$v\in V(G)$, we denote by $G \setminus \{v\}$ the subgraph obtained by removing from $G$ the vertex $v$ and the edges incident to $v$. We say that $v$ is a *cut vertex* if $G \setminus \{v\}$ has more connected components than $G$. Besides, if $p$ is a polynomial we denote by $\operatorname{Deg}_{min}(p)$ the minimum degree of their non-zero coefficients.

\[t:PropRegular\] For any ${\Delta}$-regular graph $G$, its alliance polynomial $A(G;x)$ satisfies the following properties:

i)  [$A_{-{\Delta}+2i}(G)$ is the number of connected induced subgraphs of $G$ with minimum degree $i$ $(0\le i\le{\Delta})$.]{}

ii) [$\operatorname{Deg}_{\min}\big(A(G;x)\big)= n - {\Delta}$ and $A_{-{\Delta}}(G) = n$.]{}

iii) $\operatorname{Deg}\big(A(G;x)\big) = n + {\Delta}$. Furthermore,

    $$\label{eq:RegOrderV}
             n = \frac{\operatorname{Deg}_{\min}\big(A(G;x)\big) + \operatorname{Deg}\big(A(G;x)\big)}2$$

    and $$m = A_{-{\Delta}}(G) \frac{\operatorname{Deg}\big(A(G;x)\big) - \operatorname{Deg}_{\min}\big(A(G;x)\big)}4 = \frac{\operatorname{Deg}^2\big(A(G;x)\big) - \operatorname{Deg}_{\min}^2\big(A(G;x)\big)}8.$$

iv) [$1\le A_{{\Delta}}(G) \le n/({\Delta}+1)$. Furthermore, $G$ is a connected graph if and only if $A_{{\Delta}}(G) = 1$.]{}

v)  [If ${\Delta}>0$, then $A_{-{\Delta}+2}(G) \ge m$ and $A_{{\Delta}-2}(G) \ge n+n_0$ with $n_0$ the number of cut vertices; in particular, $A_{{\Delta}-2}(G)\ge n$.]{}

vi) [$A(G;x)$ is either an even or an odd function of $x$. Furthermore, $A(G;x)$ is an even function of $x$ if and only if $n + {\Delta}$ is even.]{}

vii) [The unique real zero of $A(G;x)$ is 