Computing the alliance polynomial of a graph
Abstract
The alliance polynomial of a graph $\Gamma$ with order $n$ and maximum degree $\delta_1$ is the polynomial $A(\Gamma; x) = \sum_{k=\delta_1}^{\delta_1} A_{k}(\Gamma) \, x^{n+k}$, where $A_{k}(\Gamma)$ is the number of exact defensive $k$alliances in $\Gamma$. We provide an algorithm for computing the alliance polynomial. Furthermore, we obtain some properties of $A(\Gamma; x)$ and its coefficients. In particular, we prove that the path, cycle, complete and star graphs are characterized by their alliance polynomials. We also show that the alliance polynomial characterizes many graphs that are not distinguished by other usual polynomials of graphs.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.2940
 Bibcode:
 2014arXiv1410.2940C
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 Accepted for publication in Ars Combinatoria