Common neighbor polynomial of graphs

Graph theory is one of the well-flourished branches of Mathematics. Among various branches of graph theory, graph polynomials are one of the well-studied concepts as they are used to unveil the structural properties of graphs. Roughly speaking, a graph polynomial is a polynomial assigned to a graph whose coefficients are the indicators of some graph-theoretic parameters. It can be defined as a function from the set of all finite graphs to the polynomial ring over the set of real numbers such that isomorphic graphs are assigned to the same polynomial. In the present work, emphasizing the structural similarity of pairs of nodes in a network system, a new graph polynomial is introduced named as `Common neighbor polynomial of graphs'.