First irregularity Sombor index of trees with fixed maximum degree

ABSTRACT

The Sombor index as a new degree-based topological index was first put forward in 2021 by Gutman. Recently, Kulli introduced a variant of the Sombor index called first irregularity Sombor index. The first irregularity Sombor index of a graph G, denoted by $\mathit{IS}{O}_1(\mathit{G})$, is defined as the sum of weights $|d_G^2(\mathit{v})-d_G^2(\mathit{w})|$ of all edges vw of E(G), where $d_G(\mathit{v})$ denotes the degree of a node v in G. In this paper we show that for any tree T of order n with maximum degree Δ,

$\mathit{IS}{O}_1\left(\mathit{T}\right)\ge \{\begin{array}{l}(\mathrm{△}-1)\sqrt{{\mathrm{△}}^2-1}+\sqrt{{\mathrm{△}}^2-4}+\sqrt{3}\mathit{if}\mathrm{△}<\mathit{n}-1,\\ \mathrm{△}\sqrt{{\mathrm{△}}^2-1}\mathit{if}\mathrm{△}=\mathit{n}-1.\end{array}$

We also characterize the extremal trees.

KEYWORDS:

• First irregularity Sombor index
• lower bound
• trees

MATHEMATICS SUBJECT CLASSIFICATION:

• 05C07

Disclosure statement

No potential conflict of interest was reported by the author(s).

Author contribution statement

Nasrin Dehgardi: Conceptualization; Investigation; Methodology; Validation; Writing original draft. Yilun Shang: Visualization; Investigation; Validation; Resources; Funding acquisition; Writing review and editing.

Data availability statement

The paper does not include or use any datasets.

Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/27684830.2023.2291933