In the framework of P systems introduced by Paun (1998), the generation of rectangular arrays and hexagonal arrays has been studied in the literature. In this paper, we introduce a new P system generating a family of hexagonal array languages. We compare this new family with the existing families of hexagonal array languages.

Hypertournaments are generalizations of tournaments. We discuss the concept of scores, losing scores, total scores, and degrees in \(k\)-hypertournaments and present characterizations of sequences to be score, losing score, total score, and degree sequences of some \(k\)-hypertournaments. We further discuss stronger upper and lower bounds for scores and losing scores. We extend the concept of scores, losing scores, and degrees to bipartite hypertournaments. In the end, we list some open problems in hypertournaments.

We introduce \( k \)-ctrees, which are a natural generalization of trees. A \( k \)-ctree can be constructed by recursion as follows: Any set of \( k \) independent vertices is a \( k \)-ctree, and a \( k \)-ctree of order \( n + 1 \) is obtained by inserting an \( (n + 1) \)-th vertex, and joining it to each of any \( k \) independent vertices in a \( k \)-ctree of order \( n \). We obtain basic properties and characterizations of \( k \)-ctrees involving \( k \)-degeneracy, triangle-free properties, and number of edges. Further, we determine the conditions under which \( k \)-ctrees are line, middle, or total graphs. Finally, we pose some open problems, all of them related to the characterization of \( k \)-ctrees.

An \( (a, d) \)-edge-antimagic total labeling of a graph \( G \) with \( p \) vertices and \( q \) edges is a bijection \( f \) from the set of all vertices and edges to the set of positive integers \( \{1, 2, 3, \dots, p+q\} \) such that all the edge-weights \( w(uv) = f(u) + f(v) + f(uv) \) for \( uv \in E(G) \), form an arithmetic progression starting from \( a \) and having common difference \( d \). An \( (a, d) \)-edge-antimagic total labeling is called a super \( (a, d) \)-edge-antimagic total labeling (\((a,d)\)-SEAMT labeling) if \( f(V(G)) = \{1, 2, 3, \dots, p\} \). The graph \( F_n \), consisting of \( n \) triangles with a common vertex, is called the friendship graph. The generalized friendship graph \( F_{m_1, m_2, \dots, m_n} \) consists of \( n \) cycles of orders \( m_1 \leq m_2 \leq \dots \leq m_n \) having a common vertex. In this paper, we prove that the friendship graph \( F_{16} \) does not admit a \( (a, 2) \)-SEAMT labeling. We also investigate the existence of \( (a, d) \)-SEAMT labeling for several classes of generalized friendship graphs.

Let \( G = (V, E) \) be a connected graph with domination number \( \gamma \geq 2 \). In this paper, we discuss the construction of a visual cryptography scheme for the mindom access structure \( \Gamma_D(G) \) with a basis consisting of all \( \gamma \)-sets of \( G \). We prove that the access structure \( \Gamma_D(G) \) is a \( (2, n) \)-threshold access structure if and only if \( n \) is even and \( G = K_n – M \), where \( M \) is a perfect matching in \( K_n \). Further, the \( (k, n) \)-VCS with \( k < n \) can be realized as a \( \Gamma_D(G) \)-VCS if and only if \( k = 2 \) and \( n \) is even. We also construct \( \Gamma_D(G) \)-VCS for several classes of graphs such as complete bipartite graphs, cycles \( C_n \), and \( K_n – C_n \), and we have achieved substantial reduction in the pixel expansion when compared to the VCS constructed by using other known methods.

Let \( G = (V, E) \) be a graph of order \( n \). Let \( f: V \to \{1, 2, \dots, n\} \) be a bijection. For any vertex \( v \in V \), the neighbor sum \( \sum_{u \in N(v)} f(u) \) is called the weight of the vertex \( v \) and is denoted by \( w(v) \). If \( w(x) \neq w(y) \) for any two distinct vertices \( x \) and \( y \), then \( f \) is called a distance antimagic labeling. In this paper, we present several results on distance antimagic graphs along with open problems and conjectures.

In this paper, we focus our study on finding necessary and sufficient conditions required for the existence of an \( \hat{S}_k \)-factorization of \( (K_m \circ \overline{K}_n)^* \) and \( (C_m \circ \overline{K}_n)^* \). In particular, we show that the necessary conditions for the existence of an \( \hat{S}_k \)-factorization of \( (K_m \circ \overline{K}_n)^* \) are sufficient except when none of \( m \) or \( n \) is a multiple of \( k \). In fact, our results deduce some of the results of Ushio on \( \hat{S}_k \)-factorizations of complete bipartite and tripartite symmetric digraphs.

A bipartite \( r \)-digraph is an orientation of a bipartite multigraph that is without loops and contains at most \( r \) edges between any pair of vertices from distinct parts. In this paper, we obtain necessary and sufficient conditions for a pair of sequences of non-negative integers in non-decreasing order to be a pair of sequences of numbers, called marks (or \( r \)-scores), attached to the vertices of a bipartite \( r \)-digraph. These characterizations provide algorithms for constructing the corresponding bipartite multi-digraph.

Let \( \Gamma \) be a Cayley graph generated by a transposition tree \( T \) on \( n \) vertices. In an oft-cited paper [1] (see also (9)), it was shown that the diameter of the Cayley graph \( T \) on \( n \) vertices is bounded as

\[
\text{diam}(\Gamma) \leq \max_{\pi \in S_n} \left\{ c(\pi) -n+\sum_{i=1}^{n} dist_T(i,\pi(i)) \right\},
\]

where the maximization is over all permutations \( \pi \) in \( S_n \), \( e(\pi) \) denotes the number of cycles in \( \pi \), and \( \text{distr} \) is the distance function in \( T \). It is of interest to determine for which families of trees this inequality holds with equality. In this work, we first investigate the sharpness of this upper bound. We prove that the above inequality is sharp for all trees of maximum diameter (i.e., all paths) and for all trees of minimum diameter (i.e., all stars), but the bound can still be strict for trees that are non-extremal. We also show that a previously known inequality on the distance between vertices in some families of Cayley graphs holds with equality and we prove that for some families of graphs an algorithm related to these bounds is optimal.

Let \( G = (V, E) \) be a connected graph. Two vertices \( u \) and \( v \) are said to be distance similar if \( d(u, x) = d(v, x) \) for all \( x \in V – \{u, v\} \). A nonempty subset \( S \) of \( V \) is called a pairwise distance similar set (in short `pds-set’) if either \( |S| = 1 \) or any two vertices in \( S \) are distance similar. The maximum (minimum) cardinality of a maximal pairwise distance similar set in \( G \) is called the pairwise distance similar number (lower pairwise distance similar number) of \( G \) and is denoted by \( \Phi(G) \) (\( \Phi^-(G) \)). The maximal pds-set with maximum cardinality is called a \( \Phi \)-set of \( G \). In this paper, we initiate a study of these parameters.