The total group of a graph

Abstract

We consider ordinary graphs (finite, undirected, with no loops or multiple lines). A well-known concept in the theory of graphs is that of the (point) group 1(G) of a graph G, which is the group of all adjacency-preserving permutations of points of G. The elements of are called (point-) automorphisms of G. In contrast with the notion of is that of the line group r'(G) of G consisting of all adjacencypreserving permutations of lines of G. This group has been considered in [6]. The purpose of this paper is to treat another natural concept, called total group, and to prove that for any graph G having more than one point, the total group of G is isomorphic to the group of G if and only if no component of G is either a cycle or a complete graph. 1. Preliminaries. Denote the point set of G by V(G) and its line set by X(G). Each member of V(G)UX(G) will be called an element of G. We say two elements of G are associated if they are either adjacent or incident. The group of all permutations of elements of G which preserve association will be called the total group of G and denoted by r(G). The line graph [6] of a graph G, denoted by L(G), is that graph whose point set is X(G), and in which two points are adjacent if and only if they are adjacent in G. It is worth observing that r'(G) -r(L(G)). The notion of total graphs introduced by one of the authors [1] is a convenient tool for our purposes. The total graph T(G) of G is that graph whose point set is V(G)UX(G), and in which two points are adjacent if and only if they are associated in G. We should note that r(G) is isomorphic to r(T(G)). The graphs G and L(G) are disjoint subgraphs of T(GJ. For illustration two graphs G