**Abstract:**

Graph polynomials have captured the attention of applied mathematicians because of the applications of these polynomials in other fields of sciences such as Chemistry, Biology, and Physics \cite{ELL}. In 2014, Artes and Laja \cite{LAJ1} introduced a pioneering work on convex subgraph polynomials which represents the number of convex subgraphs in a given graph structure. In 2022, Artes, Langamin, and Calib-og \cite{ART} introduced a bivariate graph polynomial called the clique common neighborhood polynomial of a graph by considering the common neighborhood system of a clique in a graph. In 2023, Arriesgado, Salim and Artes Jr. \cite{ARR} investigated the clique connected common neighborhood polynomial of the join of graphs.