Degree Sequence of Albertson and $\sigma$-Indices on Trees of Order $n\geqslant 3$

TL;DR

This paper explores the relationships between topological indices, specifically the Albertson and sigma indices, on trees of order n≥3, highlighting their connection to degrees and irregularity.

Contribution

It introduces a new sigma index for trees and analyzes its relationship with the Albertson index and degree irregularity.

Findings

Established a relationship between Albertson index and degrees of trees.

Introduced and analyzed the sigma index for trees.

Connected topological indices with graph irregularity measures.

Abstract

In this paper, we presented a study of topological indices on trees, where we show a relationship with irregularity of Albertson index and minimum, maximum degrees $\delta,\Delta$ of graph $G$, where contribute vital roles in determining connection, shading, component incorporation, and realisability where well-known Albertson index as: $\operatorname{irr}(G)=\sum_{uv\in E(G)}\lvert d_u(G)-d_v(G) \rvert$. The sigma index on trees that we introduced as $\sigma(T)=(d_1-1)^3+\sum_{i=1}^{3}(d_i-1)(d_i-2)+(d_3-1)^3$.