Laplacian state transfer in total graphs

Abstract

The total graph of a graph G, denoted by T(G), is defined to be the graph whose vertices are the vertices and edges of G, with two vertices of T(G) adjacent if and only if the corresponding elements of G are adjacent or incident. In this paper, we investigate the existence of Laplacian perfect state transfer and Laplacian pretty good state transfer in the total graph of an r-regular graph, where r≥2. We prove that if r+1 is not a Laplacian eigenvalue of an r-regular graph G, then there is no Laplacian perfect state transfer in T(G). In contrast, we give a sufficient condition for total graphs of regular graphs to have Laplacian pretty good state transfer.