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-index H-f(G) of a graph G is defined as H-f(G) = Sigma(v is an element of V(G)) f(d(v)), where f is a real function. Recently, Tomescu considered the upper bound of H-f(G) and got the connected (n, m)-graph G with m = n which maximizes Hf (G) if f(x) is strictly convex with two special properties. He also characterized all (n, m)-graphs G with 1 <= m <= n satisfying that H-f(G) <= f(m) + mf(1) + (n - m - 1)f(0) if f(x) is 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) + (n - r)f(k) if f(x) is strictly convex, where k = left perpendicular 2m/n right perpendicular. and r = 2m - nk. Moreover, the equality holds if and only if G is an element of G(n, m), where G(n, m) is the family of all (n, m)-graphs G satisfying that the vertex-degree d(v) is an element of{left perpendicular 2m/n right perpendicular, inverted left perpendicular 2m/n inverted right perpendicular} for all v is an element of V (G). Under the same condition on f we also obtain a result for the minimum of Hf (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).