Inverse Degree Index of Graphs With a Given Cyclomatic Number

Abstract

We investigate how the inverse degree index of graphs depends on their
cyclomatic number. In particular, we provide sharp lower bounds on the
inverse degree index over all graphs on a given number of vertices with a
given cyclomatic number. We also deduce some structural properties of
extremal graphs. Some open questions regarding the upper bound over the
same class of graphs are discussed and some possible further developments
are indicated.