Geodesic
Domination polynomial of~clique~cover~product~of~graphs
Algebra and discrete mathematics, Tome 28 (2019) no. 2, pp. 248-259

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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: $\mathcal{D}$-equivalence class, friendship graphs.
Mots-clés : domination polynomial, clique cover 
@article{ADM_2019_28_2_a8,
     author = {Somayeh Jahari and Saeid Alikhani},
     title = {Domination polynomial of~clique~cover~product~of~graphs},
     journal = {Algebra and discrete mathematics},
     pages = {248--259},
     publisher = {mathdoc},
     volume = {28},
     number = {2},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2019_28_2_a8/}
}
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Somayeh Jahari; Saeid Alikhani. Domination polynomial of~clique~cover~product~of~graphs. Algebra and discrete mathematics, Tome 28 (2019) no. 2, pp. 248-259. http://geodesic.mathdoc.fr/item/ADM_2019_28_2_a8/