On Sombor index of extremal graphs

Rouhani, Soheir; Habibi, Mohammad; Mehrpouya, Mohammad Ali

doi:10.22061/jdma.2024.11328.1101

On Sombor index of extremal graphs

Document Type : Full Length Article

Authors

¹ Department of Mathematics Tafresh University Tafresh

² Tafresh University

10.22061/jdma.2024.11328.1101

Abstract

Let $ G $ be a finite simple graph. The Sombor index of $ G $ is
defined as $ \sum\nolimits_{uv\in E(G)} \sqrt{d_{u}^{2}+d_{v}^{2}} $
where $d_{u}$ and $d_{v}$ represent the degrees of vertices $ u$ and
$v$ in $ G $, respectively. The sum of the absolute values of the
adjacency eigenvalues defines the energy of a graph. This paper aims
to enhance the current connections between the Sombor index and the
energy of graphs. Additionally, we provide some upper bounds for the
Sombor index of triangle-free, square-free, $K_r$-free and
tripartite graphs in terms of order, size and minimum degree.

Graphical Abstract