Properties of Connected (n, m)-Graphs Extremal Relatively to Vertex Degree Function Index for Convex Functions

In this paper some structural properties of connected (n,m)-graphs which are maximum (minimum) with respect to vertex-degree function index H-f(G), when f is a strictly convex (concave) function are deduced. Also, it is shown that the unique graph obtained from the star S-n by adding -y edges between a fixed pendent vertex v and gamma other pendent vertices, has the maximum general zeroth-order Randic index R-0(alpha) in the set of all n-vertex connected graphs having cyclomatic number gamma when 1 <= gamma <= n - 2 and alpha >= 2. A conjecture concerning connected (n, m)-graphs G having maximum R-0(alpha)(G) for every n - 1 <= m <= 1/2(n-1 2) and alpha >= 2 was proposed, which completes the characterization of maximal graphs in the case alpha < 0.