A graph \( G \) of order \( |V(G)| \) and size \( |E(G)| \) is called edge-magic if there exists a bijection \( f : V(G) \cup E(G) \to \{1, 2, 3, \dots, |V(G)| + |E(G)|\} \) such that \( f(x) + f(xy) + f(y) \) is a constant for every edge \( xy \in E(G) \). An edge-magic graph \( G \) is said to be super if \( f(V(G)) = \{1, 2, 3, \dots, |V(G)|\} \). Furthermore, the edge-magic deficiency of a graph \( G \), denoted \( \mu(G) \), is defined as the minimum nonnegative integer \( n \) such that \( G \cup nK_1 \) is edge-magic. Similarly, the \emph{super edge-magic deficiency} of a graph \( G \), denoted \( \mu_s(G) \), is either the minimum nonnegative integer \( n \) such that \( G \cup nK_1 \) is super edge-magic or \( +\infty \) if there exists no such integer \( n \). In this paper, we investigate the (super) edge-magic deficiency of chain graphs. Based on these, we propose some open problems.

Given a large finite point set, \( P \subset \mathbb{R}^2 \), we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair of nonzero real numbers, \( (\alpha, \beta) \), we bound the size of the set \[ \{(p, q, r) \in P \times P \times P : p \cdot q = \alpha, p \cdot r = \beta\}. \]

An explicit formula for the number of spanning trees of the lexi- cographic product GLH] of two arbitrary graphs G and H is deduced in terms of structure parameters of G and H. Some properties on the number of spanning trees of G[H] are revealed. Sharp lower and upper bounds for the number of spanning trees of lexicographic product of graphs are established. In particular, simple formulae for the number of spanning trees of the lexicographic product of some special graphs are derived, which extend some previously known results
in the literature.

For a graph \( G \), the expression \( G \overset{v}{\rightarrow} (a_1,\ldots,a_s) \) means that for any \( s \)-coloring of the vertices of \( G \), there exists \( i \in \{1,\ldots,s\} \) such that there is a monochromatic \( a_i \)-clique of color \( i \). The vertex Folkman numbers

\[ F_v(a_1,\ldots,a_s;m-1) = \min\{|V(G)|: G \overset{v}{\rightarrow} (a_1,\ldots,a_s) \text{ and } K_{m-1} \nsubseteq G\} \]

are considered, where \( m = \sum_{i=1}^s (a_i – 1) + 1 \).

With the help of a computer, we show that \( F_v(2,2,5;6) = 16 \), and then we prove

\[ F_v(a_1,\ldots,a_s;m-1) = m+9, \]

if \( \max\{a_1,\ldots,a_s\} = 5 \).

We also obtain the bounds

\[ m+9 \leq F_v(a_1,\ldots,a_s;m-1) \leq m+10, \]

if \( \max\{a_1,\ldots,a_s\} = 6 \).

The diameter of a graph can be affected by the addition or the deletion of some edges. In [3], we have studied the diameter variability of the Cartesian product of graphs. In this paper, we discuss about two fundamental products, strong and lexicographic products of graphs, whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the product graphs with respect to its diameter are also solved. These problems are motivated by the fact that these graph products are good interconnection networks.

The concept of the skew energy of a digraph was introduced by Adiga, Balakrishnan and So in 2010. An oriented graph \( G^{\sigma} \) is a simple undirected graph \( G \) with an orientation, which assigns to each edge a direction so that \( G^{\sigma} \) becomes a directed graph. Then \( G \) is called the underlying graph of \( G^{\sigma} \). Let \( S(G^{\sigma}) \) be the skew-adjacency matrix of \( G^{\sigma} \) and \( \lambda_1, \lambda_2, \ldots, \lambda_n \) denote all the eigenvalues of \( S(G^{\sigma}) \). The skew energy of \( G^{\sigma} \) is defined as the sum of the absolute values of all eigenvalues of \( S(G^{\sigma}) \). Recently, Gong, Li and Xu determined all oriented graphs with minimal skew energy among all connected oriented graphs on \( n \) vertices with \( m \) (\( n \leq m \leq 2(n-2) \)) arcs. In this paper, we determine all oriented graphs with the second and the third minimal skew energy among all connected oriented graphs with \( n \) vertices and \( m \) (\( n \leq m < 2(n-2) \)) arcs. In particular, when the oriented graphs are unicyclic digraphs or bicyclic digraphs, the second and the third minimal skew energy is determined.

In this paper, according to the symmetric Lanczos algorithm and general Gauss-type quadrature rule, we give some lower bounds on the Resolvent Estrada index \( EE_r(G) \) and the Resolvent energy \( ER(G) \).

The chordal-(\(k,l\)) sandwich and strongly chordal-(\(k,l\)) sandwich problems were considered in recent work [8, 9] where classification of the complexities of the problems for all possible nonnegative integer values for \(k, l\) was considered. We extend the classification in [8, 9] by presenting polynomial time algorithms for some cases that remained open; currently, very few graph sandwich problems are known to be solvable in polynomial time.

An odd open dominating set of a graph is a subset of the graph’s vertices with the property that the open neighborhood of each vertex in the graph contains an odd number of vertices in the subset. An odd closed \( r \)-dominating set is a subset of the graph’s vertices with the property that the closed \( r \)-ball centered at each vertex in the graph contains an odd number of vertices in the subset.

We show that the \( n \)-fold direct product of simple graphs has an odd open dominating set if and only if each factor has an odd open dominating set. Secondly, we show that the \( n \)-fold strong product of simple graphs has an odd closed \( r \)-dominating set if and only if each factor has an odd closed \( r \)-dominating set.

Multireceiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify the authenticity of the received message. In this paper, we construct one multireceiver authentication code from pseudo-symplectic geometry over finite fields. The parameters and the probabilities of deceptions of the codes are also computed. The smaller the probability of successful attack, the higher the security of the authentication codes.