Domination polynomial of clique cover product of graphs

Abstract

Let \(G\) be a simple graph of order \(n\). We prove that the domination
polynomial of the clique cover product \(G^\mathcal{C} \star H^{V(H)}\) is
\[
D(G^\mathcal{C} \star H,x)=\prod_{i=1}^k\Big [\big((1+x)^{n_i}-1\big)(1+x)^{|V(H)|}+D(H,x)\Big],
\]
where each clique \(C_i \in \mathcal{C}\) has \(n_i\) vertices. As an
application, we study the \(\mathcal{D}\)-equivalence classes of some
families of graphs and, in particular, describe completely the
\(\mathcal{D}\)-equivalence classes of friendship graphs constructed by
coalescing \(n\) copies of a cycle graph of length 3 with a common vertex.