of graphs with many applications (see, e.g., [@B; @Cu; @DM; @RST]). In this paper we study the alliance polynomials of regular graphs and their coefficients, see Section \[sect:reg\]. In Section \[sect:con\_reg\] we focus on the alliance polynomials of connected regular graphs; besides, we prove that the family of alliance polynomials of connected ${\Delta}$-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not connected ${\Delta}$-regular (see Theorems \[t:Reg0-3\] and \[t:Reg4-6\]). Finally, by using Theorem \[t:Reg0-3\] and direct computation we find that the alliance polynomial determines uniquely each cubic graph of order less than or equal to $10$.

Alliance polynomials for regular graphs {#sect:reg}
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In this section we deal with regular graphs, in particular, we obtain some properties of the alliance polynomial of regular graph and its coefficients. Below, a quick reminder of some previous results for general graphs (not necessarily regular) which will be useful, see [@CRST]. We denote by $\operatorname{Deg}(p)$ the degree of the polynomial $p$.

\[p:AlliPoly\] Let $G$ be any graph. Then $A(G;x)$ satisfies the following properties:

i)  [$A(G;x)$ does not have zeros in the interval $(0,\infty)$.]{}

ii) [$A(G;1) < 2^{n}$, and it is the number 