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One of the most recent developments in the field of graph theory is the analysis of networks such as Butterfly networks, Benes networks, Interconnection networks, and David-derived networks using graph theoretic parameters. The topological indices (TIs) have been widely used as graph invariants among various graph theoretic tools. Quantitative structure activity relationships (QSAR) and quantitative structure property relationships (QSPR) need the use of TIs. Different structure-based parameters, such as the degree and distance of vertices in graphs, contribute to the determination of the values of TIs. Among other recently introduced novelties, the classes of ev-degree and ve-degree dependent TIs have been extensively explored for various graph families. The current research focuses on the development of formulae for different ev-degree and ve-degree dependent TIs for s-dimensional Benes network and certain networks derived from it. In the end, a comparison between the values of the TIs for these networks has been presented through graphical tools. [ABSTRACT FROM AUTHOR]

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, υ ∈ V ( G) there is a vertex w ∈ W such that d( u,w) ≠ d( υ,w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim( G). For a vertex u of G and a subset S of V ( G), the distance between u and S is the number min d( u, s). A k-partition Π = { S, S, ..., S} of V ( G) is called a resolving partition if for every two distinct vertices u, v ∈ V ( G) there is a set S in Π such that d( u, S) ≠ d( v, S). The minimum k for which there is a resolving k-partition of V ( G) is called the partition dimension of G, denoted by pd( G). The circulant graph is a graph with vertex set ℤ, an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i − j (mod n) ∈ C, where C ∈ ℤ has the property that C = −C and 0 ∉ C. The circulant graph is denoted by X where Δ = | C|. In this paper, we study the metric dimension of a family of circulant graphs X with connection set $$C = \left\{ {1,\tfrac{n} {2},n - 1} \right\}$$ and prove that dim( X) is independent of choice of n by showing that We also study the partition dimension of a family of circulant graphs X with connection set C = {±1,±2} and prove that pd( X) is independent of choice of n and show that pd( X) = 5 and . [ABSTRACT FROM AUTHOR]

Let G 1 and G 2 be disjoint copies of a graph G and g : V (G 1) → V (G 2) be a function. A functigraph F G consists of the vertex set V (G 1) ∪ V (G 2) and the edge set E (G 1) ∪ E (G 2) ∪ { u v : g (u) = v } . In this paper, we extend the study of distinguishing numbers of a graph to its functigraph. We discuss the behavior of distinguishing number in passing from G to F G and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs. [ABSTRACT FROM AUTHOR]

Abstract: Let be a connected graph and be the distance between the vertices and in . A subset of vertices is called a resolving set or locating set for if for every two distinct vertices , there is a vertex such that for . A resolving set containing the minimum number of vertices is called a metric basis for and the number of vertices in a metric basis is its metric dimension, denoted by . Let be a family of connected graphs depending on as follows: the order and . If there exists a constant such that for every then we shall say that has bounded metric dimension. The metric dimension of a class of circulant graphs has been determined by Javaid and Rahim (2008) . In this paper, we extend this study to an infinite class of circulant graphs . We prove that the circulant graphs have metric dimension equal to 4 for . For only 5 vertices appropriately chosen suffice to resolve all the vertices of , thus implying that except when . [Copyright &y& Elsevier]