On the vertex degree function of graphs

The vertex-degree function index, denoted as Hf(G), is defined for a graph G with vertex set V(G) as Hf(G)=∑v∈V(G)f(d(v)) where f(x) is a function defined on non-negative real numbers, and dG(vi) represents the degree of the vertex vi 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 Hf(G) and provide explicit characterizations of these extremal graphs. Additionally, we establish a lower bound on Hf(G) that depends on the number of vertices n and the clique number ω. The extremal graphs that reach this lower bound are also characterized. Finally, we derive an upper bound for Hf(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 Hf(G) across various graph classes and contributes to the understanding of the structural properties of graphs that influence this index.