M[sub.ve]--Polynomial of Cog-Special Graphs and Types of Fan Graphs

The study of topological indices in graph theory is one of the more important topics, as the scientific development that occurred in the previous century had an important impact by linking it to many chemical and physical properties such as boiling point and melting point. So, our interest in this paper is to study many of the topological indices “generalized indices' network” for some graphs that have somewhat strange structure, so it is called the cog-graphs of special graphs “molecular network”, by finding their polynomials based on vertex - edge degree then deriving them with respect to x, y, and xy, respectively, after substitution x=y=1 of these special graphs are cog-path, cog-cycle, cog-star, cog-wheel, cog-fan, and cog-hand fan graphs; the importance of some types of these graphs is the fact that some vertices have degree four, which corresponds to the stability of some chemical compounds. These topological indices are first and second Zagreb, reduced first and second Zagreb, hyper Zagreb, forgotten, Albertson, and sigma indices.