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, a new multisender authentication codes with simultaneous model is constructed base on singular symplectic geometry over finite fields. The parameters and the maximum probabilities of deceptions are also computed.

Let \(D = (V, A)\) be a digraph with vertex set \(V\) and arc set \(A\). An absorbant of \(D\) is a set \(S \subseteq V\) such that for each \(v \in V \setminus S\), \(O(v) \cap S \neq \emptyset\), where \(O(v)\) is the out-neighborhood of \(v\). The absorbant number of \(D\), denoted by \(\gamma_a(D)\), is defined as the minimum cardinality of an absorbant of \(D\). The generalized de Bruijn digraph \(G_B(n, d)\) is a digraph with vertex set \(V(G_B(n, d)) = \{0, 1, 2, \ldots, n-1\}\) and arc set \(A(G_B(n, d)) = \{(x, y) \mid y = dx + i \, (\text{mod} \, n), 0 \leq i < d\}\). In this paper, we determine \(\gamma_a(G_B(n, d))\) for all \(d \leq n \leq 4d\).

We provide a concise combinatorial proof for the solution of the general two-term recurrence \(u(n, k) = u(n-1, k-1) + (a_{n-1}+b_{k})u(n-1, k)\), initially discovered by Mansour et al. \([4]\).

The vulnerability value of a communication network is the resistance of this communication network until some certain stations or communication links between these stations are disrupted and, thus communication interrupts. A communication network is modeled by a graph to measure the vulnerability as stations corresponding to the vertices and communication links corresponding to the edges, There are several types of vulnerability parameters depending upon the distance for each pair of two vertices. In this paper. closeness, vertex residual closeness (\(VRC\)) and normalized vertex residual closeness (\(NV RC\)) of some Mycielski graphs are calculated, furthermore upper and lower bounds are obtained.

A graph \(G\) is an {\([s, t]\)-graph if every subgraph induced by \(s\) vertices of \(G\) has at least \(t\) edges. This concept extends the independent number. In this paper, we prove that:

(1) if \(G\) is a \(k\)-connected \([k+2, 2]\)-graph, then \(G\) has a Hamilton cycle or \(G\) is isomorphic to the Petersen graph or \(\overline{K_{k+1}} \vee G_k\),

(2) if \(G\) is a \(k\)-connected \([k+3, 2]\)-graph, then \(G\) has a Hamilton path or \(G\) is isomorphic to \(\overline{K_{k+1}} \vee G_k\),
where \(G_r\) is an arbitrary graph of order \(k\). These two results generalize the following known results obtained by Chvátal-Erdős and Bondy, respectively:

(a) if \(\alpha(G)\leq \kappa(G) \) of order \(n \geq 3\), then \(G\) has a Hamilton cycle,

(b) if \(\alpha(G) – 1 \leq \kappa(G)\) , then \(G\) has a Hamilton path.

In this paper we define new generalizations of Fibonacci numbers and Lucas numbers in the distance sense. These generalizations are closely related to the concept of \((2,k )\)-distance Fibonacci numbers presented in \([10]\). We show some applications of these numbers in number decompositions and we also define a new type of Lucas numbers.

For a vector \({R} = (r_1, r_2, \ldots, r_m)\) of non-negative integers, a mixed hypergraph \(\mathcal{H}\) is a realization of \({R}\) if its chromatic spectrum is \({R}\). In this paper, we determine the minimum number of vertices of realizations of a special kind of vectors \({R}_2\). As a result, we partially solve an open problem proposed by Král in \(2004\).

