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Polynomial graph invariants from homomorphism numbers

4 years 7 months ago
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-GodlinMakowsky 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 ...
Delia Garijo, Andrew J. Goodall, Jaroslav Nesetril
Added 01 Apr 2016
Updated 01 Apr 2016
Type Journal
Year 2016
Where DM
Authors Delia Garijo, Andrew J. Goodall, Jaroslav Nesetril
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