Alliance polynomial of regular graphs

TL;DR

This paper studies the alliance polynomial of regular graphs, revealing how it characterizes regularity and uniquely identifies small cubic graphs, thus advancing understanding of graph alliances.

Contribution

It introduces properties of the alliance polynomial for regular graphs and shows it uniquely determines small cubic graphs, highlighting its potential as a graph invariant.

Findings

Alliance polynomial characterizes regular graphs by the number of non-zero coefficients.

Family of alliance polynomials for small degree regular graphs is highly restricted.

Alliance polynomial uniquely identifies each small cubic graph up to order 10.

Abstract

The alliance polynomial of a graph $G$ with order $n$ and maximum degree $\Delta$ is the polynomial $A(G; x) = \sum_{k=-\Delta}^{\Delta} A_{k}(G) \, x^{n+k}$, where $A_{k}(G)$ is the number of exact defensive $k$-alliances in $G$. We obtain some properties of $A(G; x)$ and its coefficients for regular graphs. In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of $\Delta$-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not $\Delta$-regular. By using this last result and direct computation we find that the alliance polynomial determines uniquely each cubic graph of order less than or equal to $10$.