A strong edge-coloring is a proper edge-coloring such that two edges with the same color are not allowed to lie on a path of length three. The strong chromatic index of a graph \(G\), denoted by \(s'(G)\), is the minimum number of colors in a strong edge-coloring. We denote the degree of a vertex \(v\) by \(d(v)\). Let the \({Ore-degree}\) of a graph \(G\) be the maximum value of \(d(u) + d(v)\), where \(u\) and \(v\) are adjacent vertices in \(G\). Let \(F_3\) denote the graph obtained from a \(5\)-cycle by adding a new vertex and joining it to a pair of nonadjacent vertices of the \(5\)-cycle. In \(2008\), Wu and Lin [J. Wu and W. Lin, The strong chromatic index
of a class of graphs, Discrete Math., \(308 (2008), 6254-6261]\) studied the strong chromatic index with respect to the Ore-degree. Their main result states that if a connected graph \(G\) is not \(F_3\) and its Ore-degree is \(5\), then \(s'(G) \leq 6\). Inspired by the result of Wu and Lin, we investigate the strong edge-coloring of graphs with Ore-degree 6. We show that each graph \(G\) with Ore-degree \(6\) has \(s'(G) \leq 10\). With the further condition that \(G\) is bipartite, we have \(s'(G) \leq 9\). Our results give general forms of previous results about strong chromatic indices of graphs with maximum degree \(3\).

For a graph \(G\), an edge labeling of \(G\) is a bijection \(f: E(G) \to \{1, 2, \ldots, |E(G)|\}\). The \emph{induced vertex sum} \(f^*\) of \(f\) is a function defined on \(V(G)\) given by \(f^+(u) = \sum_{uv \in E(G)} f(uv)\) for all \(u \in V(G)\). A graph \(G\) is called \emph{antimagic} if there exists an edge labeling of \(G\) such that the induced vertex sum of the edge labeling is injective. Hartsfield and Ringel conjectured in 1990 that all connected graphs except \(K_2\) are antimagic. A spider is a connected graph with exactly one vertex of degree exceeding \(2\). This paper shows that all spiders are antimagic.

In this paper, we consider the problem of determining precisely which graphic matroids \(M\) have the property that the splitting operation,by every pair of elements, on \(M\) yields a cographic matroid. This problem is solved by proving that there are exactly three minorminimal graphs that do not have this property.

In this paper, we give a new and interesting identities of Boole and Euler polynomials which are derived from the symmetry properties of the \(p\)-adic fermionic integrals on \(\mathbb{Z}_p\).

In this paper we address the problem of construction of critical sets
in \(F\)-squares of the form \(F(2n; 2, 2,……… ,2)\). We point out that the
critical set in \(F(2n; 2,2, ……… ,2)\) obtained by Fitina, Seberry and
Sarvate \((1999)\) is not correct and prove that in the given context a
proper subset is a critical set.

A connected graph \(G = (V(G), E(G))\) is called a quasi-tree graph if there exists a vertex \(u_0 \in V(G)\) such that \(G – u_0\) is a tree. Let \(\mathcal{P}(2k) := \{G: G \text{ is a quasi-tree graph on } 2k \text{ vertices with perfect matching}\}\), and \(\mathcal{P}(2k, d_0) := \{G: G \in \mathcal{P}(2k), \text{ and there is a vertex } u_0 \in V(G) \text{ such that } G – u_0 \text{ is a tree with } d_G(u_0) = d_0\}\). In this paper, the maximal indices of all graphs in the sets \(\mathcal{P}(2k)\) and \(\mathcal{P}(2k, d_0)\) are determined, respectively. The corresponding extremal graphs are also characterized.

A combinatorial sum for the Stirling numbers of the second kind is generalized. This generalization provides a new explicit formula for the binomial sum \(\sum_{k=0}^{n}k^ra^kb^{n-k} \binom{n}{k}\), where \(a, b \in \mathbb{R} – \{0\}\) and \(n, r \in \mathbb{N}\). As relevant special cases, simple explicit expressions for both the binomial sum \(\sum_{k=0}^{n} k^r\binom{n}{k} \) and the raw moment of order \(r\) of the binomial distribution \(B(n, p)\) are given. All these sums are expressed in terms of generalized \(r\)-permutations.

