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

TL;DR

The paper investigates how the Albertson irregularity index $irr(G)$ and the sigma index $\sigma(G)$ behave on trees, with emphasis on how degree sequences govern irregularity. Using caterpillar trees $C(n,m)$ as a structured testbed, it derives closed forms $irr(C(n,m)) = 2m^2 + m(m+1)(n-2) + 2$ for $n\ge 3$ (with $n=2$ giving $2m^2$) and $\sigma(C(n,m)) = 2m^3$ under the proposed parametrization. The authors also analyze extremal configurations for $\sigma$ in terms of degree sequences, presenting exact expressions for $\sigma$ with top-degree triples $(d_1,d_2,d_3)$ and extending to quadruples $(d_1,d_2,d_3,d_4)$, including a condition $d_2=d_1+1$, $d_3=d_2+1$, $d_4=d_3+1$ for equality. A comparative study on normal vs caterpillar trees is reported, with tables and numeric examples illustrating regimes where $\sigma$ surpasses $irr$ and vice versa, complemented by a small Python demonstration for four-degree sequences. Overall, the work sharpens understanding of how degree-based irregularity measures reflect tree structure and provides exact formulas that could inform chemical graph theory and network design.

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 $δ,Δ$ 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 $σ(T)=(d_1-1)^3+\sum_{i=1}^{3}(d_i-1)(d_i-2)+(d_3-1)^3$.

Figures (4)

• Figure 1: Example of Caterpillars in graph theory.
• Figure 2: Caterpillars tree with $(n,3)$ vertices
• Figure 3: General case of Caterpillar tree
• Figure 4: describe sequence $d(d_1,d_2,d_3)$.

Theorems & Definitions (20)

• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 2.1
• Lemma 2.4
• Lemma 2.5
• Theorem 2.6
• Proposition 3.1
• proof
• Proposition 3.2