Connectivity of Total Graphs

Abstract

We associate with a graph (finite, undirected, without loops and multiple lines) a graph T(G), called the total graph of G. This new graph has the prop- erty that a one-to-one correspondence can be established between its points and the elements (points and lines) of G such that two points of T(G) are adjacent if and only if the corresponding elements of G are adjacent or incident. Connectivity is a basic concept of graph theory. It defines whether a graph is connected or disconnected. Without connectivity, it is not possible to traverse a graph from one vertex to another vertex. A graph is said to be connected graph if there is a path between every pair of vertex. From every vertex to any other vertex there must be some path to traverse. This is called the connectivity of a graph. In graph theory, an automorphism of a graph G is a permutation of the vertices that preserves the adjacency structure of the graph. In simpler terms, it’s a re-labeling of the vertices such that the overall shape or structure of the graph remains unchanged. Formally, let G=(V,E) be a graph with vertex set V and edge set E. An automorphism of G is a bijective function f:V → V such that for any two vertices u,v in V ,u is adjacent to v if and only if f(u) is adjacent to f(v).