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 \(m \geq 14\).