USA'
author:
- Walter Carballosa
- 'José M. Rodr[í]{}guez'
- 'José M. Sigarreta'
- 'Yadira Torres-Nuñez'
title: Alliance polynomial of regular graphs
---

Regular Graphs ,Cubic Graphs ,Defensive Alliances ,Alliance Polynomials

05C69 ,11B83

Introduction.
=============

Graph polynomials have been widely study since George D. Birkhoff introduced the chromatic polynomial (1912) in an attempt to prove the four color theorem (see [@Bi]). Although the original motivation for the study of this invariant (chromatic number) is still important, much of the current interest, for example, in the Tutte polynomial is not related to any of its applications. In particular, graph polynomials are considered interesting when they encode much information about the underlying graph. Some parameters of a graph $G$ allow one to define polynomials on the graph $G$, for instance, the parameters associated to matching sets [@F; @GG], independent sets [@BDN; @GH; @HL], domination sets [@AAP; @AP], chromatic numbers [@Bi; @R; @T] and many others. In recent years there has been an increase in the number of papers introducing new properties of graph polynomials [@KM; @MR; @MRB].

The study of defensive alliances in graphs, together with a variety of other kinds of alliances, was introduced in [@KHH]. In the cited paper authors 