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...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2016 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/98072 |
| Acceso en línea: | https://hdl.handle.net/11441/98072 https://doi.org/10.1016/j.disc.2015.11.022 |
| Access Level: | acceso abierto |
| Palabra clave: | Graph polynomial Graph homomorphism Graph sequence |
| Sumario: | 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. |
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