Abstract
In this paper it is shown that the unique graph obtained from the star
\(S_n\) by adding \(\gamma\) edges between a fixed pendant vertex \(v\)
and \(\gamma\) 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 \(\gamma\) when \(1 \leq\gamma\leq n − 2\)
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 Randić index
\(^0R_{\alpha}(G)\) is strictly convex and verifies this property for
\(\alpha > 2\).