Distinguishing threshold for some graph operations

A vertex coloring of a graph $G$ is distinguishing if non-identity automorphisms do not preserve it. The distinguishing number, $D(G)$, is the minimum number of colors required for such a coloring and the distinguishing threshold, $θ(G)$, is the minimum number of colors~$k$ such that any arbitrary $k$-coloring is distinguishing. Moreover, $Φ_k (G)$ is the number of distinguishing coloring of $G$ using at most $k$ colors. In this paper, for some graph operations, namely, vertex-sum, rooted product, corona product and lexicographic product, we find formulae of the distinguishing number and threshold using $Φ_k (G)$.

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