4 research outputs found
Polynomial graph invariants from homomorphism numbers
We give a new method of generating strongly polynomial sequences of graphs, i.e., sequences
(Hk) indexed by a tuple k = (k1, . . . , kh) of positive integers, with the property
that, for each fixed graph G, there is a multivariate polynomial p(G; x1, . . . , xh) such that
the number of homomorphisms from G to Hk is given by the evaluation p(G; k1, . . . , kh).
A classical example is the sequence of complete graphs (Kk), for which p(G; x) is the chromatic
polynomial of G. Our construction is based on tree model representations of graphs. It
produces a large family of graph polynomials which includes the Tutte polynomial,
the Averbouch–Godlin–Makowsky polynomial, and the Tittmann–Averbouch–Makowsky
polynomial. We also introduce a new graph parameter, the branching core size of a simple
graph, derived from its representation under a particular tree model, and related to
how many involutive automorphisms it has. We prove that a countable family of graphs of
bounded branching core size is always contained in the union of a finite number of strongly
polynomial sequences.Ministerio de EconomĂa y Competitividad MTM2014-60127-