Reza Sharafdini

Let G be a graph with the vertex set V(G) = {v1, . . . , vn}. A symmetric matrix L of order n is called a generalized Laplacian of G if Lvivj < 0 when vi and vi are adjacent vertices of G and Lvivj = 0 when vi and vj are distinct and not adjacent. The transmission of the vertex vi ∈ V(G), denoted by σG(vi), is defined to be the sum of distances between vi and any other ver- tices in G , i.e., σG(vi) = ∑n j=1 dG(vi, vj). Let A(G) be the adjacency matrix of a connected graph G. The transmission Laplacian matrix of G, which is defined as LTr(G) = diag(σG(v1), · · · , σG(vn)) − A(G), is a generalized Laplacian of G. In this paper, we first recall some basic properties of this generalized Laplacian matrix. Then using Sagemath we list all connected graphs of order at most 8 being cospectral with respect to the transmission Laplacian matrix.