Graphs corresponding to reference polynomial or to circuit characteristic polynomial

Abstract

A vertex-weighted graphG* is studied which is obtained by deleting edgee rs in a circuit of a graphG and giving two vertices νr and es weightsh r = 1 andh s = -1, respectively. It is shown that if subgraphG - νr is identical with subgraphG - νs, then the reference polynomial ofG* is identical with that ofG and the characteristic polynomial ofG* contains the contributions due to only a certain part of the circuits found in the original graphG. This result gives a simple way to find a graph whose characteristic polynomial is equal to the reference polynomial in the topological resonance energy theory or to the circuit characteristic polynomial in the circuit resonance energy theory. This approach can be applied not only to Hilckel graphs but also to Mobius graphs, provided that they satisfy a certain condition. The significances of this new type of “reference” graph thus obtained are pointed out.