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.