Let \(G\) be a simple connected graph with vertex set \(V(G)\). The Gutman index \(\text{Gut}(G)\) of \(G\) is defined as \(\text{Gut}(G) = \sum\limits_{\{x,y\} \subseteq V(G)} d_G(x) d_G(y) d_G(x,y)\), where \(d_G(x)\) is the degree of vertex \(v\) in \(G\) and \(d_G(x,y)\) is the distance between vertices \(x\) and \(y\) in \(G\). In this paper, the second-minimum Gutman index of unicyclic graphs on \(n\) vertices and girth \(m\) is characterized.

The clique-chromatic number of a graph is the least number of colors on the vertices of the graph without a monocolored maximal clique of size at least two.In \(2004\), Bacsé et al. proved that the family of line graphs has no bounded clique-chromatic number. In particular, the Ramsey numbers provide a sequence of the line graphs of complete graphs with no bounded clique-chromatie number. We
complete this result by giving the exact values of the clique-chromatic numbers of the line graphs of complete graphs in terms of Ramsey numbers. Furthermore, the clique-chromatic numbers of the line graphs of triangle-free graphs are characterized.

The current article focuses on the generalized \(k\)-Pell \((p, i)\)-numbers for \(k = 1, 2, \ldots\) and \(0 \leq i \leq p\). It introduces the generalized \(k\)-Pell \((p, i)\)-numbers and their generating matrices and generating functions. Some interesting identities are established.

For a graph \(G\), let \({Z}(G)\) be the total number of matchings in \(G\). For a nontrivial graph \(G\) of order \(n\) with vertex set \(V(G) = \{v_1, \ldots, v_n\}\), Cvetković et al. \([2]\) defined the triangle graph of \(G\), denoted by \(R(G)\), to be the graph obtained by introducing a new vertex \(v_{ij}\) and connecting \(u_{ij}\) both to \(v_i\) and to \(v_j\) for each edge \(v_iv_j\) in \(G\). In this short paper, we prove that for a connected graph \(G\), if \({Z}(R(G)) \geq (\frac{1}{2}n-\frac{1}{2}+\frac{5}{2n})^2\), then \(G\) is traceable. Moreover, for a connected graph \(G\), we give sharp upper bounds for \({Z}(R(G))\) in terms of clique number, vertex connectivity, and spectral radius of \(G\), respectively.

We prove a two-point concentration for the tree domination number of the random graph \(G_{n,p}\) provided \(p\) is constant or \(p \to 0\) sufficiently slow.

A 2-independent set in a graph \(G\) is a subset \(J\) of the vertices such that the distance between any two vertices of \(J\) in \(G\) is at least three. The number of 2-independent sets of a graph \(G\) is denoted by \(i_2(G)\). For a forest \(F\), \(i_2(F – e) > i_2(F)\) for each edge \(e\) of \(F\). Hence, we exclude all forests having isolated vertices. A forest is said to be extra-free if it has no isolated vertex. In this paper, we determine the \(k\)-th largest number of 2-independent sets among all extra-free forests of order \(n \geq 2\), where \(k = 1, 2, 3\). Extremal graphs achieving these values are also given.

The notion of multiparameter \(q\)-noncentral Stirling numbers is introduced by means of a triangular recurrence relation. Some properties for these \(q\)-analogues such as vertical and horizontal recurrence relations, horizontal generating functions, explicit formula, orthogonality and inverse relations are established. Moreover, we introduce the multiparameter Bell numbers and Bell polynomials, their connection to factorial moments and their respective \(q\)-analogues.

Let \(a, b\), and \(k\) be nonnegative integers with \(2 \leq a \leq 6\) and \(b \equiv 0 \pmod{a-1}\). Let \(G\) be a graph of order \(n\) with \(n \geq \frac{(a+b-1)(2a+b-4)-a+1}{b} + k\). A graph \(G\) is called an \((a, b, k)\)-critical graph if after deleting any \(k\) vertices of \(G\), the remaining graph has an \([a, b]\)-factor. In this paper, it is proved that \(G\) is an \((a, b, k)\)-critical graph if and only if \[|N_G(X)| >\frac{(a-1)n + |X| + bk-1}{a+b-1} \] for every non-empty independent subset \(X\) of \(V(G)\), and \[\delta(G) > \frac{(a-1)n + b + bk}{a+b-1}.\] Furthermore, it is shown that the result in this paper is best possible in some sense.

