Super Domination Polynomial of a Graph

In this paper, a super domination polynomial of a simple graph \(G = (V, E)\) of order \(|V| = n\) is introduced as the polynomial \(D_{sp}(G, x)=\sum \limits _{i=\gamma _{sp}(G)}^{n} d_{sp}(G, i)x^i\) , where \(\gamma _{sp}(G)\) is the minimum cardinality of a super dominating set in G and \(d_{sp}(G, i)\) is the number of super dominating sets \(S_{sp}\) of G of size i. Some properties of \(D_{sp}(G, x)\) and its coefficients for a given graph G are obtained. Furthermore, explicit formulas of the super domination polynomial of some families of graphs are presented.