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.

Betweenness is a centrality measure based on shortest paths, widely used in complex network analysis. The betweenness centrality of a vertex is defined as the fraction of shortest paths that pass through that vertex over all pairs of vertices. It measures the control a vertex has over communication in the network, and can be used to identify key vertices in the network. High centrality indices indicate that a vertex can reach other vertices on relatively short paths, or that a vertex lies on a considerable fraction of shortest paths connecting pairs of other vertices. In this paper, we find the betweenness centrality of the honeycomb mesh, which has important applications in mobile networks.

Tree replacement / rewriting systems are an interesting model of computation. They are used in theorem proving, algebraic simplification, and language theory. A fundamental property of tree replacement systems is the Church-Rosser property, which expresses the fact that interconvertability of two trees can be checked by mere simplification to a common tree. In this paper, we give a learning algorithm for a subclass of the class of Church-Rosser tree replacement systems.