The vertex-degree function index, denoted as \(H_{f}(G)\) , is defined for a graph G with vertex set V(G) as \(H_{f}(G)=\sum _{v\in V(G)}f(d(v))\) where f(x) is a function defined on non-negative real numbers, and \(d_G(v_i)\) represents the degree of the vertex \(v_i\) in G. In this paper, we investigate the extremal graphs that maximize or minimize the vertex-degree function index within specific classes of graphs, namely n-vertex quasi-trees, unicyclic graphs, and bicyclic graphs. We identify the graphs that achieve these extremal values of \(H_{f}(G)\) and provide explicit characterizations of these extremal graphs. Additionally, we establish a lower bound on \(H_{f}(G)\) that depends on the number of vertices n and the clique number \(\omega \) . The extremal graphs that reach this lower bound are also characterized. Finally, we derive an upper bound for \(H_{f}(G)\) , which is expressed in terms of n and the vertex (or edge) connectivity of the graphs. We also identify the specific graphs that attain this upper bound. This study provides a comprehensive analysis of the vertex-degree function index \(H_{f}(G)\) across various graph classes and contributes to the understanding of the structural properties of graphs that influence this index.
