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.

We show that the butterfly network and Benes network can be embedded into generalized fat trees with minimum dilation.

The crossing number of a graph \( G \) is the minimum number of crossings of its edges among the drawings of \( G \) in the plane and is denoted by \( \operatorname{cr}(G) \). In this paper, we obtain bounds for the crossing number for two different honeycomb tori, namely, the honeycomb rectangular torus and the honeycomb rhombic torus, which are obtained by adding wraparound edges to honeycomb meshes.

In cellular radio communication systems, the concept of maximum packing is used for dynamic channel assignment. An \( H \)-packing of a graph \( G \) is a set of vertex-disjoint subgraphs of \( G \), each of which is isomorphic to a fixed graph \( H \). The maximum \( H \)-packing problem is to find the maximum number of vertex-disjoint copies of \( H \) in \( G \), called the packing number, denoted by \( \lambda(G, H) \). In this paper, we determine the maximum \( H \)-packing number of hexagonal networks when \( H \) is isomorphic to \( P_6 \) as well as \( K_{1,3} \).