Novel Zagreb Indices-Based Inequalities with Particular Regard to Semiregular and Generalized Semiregular Graphs

Topological relations between three degree-based invariants of a connected graph G are investigated. We present novel inequalities including M1(G), M2(G) and F(G), and show that in all cases equality holds if G is a regular or a semiregular graph. Additionally, the notion of so called weakly semiregular graphs is introduced, they are considered as a possible generalization of traditional bidegreed semiregular graphs. Based on the use of Zagreb indices based graph irregularity indices, for purposes of fullerene stability prediction, comparative tests have been performed on a finite set of dual graphs of C40 fullerene isomers. By using the findings obtained, the traditional concept of graph irregularity characterization has been critically reevaluated.