On the diameter and incidence energy of iterated total graphs

Abstract

The total graph of G, T (G) is the graph whose vertex set is the union of the sets of vertices and edges of G, where two vertices are adjacent if and only if they stand for either incident or adjacent elements in G. For k ≥ 2, the k-th iterated total graph of G, T k(G), is defined recursively as T k(G) = T (T k-1(G)), where T 1(G) = T (G) and T 0(G) = G. If G is a connected graph, its diameter is the maximum distance between any pair of vertices in G. The incidence energy IE(G) of G is the sum of the singular values of the incidence matrix of G. In this paper, for a given integer k we establish a necessary and sufficient condition under which diam(T r+1(G)) > k-r, r ≥ 0. In addition, bounds for the incidence energy of the iterated graph T r+1(G) are obtained, provided G is a regular graph. Finally, new families of non-isomorphic cospectral graphs are exhibited.