Abstract
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 \( H_f(G) = \sum_{v \in V(G)} f(d(v)) \), 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 \( H_f(G) \leq (n - k - 1) f(0) + f(p) + (k - p) f(k - 1) + p f(k) \) for any connected \( (n,m) \)-graph \( G \) with \( m = n + k(k - 3) / 2 + p \), where \( 2 \leq k \leq n - 1 \) and \( 0 \leq p \leq k - 2 \). 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 \( \alpha \leq - t , - 1 < q \leq 2.038t - 0.038 \) and \( t \geq 1 \) or when \( \alpha < 0, - 1 < q \leq 0 \).
We also prove that if \( f(x) \) is strictly convex and strictly monotonically decreasing and satisfies some additional properties, then \( H_f(G) \leq (n - k - 1) f(0) + f(p) + (k - p) f(k - 1) + p f(k) \) for any \( (n, m) \)-graph \( G \) with \( m=k(k - 1) / 2 + p \), where \( 2 \leq k \leq n - 1 \) and \( 0 \leq p \leq k - 1 \). 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 \( \alpha \leq - t \) and \( 0 < q \leq 1.413t + 0.587 \) and \( t \geq 1 \).
