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.

A signed graph (digraph) \( \Sigma \) is an ordered triple \( (V, E, \sigma) \) (respectively, \( (V, \mathcal{A}, \sigma) \)), where \( |\Sigma| := (V, E) \) (respectively, \( (V, \mathcal{A}) \)) is a graph (digraph), called the underlying graph (underlying digraph) of \( \Sigma \), and \( \sigma \) is a function that assigns to each edge (arc) of \( |\Sigma| \) a weight \( +1 \) or \( -1 \). Any edge (arc) \( e \) of \( \Sigma \) is said to be positive or negative according to whether \( \sigma(e) = +1 \) or \( \sigma(e) = -1 \). A subset \( D \subseteq V \) of vertices of \( \Sigma \) is an absorbent (respectively, a dominating set) of \( \Sigma \) if there exists a marking \( \mu: V \to \{+1, -1\} \) of \( \Sigma \) such that every vertex \( u \) of \( \Sigma \) is either in \( D \) or
\[
O(u) \cap D \neq \emptyset \quad \text{and} \quad \sigma(u, v) = \mu(u) \mu(v) \quad \forall \quad v \in O(u) \cap D,
\]
(respectively,
\[
I(u) \cap D \neq \emptyset \quad \text{and} \quad \sigma(u, v) = \mu(u) \mu(v) \quad \forall \quad v \in I(u) \cap D),
\]
where \( O(u) \) (\( I(u) \)) denotes the set of vertices \( v \) of \( \Sigma \) that are joined by the outgoing arcs \( (u, v) \) from \( u \) (incoming arcs \( (v, u) \) at \( u \)). Further, an absorbent (dominating set) of \( \Sigma \) that is independent is called a kernel (solution) of \( \Gamma \). The main aim of this paper is to initiate a study of absorbents and dominating sets in a signed graph (signed digraph), extending the existing studies on these special sets of vertices in a graph (digraph).