On the diameter and incidence energy of iterated total graphs

Abstract

The total graph of $G$, $\mathcal{T}(G)$ is the graph whose set of vertices 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 $G$. Let ${\mathcal{T}}^1(G)=\mathcal{T}(G)$, the total graph of $G$. For $k\ge 2$, the $k\text{-}th$ iterated total graph of $G$, ${\mathcal{T}}^k(G)$, is defined recursively as ${\mathcal{T}}^k(G)=\mathcal{T}({\mathcal{T}}^{k-1}(G)).$ 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 $diam({\mathcal{T}}^{r+1}(G))>k-r,$ $r\ge 0$. In addition, bounds for the incidence energy of the iterated graph ${\mathcal{T}}^{r+1}(G)$ are obtained, provided $G$ to be a regular graph. Finally, new families of non-isomorphic cospectral graphs are exhibited.