On a Conjecture About the Sombor Index of Graphs

TL;DR

This paper proves a conjecture that a specific graph construction maximizes the Sombor index among connected graphs with a given number of cycles, and extends the result to the reduced Sombor index, also exploring related indices.

Contribution

It confirms the conjecture about the maximum Sombor index for a class of graphs and extends the result to the reduced Sombor index, providing new insights into graph index relationships.

Findings

The graph $H_{n, u}$ maximizes the Sombor index among connected $ u$-cyclic graphs.

The conjecture holds for the reduced Sombor index as well.

Relationships between Sombor, reduced Sombor, and first Zagreb indices are established.

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The Sombor and reduced Sombor indices of $G$ are defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{deg_G(u)^2+deg_G(v)^2}$ and $SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(deg_G(u)-1)^2+(deg_G(v)-1)^2}$, respectively. We denote by $H_{n,\nu}$ the graph constructed from the star $S_n$ by adding $\nu$ edge(s) $(0\leq \nu\leq n-2)$, between a fixed pendent vertex and $\nu$ other pendent vertices. R\'eti et al. [T. R\'eti, T. Do\v{s}li\'c and A. Ali, On the Sombor index of graphs, $\textit{Contrib. Math. }$ $\textbf{3}$ (2021) 11-18] proposed a conjecture that the graph $H_{n,\nu}$ has the maximum Sombor index among all connected $\nu$-cyclic graphs of order $n$, where $5\leq \nu \leq n-2$. In this paper we confirm that the former conjecture is true. It is also shown that this conjecture is valid for the reduced Sombor index. The relationship between Sombor, reduced Sombor and first Zagreb indices of graph is also investigated.