A new construction of authentication codes with arbitration from \((2\nu-2+2+1)\)-dimensional singular pseudo-symplectic geometry on finite fields is given. Assuming that the encoding rules are chosen according to a uniform probability distribution, the parameters and the probabilities of success for different types of deceptions are also computed.

By a defensive alliance in a graph \(G\) we mean any set \(S\) of vertices in \(G\) such that each vertex in \(S\) is adjacent to at least as many vertices inside \(S\), including the vertex itself, as outside \(S\). If, in addition, we require that every vertex outside a defensive alliance \(S\) is adjacent to at least one vertex in \(S\), then \(S\) becomes a global defensive alliance. The minimum cardinality of a global defensive alliance is the global defensive alliance number of \(G\). In this paper, we determine bounds for the global defensive alliance numbers in the join, corona, and composition of graphs.

Let \(P_{k+1}\) denote a path of length \(k\) and let \(C_k\) denote a cycle of length \(k\). A triangle is a cycle of length three. As usual, \(K_n\) denotes the complete graph on \(n\) vertices. It is shown that for all nonnegative integers \(p\) and \(q\) and for all positive integers \(n\), \(K_n\) can be decomposed into \(p\) copies of \(P_4\) and \(q\) copies of \(C_3\) if and only if \(3(p+q) = e(K_n)\), \(p \neq 1\) if \(n\) is odd, and \(p \geq \frac{n}{2}\) if \(n\) is even.

Motivated by Kotzig and Rosa’s concept of edge magic deficiency, Figueroa-Centeno, Ichishima, and Muntaner-Batle defined a similar concept for super edge magic total labelings. The super edge magic deficiency of a graph \(G\), which is denoted by \(\mu_s(G)\), is the minimum nonnegative integer \(n\) \(+\infty\) if there exists no such \(n\). In this paper, we study the super edge magic deficiency of kite graphs.

The corona of two graphs \(G\) and \(H\), written as \(G \odot H\), is the graph obtained by taking one copy of \(G\) and \(|V(G)|\) copies of \(H\), and then joining the \(i\)th vertex of \(G\) to every vertex in the \(i\)th copy of \(H\). In this paper, we present the explicit formulae for the Wiener, hyper-Wiener and reverse-Wiener indices of the corona of two graphs.

The energy of a graph \(G\), denoted by \(E(G)\), is defined to be the sum of absolute values of all eigenvalues of the adjacency matrix of \(G\). Let \(\mathcal{B}(p, q)\) denote the set of bipartite unicyclic graphs with a \((p, q)\)-bipartition, where \(q \geq p \geq 2\). Recently, Li and Zhou [MATCH Commun. Math. Comput. Chem. \(54 (2005) 379-388]\) conjectured that for \(q \geq 3\), \(E(B(3, q)) > E(H(3, q))\), where \(B(3, q)\) and \(H(3, q)\) are respectively graphs as shown in Fig. 1. In this note, we show that this conjecture is true for \(3 \leq q \leq 217\). As a byproduct, we determined the graph with minimal energy among all graphs in \(\mathcal{B}(3, q)\).

In this work, infinite similarities of permutation groups are investigated by means of new methods. For this purpose, we handle distinct groups on the set of natural numbers and we give the separation of the subgroups of them. Afterwards, we give the matrix representation of this groups.

This paper studies edge- and total-colorings of graphs in which (all or only adjacent) vertices are distinguished by their sets of colors. We provide bounds for the minimum number of colors needed for such colorings for the Cartesian product of graphs along with exact results for generalized hypercubes. We also present general bounds for the direct, strong and lexicographic products.

The pebbling number \(f(G)\) of a graph \(G\) is the smallest number \(k\) such that, however \(n\) pebbles are placed on the vertices of \(G\), we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Graham conjectured that for any connected graphs \(G\) and \(H\), \(f(G \times H) \leq f(G)f(H)\), where \(G \times H\) represents the Cartesian product of \(G\) and \(H\). In this paper, we prove that \(f(G \times H) \leq f(G)f(H)\) when \(G\) has the two-pebbling property and \(H = K_{r,s}^{ – k}\), a graph obtained from the \(r \times s\) complete bipartite graph \(K_{r,s}\) by deleting \(k\) edges which form a matching. We also show that Graham’s conjecture holds for \(K_{r,s}^{-k_1} \times K_{m,n}^{-k_2}\).

The Hosoya polynomial of a graph \(G\) is defined as \(H(G,x) = \sum\limits_{k\geq 0} d(G,k)x^k,\)
where \(d(G, k)\) is the number of vertex pairs at distance \(k\) in \(G\). The calculation of Hosoya polynomials of molecular graphs is a significant topic because some important molecular topological indices such as Wiener index, hyper-Wiener index, and Wiener vector, can be obtained from Hosoya polynomials. Hosoya polynomials of zig-zag open-ended nanotubes have been given by Xu and Zheng et al. A capped zig-zag nanotube \(T(p, q)[C, D; a]\) consists of a zig-zag open-ended nanotube \(T(p, q)\) and two caps \(C\) and \(D\) with the relative position \(a\) between \(C\) and \(D\). In this paper, we give a general formula for calculating the Hosoya polynomial of any capped zig-zag nanotube. By the formula, the Hosoya polynomial of any capped zig-zag nanotube can be deduced. Furthermore, it is also shown that any two non-isomorphic capped zig-zag nanotubes \(T(p, q)[C, D; a_1]\), \(T(p, q’)[C, D; a_2]\) with \(q’ \geq q^* \geq p+1\) have the same Hosoya polynomial, where \(q^*\) is an integer that depends on the structures of \(C\) and \(D\).