(n, m)-Graphs with Maximum 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. In this paper, we show that if f(x) is strictly convex and strictly monotonically decreasing and satisfies some additional properties, then H-f(G) <= (n - k - 1)f(0) + f(p) + (k - p)f(k - 1) + pf(k) for any connected (n, m)-graph G with m = n + k(k - 3)/2 + p, where 2 <= k <= n - 1 and 0 <= p <= k - 2. The unique graph that satisfies the above equality is characterized. As an instance, the function f(x) = (x + q)(alpha) is such a function when a = -t, -1 < q = 2.038t - 0.038 and t >= 1 or when alpha< 0, -1 < q <= 0. We also prove that if f(x) is strictly convex and strictly mono-tonically decreasing and satisfies some additional properties, then H-f(G) <= (n - k - 1)f(0) + f(p) + (k - p)f(k - 1) + pf(k) for any (n, m)-graph G with m = k(k - 1)/2 + p, where 2 <= k <= n - 1 and 0 <= p <= k - 1. The unique graph that satisfies the above equality is characterized. As an instance, the function f(x) = (x + q)(alpha) has the properties as described above when alpha <= -t and 0 < q <= 1.413t + 0.587 and t >= 1.