graphs with small degree {#sect:con_reg}
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The theorems in this section can be seen as a natural continuation of the study in [@CRST] in the sense of showing the distinctive power of the alliance polynomial of a graph. In particular, we show that the family of alliance polynomials of ${\Delta}$-regular graphs with small degree ${\Delta}$ is a special family of alliance polynomials since there not exists a non ${\Delta}$-regular graph with alliance polynomial equal to one of their members, see Theorems \[t:Reg0-3\] and \[t:Reg4-6\].

\[t:Reg0-3\] Let $G$ be a ${\Delta}$-regular graph with $0\le{\Delta}\le3$ and $G^*$ another graph. If $A(G^*;x) = A(G;x)$, then $G^*$ is a ${\Delta}$-regular graph with the same order, size and number of connected components of $G$.

In [@CRST Theorem 2.7] the authors obtain the uniqueness of the alliance polynomials of $0$-regular graphs (the empty graphs).

Theorems \[p:AlliPoly\] iii) and \[t:PolyReg\] give that $1$-regular graphs are the unique graphs which have exactly two non-zero terms in their alliance polynomial; besides, Theorems \[p:AlliPoly\] vi) and \[t:PropRegular\] ii) give the uniqueness of these alliance polynomials.

In order to obtain the result for $2\le{\Delta}\le3$, denote by $n,n_1$, the orders of $G,G^*$, respectively, and let ${\delta}_1,{\Delta}_1$ be the minimum and maximum degree 