1)(n-1)+1 & \text{if \(n\) and \(t\) are even.}
\end{cases}\]

In this paper, we prove that the above equality holds for \(n \geq t+4\).

A method called the standard construction generates an algebra from a \(K\)-perfect \(m\)-cycle system. Let \({C}_m^K\) denote the class of algebras generated by \(K\)-perfect \(m\)-cycle systems. For each \(m\) and \(K\), there is a known set \(\Sigma_m^K\) of identities which all the algebras in \({C}_m^K\) satisfy. The question of when \({C}_m^K\) is a variety is answered in [2]. When \({C}_m^K\) is a variety, it is defined by \(\Sigma_m^K\). In general, \({C}_m^K\) is a proper subclass of \({V}(\Sigma_m^K)\), the variety of algebras defined by \(\Sigma_m^K\).

If the standard construction is applied to partial \(K\)-perfect \(m\)-cycle systems, then partial algebras result. Using these partial algebras, we are able to investigate properties of \({V}(\Sigma_m^K)\). We show that the free algebras of \({V}(\Sigma_m^K)\) correspond to \(K\)-perfect \(m\)-cycle systems, so \({C}_m^K\) generates \({V}(\Sigma_m^K)\). We also answer two questions asked in [5] concerning subvarieties of \({V}(\Sigma_m^K)\). Many of these results can be unified in the result that for any subset \(K’\) of \(K\), \({V}(\Sigma_m^{K’})\) is generated by the class of algebras corresponding to finite \(K\)-perfect \(m\)-cycle systems.

We examine designs \( \mathcal{D}_i \) and ternary codes \( C_i \), where \( i \in \{112, 113, 162, 163, 274\} \), constructed from a primitive permutation representation of degree 275 of the sporadic simple group \( M^cL \). We prove that \( \dim(C_{113}) = 22, \quad \dim(C_{162}) = 21, \quad C_{113} \supset C_{162}\) and \( M^cL:2 \) acts irreducibly on \( C_{162} \). Furthermore, we have \( C_{112} = C_{163} = C_{274} = V_{27_5}(GF(3)),\) \(
\text{Aut}(\mathcal{D}_{112}) = \text{Aut}(\mathcal{D}_{163})\) = \(
\text{Aut}(\mathcal{D}_{113}) = \text{Aut}(\mathcal{D}_{162}) =
\text{Aut}(C_{113}) = \text{Aut}(C_{162}) = M^{c}L:2 \) while \( Aut(\mathcal{D}_{274}) = Aut(C_{112}) = Aut(C_{163}) = Aut(C_{274}) = S_{275}. \)
We also determine the weight distributions of \( C_{113} \) and \( C_{162} \) and that of their duals.