Two-dimensional codes in \(LRTJ\) spaces are subspaces of the space \(Mat_{m\times s}(\mathbb{Z}_q)\), the linear space of all \(m \times s\)-matrices with entries from a finite ring \(\mathbb{Z}_q\), endowed with the \(LRTJ\)-metric \([3,9]\). Also, the error-correcting capability of a linear code depends upon the number of parity-check symbols. In this paper, we obtain a lower bound on the number of parity checks of two-dimensional codes in \(LRTJ\)-spaces correcting both independent as well as cluster array errors.

Let \(G = (V, E)\) be a graph without an isolated vertex. A set \(D \subseteq V(G)\) is a total dominating set if \(D\) is dominating and the induced subgraph \(G[D]\) does not contain an isolated vertex. The total domination number of \(G\) is the minimum cardinality of a total dominating set of \(G\). A set \(D \subseteq V(G)\) is a total outer-connected dominating set if \(D\) is total dominating and the induced subgraph \(G[V(G) – D]\) is connected. The total outer-connected domination number of \(G\) is the minimum cardinality of a total outer-connected dominating set of \(G\). We characterize all unicyclic graphs with equal total domination and total outer-connected domination numbers.

We give a characterization of strongly multiplicative graphs. First, we introduce some necessary conditions for a graph to be strongly multiplicative.Second, we discuss the independence of these necessary conditions. Third, we show that they are altogether not sufficient for a graph to be strongly multiplicative. Fourth, we add another necessary condition. Fifth, we prove that this necessary condition is stronger than the mentioned necessary conditions except one. Finally, we conjecture that the condition itself is stronger than all of them.

Let \(G = (V, E)\) be a simple connected graph with \(n\) vertices and \(m\) edges. Further, let \(\lambda_i(L)\), \(i = 1, 2, \ldots, n\), be the non-increasing eigenvalues of the normalized Laplacian matrix of the graph \(G\). In this paper, we obtain the following result: For a connected graph \(G\) of order \(n\), \(lambda_2(L) = \lambda_3(L) = \cdots = \lambda_{n-1}(L)\) if and only if \(G\) is a complete graph \(K_n\) or \(G\) is a complete bipartite graph \(K_{p,q}\). Moreover, we present lower and upper bounds for the normalized Laplacian spectral radius of a graph and characterize graphs for which the lower or upper bounds are attained.

Let \(k \geq 3\) be an integer, and let \(G\) be a graph of order \(n\) with \(n \geq \max\{10, 4k-3\}\) and \(\delta(G) \geq k+1\). If \(G\) satisfies \(\max\{d_G(x), d_G(y)\} \geq \frac{n}{2}\) for each pair of nonadjacent vertices \(x, y\) of \(G\), then \(G\) is a fractional \(k\)-covered graph. The result is best possible in some sense, and it improves and extends the result of C. Wang and C. Ji (C. Wang and C. Ji, Some new results on \(k\)-covered graphs, Mathematica Applicata \(11(1) (1998), 61-64)\).

