Once a graph is associated with a matrix according to some rule, we can find the spectrum of a graph with respect to a matrix. Two graphs are cospectral if they share a spectrum. The spectrum holds some structural information about the graph. Constructions of cospectral graphs help us establish patterns about graph properties not preserved by the spectrum. There a natural construction for the combinatorial Laplacian matrix using twin vertices. We relax the twin structure to a set of vertices called cousins and show an extension of the construction method produces cospectral graphs. This construction is based on a cospectral construction for the distance Laplacian. In addition, we show that the construction method extends to the adjacency matrix, signless Laplacian matrix, normalized Laplacian matrix, distance matrix, and distance Laplacian matrix.
Cospectral constructions using cousin vertices
Kate J Lorenzen, Linfield University
2022 AWM Research Symposium
Women from the Graduate Research Workshop in Combinatorics