On Diameter-Constrained Unicyclic Graphs and BID Indices

A connected graph in which the number of vertices equals the number of edges is referred to as a unicyclic graph. Let G denote such a graph with edge set E(G). For any vertex w∈V(G), its degree is denoted by d(w). This study focuses on a class of topological indices defined by Bf(G)=∑uv∈E(G)f(d(u),d(v)), where f is a symmetric, real-valued function depending on the degrees of adjacent vertices. The primary objective is to systematically identify those graphs, among all unicyclic graphs with a fixed number of vertices and a specified diameter, that either minimize or maximize Bf, under explicit assumptions on the function f. These assumptions are satisfied by a wide variety of well-known and recently introduced indices, thus rendering the resulting characterizations broadly applicable to numerous classical and modern topological indices. A principal contribution of this work is the precise characterization of unicyclic graphs that minimize various indices, including the sum-connectivity, harmonic, modified Sombor, and modified Euler–Sombor indices, within the aforementioned class of unicyclic graphs. In addition, an obtained result enables the characterization of considered graphs that maximize other indices, such as the atom-bond sum-connectivity, Sombor, reciprocal sum-connectivity, and Euler–Sombor indices. In the applications of the main results, the constraints on the function f are verified using the symbolic computation software Mathematica.