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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 $10$.'
address:
- 'National council of science and technology (CONACYT) $\&$ Autonomous University of Zacatecas, Paseo la Bufa, int. Calzada Solidaridad, 98060 Zacatecas, ZAC, Mexico'
- 'Department of Mathematics, Universidad Carlos III de Madrid, Av. de la Universidad 30, 28911 Leganés, Madrid, Spain'
- 'Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, Mexico'
- 'Humboldt International University, 4000 West Flagler Street, 33134, Miami, Fl., 