The number of different distinguishing colorings of a graph

A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphism of $G$ preserves it. The distinguishing number of $G$, denoted by $D(G)$, is the smallest number of colors required for such a coloring. We are intent to count number of different distinguishing colorings with a set of $k$ colors for some certain kinds of graphs. To do this, we first introduce a parameter, namely $\Phi_k (G)$, as the number of different distinguishing colorings of a graph $G$ with at most $k$ definite colors. We then introduce another similar parameter, namely $\varphi_k (G)$, as the number of different distinguishing colorings of a graph $G$ with exactly $k$ definite colors. Showing that it might be hard to calculate $\Phi_k (G)$ and $\varphi_k (G)$ even for small graphs, we use the \emph{distinguishing threshold of $G$}, $\theta(G)$, which is the minimum number $t$ such that for any $k\geq t$, any arbitrary coloring of $G$ with $k$ colors is distinguishing. When $k\geq \theta (G)$, calculating $\Phi_k (G)$ and $\varphi_k (G)$ showed to be much easier. Finally, we introduce the \emph{distinguishing polynomial} for a graph $G$ to be $$P_D (G)=\sum_{k=D(G)}^{n} \varphi_k (G) x^k .$$ An application to distinguishing lexicographic products of graphs is also presented. Coauthors: Bahman Ahmadi, Fatemeh AlinaghipourNon UBCUnreviewedAuthor affiliation: Shiraz UniversityPostdoctora