Isospectral genus two graphs are isomorphic

Alexander D. Mednykh, Ilya A. Mednykh

Abstract


By a graph we mean a finite connected multigraph without bridges. The genus of a graph is the dimension of its homology group. Two graphs are isospectral is they share the same Laplacian spectrum. We prove that two genus two graphs are isospectral if and only if they are isomorphic. Also, we present two bridgeless genus three graphs that are not isomorphic. The paper is motivated by the following open problem posed by Peter Buser: are isospectral Riemann surfaces of genus two isometric? 


Keywords


Graph, Laplacian spectrum, isospectral graphs, Laplacian polynomial, spanning tree

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DOI: https://doi.org/10.26493/1855-3974.550.e1a

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications