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Topological relations between three degree-based invariants of a connectedgraph 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 asemiregular graph. Additionally, the notion of so called weakly semiregulargraphs is introduced, they are considered as a possible generalization oftraditional bidegreed semiregular graphs. Based on the use of Zagreb indicesbased graph irregularity indices, for purposes of fullerene stabilityprediction, comparative tests have been performed on a finite set of dualgraphs of C40 fullerene isomers. By using the findings obtained, thetraditional concept of graph irregularity characterization has been criticallyreevaluated.

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