We answer in the affirmative a question posed by Al-Addasi and Al-Ezeh in 2008 on the existence of symmetric diametrical bipartite graphs of diameter 4. Bipartite symmetric diametrical graphs are called \( S \)-graphs by some authors, and diametrical graphs have also been studied by other authors using different terminology, such as self-centered unique eccentric point graphs. We include a brief survey of some of this literature and note that the existence question was also answered by Berman and Kotzig in a 1980 paper, along with a study of different isomorphism classes of these graphs using a \( (1,-1) \)-matrix representation which includes the well-known Hadamard matrices. Our presentation focuses on a neighborhood characterization of \( S \)-graphs, and we conclude our survey with a beautiful version of this characterization known to Janakiraman.

The achromatic number for a graph \( G = (V, E) \) is the largest integer \( m \) such that there is a partition of \( V \) into disjoint independent sets \( (V_1, \ldots, V_m) \) such that for each pair of distinct sets \( V_i, V_j \), \( V_i \cup V_j \) is not an independent set in \( G \). In this paper, we present an \( O(1) \)-approximation algorithm to determine the achromatic number of circulant graphs \( G(n; \pm\{1, 2\}) \) and \( G(n; \pm\{1, 2, 3\}) \).

Let \( G = (V, E) \) be a graph with vertex set \( V \) and edge set \( E \). Let \( diam(G) \) denote the diameter of \( G \) and \( d(u, v) \) denote the distance between the vertices \( u \) and \( v \) in \( G \). An antipodal labeling of \( G \) with diameter \( d \) is a function \( f \) that assigns to each vertex \( u \) a positive integer \( f(u) \), such that \( d(u, v) + |f(u) – f(v)| \geq d \), for all \( u, v \in V \). The span of an antipodal labeling \( f \) is \( \max\{|f(u) – f(v)| : u, v \in V(G)\} \). The antipodal number for \( G \), denoted by \( an(G) \), is the minimum span of all antipodal labelings of \( G \). Determining the antipodal number of a graph \( G \) is an NP-complete problem. In this paper, we determine the antipodal number of certain graphs with diameter equal to \( 3 \) and \( 4 \).

A radio labeling of a connected graph \( G \) is an injection \( f \) from the vertices of \( G \) to the natural numbers such that \( d(u, v) + |f(u) – f(v)| \geq 1 + \operatorname{diam}(G) \) for every pair of distinct vertices \( u \) and \( v \) of \( G \). The radio number of \( f \), denoted \( rn(f) \), is the maximum number assigned to any vertex of \( G \). The radio number of \( G \), denoted \( rn(G) \), is the minimum value of \( rn(f) \) taken over all labelings \( f \) of \( G \). In this paper, we determine bounds for the radio number of the hexagonal mesh.

In this paper, we introduce a finite graph using group characters and discuss the basic properties of the graph.

In this paper, a fuzzy inner product on a real vector space is introduced. The notion of fuzzy inner product is defined. Some of its properties are studied.

Let \( G = (V, E) \) be a simple graph. Let \( S \) be a subset of \( V(G) \). The toughness value of \( S \), denoted by \( T_S \), is defined as \( \frac{|S|}{\omega(G – S)} \), where \( \omega(G – S) \) denotes the number of components in \( G – S \). If \( S = V \), then \( \omega(G – S) \) is taken to be \( 1 \) and hence \( T_{V(G)} = |V(G)| \). A partition of \( V(G) \) into subsets \( V_1, V_2, \ldots, V_t \) such that \( T_{V_i} \), \( 1 \leq i \leq t \), is a constant is called an equi-toughness partitio of \( G \). The maximum cardinality of such a partition is called the equi-toughness partition number of \( G \) and is denoted by \( ET(G) \). The existence of \( ET \)-partition is guaranteed. In this paper, a study of this new parameter is initiated.

The parameter \( t \) of a tree \( t \)-spanner of a graph is always bounded by \( 2\lambda \) where \( \lambda \) is the diameter of the graph. In this paper, we establish a sufficient condition for graphs to have the minimum spanner at least \( 2\rho – 1 \) where \( \rho \) is the radius. We also obtain a characterization for tree \( 3 \)-spanner admissible chordal graphs in terms of tree \( 3 \)-spanner admissibility of certain subgraphs.

A connected graph \( G(V, E) \) is said to be \((a, d)\)-antimagic if there exist positive integers \( a \) and \( d \) and a bijection \( f: E \to \{1, 2, \ldots, |E|\} \) such that the induced mapping \( \text{g}_\text{f}: V \to \mathbb{N} \) defined by \( \text{g}_\text{f}(v) = \sum_{\text{e} \in \text{I}(v)} \text{f(e)} \), where \( \text{I}(v) = \{\text{e} \in E \mid \text{e} \text{ is incident to } v\} \), \( v \in V \) is injective and \( \text{g}_\text{f}(V) = \{a, a+d, a+2d, \ldots, a+(|V|-1)d\} \). In this paper, using partition, we prove that (i) the 1-sided infinite path \( P_1 \) is \((1, 2)\)-antimagic, (ii) the path \( P_{2n+1} \) is \((n, 1)\)-antimagic, and (iii) the \((n+2, 1)\)-antimagic labeling is the unique \((a, d)\)-antimagic labeling of \( C_{2n+1} \); and the graphs \( K_1 + (K_1 \cup K_2) \), \( P_{2n} \), and \( C_{2n} \) are not \((a, d)\)-antimagic. For \( a, d \in \mathbb{N} \), on an \((a, d)\)-antimagic graph \( G \), we obtain a new relation, \( a + (p-1)d \leq \frac{\Delta(2q – \Delta + 1)}{2} \). Using the results on \((a, d)\)-antimagic labeling of \( C_{2n} \) and \( C_{2n+1} \), we obtain results on the existence of \((a, d)\)-arithmetic sequences of length \( 2n \) and \( 2n+1 \), respectively.