### 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.

#### Keywords

domination polynomial, \(\mathcal{D}\)-equivalence class, clique cover, friendship graphs

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