Graphs with Minimum Vertex-Degree Function-Index for Convex Functions

An (n, m)-graph is a graph with n vertices and m edges.The vertex-degree function-indexis strictly convex and differentiable and its derivative is strictly convex.In this paper, we will consider the lower bound of H f (G) and show that every (n, m)-graph with 1 ≤ m ≤ n(n -1)/2 satisfies that H f (G) ≥ rf (k + 1) + (nr)f (k) if f (x) is strictly convex, where k = ⌊2m/n⌋ and r = 2mnk.Moreover, the equality holds if and only if G ∈ G(n, m), where G(n, m) is the family of all (n, m)-graphs G satisfying that the vertex-degree d(v) ∈ {⌊ 2m n ⌋, ⌈ 2m n ⌉} for all v ∈ V (G).Under the same condition on f we also obtain a result for the minimum of H f (G) among all connected (n, m)-graphs.It is easy to see that if f (x) is strictly concave, we can get the maximum case for H f (G).