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Keywords
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Eigenvalue, generalized Laplacian Matrix, Transmission, Transmission Lapla-
cian matrix, cospectral
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Abstract
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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 dis-
tances between vi and any other vertices 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 LT r(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
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