  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.
