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. Recently, Tomescu considered the upper bound
of \(H_f(G)\) and got the connected \((n,m)\)-graph \(G\) with \(m\geq n\) which maximizes \(H_f(G)\) if \(f(x)\) is strictly convex with two special properties. He also characterized all \((n,m)\)-graphs \(G\) with \(1\leq m\leq n\) satisfying
that \(H_f(G)\leq f(m)+mf(1)+(n-m-1)f(0)\) 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
\(1\leq m\leq n(n-1)/2\) satisfies that \(H_f(G)\geq rf(k+1)+(n-r)f(k)\) if \(f(x)\) is strictly convex, where \(k=\lfloor 2m/n \rfloor\) and \(r=2m-nk\). Moreover, the equality holds if and only if \(G\in \mathcal{G}(n,m)\), where \(\mathcal{G}(n,m)\)
is the family of all \((n,m)\)-graphs \(G\) satisfying that the vertex-degree \(d(v)\in \{\lfloor \frac{2m}{n} \rfloor,\lceil \frac{2m}{n} \rceil \}\) for all \(v\in V(G)\). 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 \(H_f(G)\).