Homomorphism polynomials of chemical graphs

Abstract

We say that a graphG ishomomorphic to a graphH if there is a mappingp from the vertices of G onto the vertices ofH such thatp(u) andp(ν) are adjacent inH wheneveru andν are adjacent in G. Thehomomorphism polynomial of a graphG is a polynomial in two variables that counts the number of homomorphisms ofG onto the complete graph of each order. This polynomial can be computed recursively in an analog to the chromatic polynomial. In this paper, we present some results regarding the homomorphism polynomials of the graphs of chemical compounds — in particular, alkane isomers. The coefficients of the homomorphism polynomial can be used to predict the rankings of compounds with respect to several chemical properties. Our results seem to refine those obtained by Randić et al. from path lengths.