of connected induced subgraphs in $G$.]{}

iii) [If $G$ has at least an edge and its degree sequence has exactly $r$ different values, then $A(G;x)$ has at least $r + 1$ terms.]{}

iv) [ $A(G;x)$ is a symmetric polynomial (i.e., an even or an odd function of $x$) if and only if the degree sequence of $G$ has either all values odd or all even.]{}

v)  [$A_{-{\Delta}}(G)$ and $A_{-{\Delta}+ 1}(G)$ are the number of vertices in $G$ with degree ${\Delta}$ and ${\Delta}-1$, respectively.]{}

vi) [$A_{{\Delta}}(G)$ is equal to the number of connected components in $G$ which are ${\Delta}$-regular.]{}

vii) [$n + {\delta}\leq \operatorname{Deg}(A(G;x)) \leq n + {\Delta}$.]{}

As usual, by *cycle* we mean a simple closed curve, i.e., a path with different vertices, unless the last one, which is equal to the first vertex. The following lemma is a well known result of graph theory.

\[l:degreem\] If $r\ge 2$ is a natural number and $G$ is any graph with ${\delta}(v) \ge r$ for every $v \in V(G)$, then there exists a cycle $\eta$ in $G$ with $L(\eta) \ge r+1$.

We show now some results about the alliance polynomial of regular graphs and their coefficients. If $G$ is a graph and 