For a positive integer \(k\), let \(\mathbb{Z}_k = (\mathbb{Z}_k, +, 0)\) be the additive group of congruences modulo \(k\) with identity \(0\), and \(\mathbb{Z}_1\) is the usual group of integers \(\mathbb{Z}\). We call a finite simple graph \(G = (V(G), E(G))\) \(\mathbb{Z}_k\)-magic if it admits an edge labeling \(\ell: E(G) \to \mathbb{Z}_k \setminus \{0\}\) such that the induced vertex sum labeling \(\ell^+: V(G) \to \mathbb{Z}_k\), defined by \(\ell^+(v) = \sum_{uv \in E(G)} \ell(uv)\), is constant. The constant is called a \emph{magic sum index}, or an \emph{index} for short, of \(G\) under \(\ell\), following R. Stanley. The \emph{null set} of \(G\), defined by E. Salehi as the set of all \(k\) such that \(G\) is \(\mathbb{Z}_k\)-magic with zero magic sum index, is denoted by \(Null(G)\). For a fixed integer \(k\), we consider the set of all possible magic sum indices \(r\) such that \(G\) is \(\mathbb{Z}_k\)-magic with magic sum index \(r\), and denote it by \(I_k(G)\). We call \(I_k(G)\) the \emph{index set} of \(G\) with respect to \(\mathbb{Z}_k\). In this paper, we investigate properties and relations of index sets \(I_k(G)\) and null sets \(Null(G)\) for \(\mathbb{Z}_k\)-magic graphs. Among others, we determine null sets of generalized wheels and generalized fans and construct infinitely many examples of \(\mathbb{Z}_k\)-magic graphs with magic sum zero. Some open problems are presented.

Packing and covering are dual problems in graph theory. A graph \(G\) is called \(H\)-equipackable if every maximal \(H\)-packing in \(G\) is also a maximum \(H\)-packing in \(G\). Dually, a graph \(G\) is called \(H\)-equicoverable if every minimal \(H\)-covering in \(G\) is also a minimum \(H\)-covering in \(G\). In 2012, Zhang characterized two kinds of equipackable paths and cycles: \(P_k\)-equipackable paths and cycles, and \(M_k\)-equipackable paths and cycles. In this paper, we characterize \(P_k\)-equicoverable (\(k > 3\)) paths and cycles, and \(M_k\)-equicoverable (\(k > 2\)) paths and cycles.

For non-negative integers \(n_1, n_2, \ldots, n_t\), let \(GL_{n_1, n_2, \ldots, n_t}(\mathbb{F}_q)\) denote the \(t\)-singular general linear group of degree \(n = n_1 + n_2 + \cdots + n_t\) over the finite field \(\mathbb{F}_q^{n_1+n_2+\ldots+n_t}\) denote the \((n_1+n_2+\ldots+n_t)\)-dimensional \(t\)-singular linear space over the finite \(\mathbb{F}\). Let \(\mathcal{M}\) be any orbit of subspaces under \(GL_{n_1, n_2, \ldots, n_t}(\mathbb{F}_q)\). Denote by \(\mathcal{L}\) the set of all intersections of subspaces in \(M\). Ordered by ordinary or reverse inclusion, two posets are obtained. This paper discusses their geometricity and computes their characteristic polynomials.

The purpose of this paper is to establish g-analogue of some identities and then generalize the result to give identities for finite sums for products of generalized q-harmonic numbers and reciprocals of \(q\)-binomial coefficients.

For \(S \subseteq V(G)\) and \(|S| \geq 2\), let \(\lambda(S)\) denote the maximum number of edge-disjoint trees connecting \(S\) in \(G\). For an integer \(k\) with \(2 \leq k \leq n\), the generalized \(k\)-edge-connectivity \(\lambda_k(G)\) of \(G\) is defined as \(\lambda_k(G) = \min\{\lambda(S) : S \subseteq V(G) \text{ and } |S| = k\}\). Note that when \(|S| = 2\), \(\lambda_2(G)\) coincides with the standard \emph{edge-connectivity} \(\lambda(G)\) of \(G\). In this paper, we characterize graphs of order \(n\) such that \(\lambda_n(G) = n – 3\). Furthermore, we determine the minimal number of edges of a graph \(G\) of order \(n\) with \(\lambda_3(G) = 1, n – 3, n – 2\) and establish a sharp lower bound for \(2 \leq \lambda_3(G) \leq n – 4\).

