On the diameter and incidence energy of iterated total graphs

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 $k\geq2$, 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 $diam(\mathcal{T}^{r+1}(G))>k-r,$ $r\geq0$. 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

Sequence mixed graphs

A mixed graph can be seen as a type of digraph containing some edges (or two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and literated line digraphs. These structures are proven to be useful in the problem of constructing dense graphs or digraphs, and this is related to the degree/diameter problem. Thus, our generalized approach gives rise to graphs that have also good ratio order/diameter. Moreover, we propose a general method for obtaining a sequence mixed diagraph by identifying some vertices of certain iterated line digraph. As a consequence, some results about distance-related parameters (mainly, the diameter and the average distance) of sequence mixed graphs are presented.Postprint (author's final draft