Graphs with given cyclomatic number extremal relatively to vertex degree function index for convex functions

Abstract

In this paper it is shown that the unique graph obtained from the star S n by adding edges between a fixed pendant vertex v and other pendant vertices, has the maximum (minimum) vertex degree function index H f (G) in the set of all n-vertex connected graphs having cyclomatic number when 1 n -2 if f (x) is strictly convex (concave) and satisfies an additional property. This property holds for example if f (x) is differentiable and its derivative is also strictly convex (concave). The general zeroth-order Randi index 0 R (G) is strictly convex and verifies this property for > 2.

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