The noncrossing partitions with fixed points have been introduced and studied in the literature. In this paper, as their continuations, we derive expressions for \(f_m(x_1, 0^\mu, x_{\mu+2},0^\rho,x_{\mu+\mu+3},0^{m-\mu-\rho-3})\),and \(f_{m}(x_1,x_2, 0^\mu, x_{\mu+3},0^\rho,x_{\mu+\mu+3},0^{\rho+\mu+4},0^{m-\rho-\mu-4}\), are given,respectively. Moreover, we introduce noncrossing partitions with fixed points having specific property \(\mathcal{P}\) and describe their enumeration through a multivariable function \(f_m^\mathcal{P}(x_1, x_2, \ldots, x_m)\). Additionally, we obtain counting formulas for \(f_m^\mathcal{P}(x_1, 0^{m-1})\) and \(f_m^\mathcal{P}(x_1, x_2, 0^{m-2})\) for various properties \(\mathcal{P}\).

Let \(G = (V(G), E(G))\) be a simple, connected, and undirected graph with vertex set \(V(G)\) and edge set \(E(G)\). A set \(S \subseteq V(G)\) is a \emph{dominating set} if for each \(v \in V(G)\), either \(v \in S\) or \(v\) is adjacent to some \(w \in S\). That is, \(S\) is a dominating set if and only if \(N[S] = V(G)\). The \emph{domination number} \(\gamma(G)\) is the minimum cardinality of minimal dominating sets. In this paper, we provide an improved upper bound on the domination number of generalized Petersen graphs \(P(c,k)\) for \(c \geq 3\) and \(k \geq 3\). We also prove that \(\gamma(P(4k,k)) = 2k + 1\) for even \(k\), \(\gamma(P(5k, k)) = 3k\) for all \(k \geq 1\), and \(\gamma(P(6k,k)) = \left\lceil \frac{10k}{3} \right\rceil\) for \(k \geq 1\) and \(k \neq 2\).

A proper coloring of a graph \(G\) assigns colors to vertices such that adjacent vertices receive distinct colors. The minimum number of colors is the chromatic number \(\chi(G)\). For a graph \(G\) and a proper coloring \(c: V(G) \to \{1, 2, \ldots, k\}\), the color code of a vertex \(v\) is \(code(v) = (c(v), S_v)\), where \(S_v = \{c(u): u \in N(v)\}\). Coloring \(c\) is \emph{singular} if distinct vertices have distinct color codes, and the \emph{singular chromatic number} \(\chi_s(G)\) is the minimum positive integer \(k\) for which \(G\) has a singular \(k\)-coloring. Thus, \(\chi(G) \leq \chi_{si}(G) \leq n\) for every graph \(G\) of order \(n\). We establish a characterization for all triples \((a, b, n)\) of positive integers for which there exists a graph \(G\) of order \(n\) with \(\chi(G) = a\) and \(\chi_{si}(G) = b\). Furthermore, for every vertex \(v\) and edge \(e\) in \(G\), we show:
\( \chi_{si}(G) – 1 \leq \chi_{si}(G – v) \leq \chi_{si}(G) + \deg(v) \) and
\( \chi_{si}(G) – 1 \leq \chi_{si}(G – e) \leq \chi_{si}(G) + 2, \)
and prove that these bounds are sharp. Additionally, we determine the singular chromatic numbers of cycles and paths.

In this paper, we construct new classes of difference systems of sets with three blocks.

Ruskey and Savage posed the question: For \(n \geq 2\), does every matching in \(Q_n\) extend to a Hamiltonian cycle in \(Q_n\)? Fink showed that the answer is yes for every perfect matching, thereby proving Kreweras’ conjecture. In this paper, we prove that for \(n \geq 3\), every matching in \(Q_n\) not covering exactly two vertices at distance \(3\) extends to a Hamiltonian cycle in \(Q_n\). An edge in \(Q_n\) is an \(i\)-edge if its endpoints differ in the \(i\)th position. We also show that for \(n \geq 2\), every matching in \(Q_n\) consisting of edges in at most four types extends to a Hamiltonian cycle in \(Q_n\).