ScholarWorks@성균관대학교

초록

A connected graph in which the number of vertices equals the number of edges is referred to as a unicyclic graph. Let G$G$ denote such a graph with edge set E(G)$E(G)$. For any vertex w is an element of V(G)$w\in V(G)$, its degree is denoted by d(w)$d(w)$. This study focuses on a class of topological indices defined by Bf(G)=& sum;uv is an element of E(G)f(d(u),d(v)),${\mathcal{B}}_{\text{mathcalligra}f}(G)={\sum }_{uv\in E(G)}\text{mathcalligra}f(d(u),d(v)),$ where f$\text{mathcalligra}f$ 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${\mathcal{B}}_{\text{mathcalligra}f}$, under explicit assumptions on the function f$\text{mathcalligra}f$. 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$\text{mathcalligra}f$ are verified using the symbolic computation software Mathematica.

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