Graphs corresponding to reference polynomial or to circuit characteristic polynomial
Journal of Mathematical Chemistry, 12 (1):
265--277 (1993)DOI: 10.1007/BF01164640
Аннотация
A vertex-weighted graph G* is studied which is obtained by deleting edge e rs in a circuit of a graphG and giving two vertices $\nu$r and $\epsilon$s weights h r = 1 andh s = -1, respectively. It is shown that if subgraphG - $\nu$r is identical with subgraphG - $\nu$s, 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 Möbius graphs, provided that they satisfy a certain condition. The significances of this new type of ``reference'' graph thus obtained are pointed out.