TOPOLOGICAL TWIN GRAPHS - SMALLEST PAIR OF ISOSPECTRAL POLYHEDRAL GRAPHS WITH 8 VERTICES

A novel pair of isospectral nonahedral graphs with eight vertices were found, which not only have the same characteristic polynomial, P(G)(x), but also have the same distance polynomial, S(G)(x), and the same polygonal (f3 = 6, f4 = 3) and vertex degree (v3 = 4, v4 = 2, v5 = 2) distributions, contrary to the observation that almost all the reported pairs of isospectral graphs do not have common topological factors other than P(G)(x). Many of their topological indices, such as Z and W, were found to have common values. The only structural difference between this pair of ''topological twin graphs'' is in the distribution of edge types, namely, the degrees of the two vertices defining an edge. The newly found pair of graphs are the smallest topological twin graphs and also the smallest isospectral polyhedral graphs. The spectra of P(G)(x) and S(G)(x) of them are also discussed.