Maximal augmented sombor index in unicyclic graphs

Qinfei Tang; Qinghai Liu; Qinfei Tang; Qinghai Liu

doi:10.3934/math.20261244

AIMS Mathematics

2026, Volume 11, Issue 9: 31538-31551. doi: 10.3934/math.20261244

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Research article

Maximal augmented sombor index in unicyclic graphs

Qinfei Tang ¹,
Qinghai Liu ^{2,3,4
,
,}

1.
Department of Mathematics Research, Fujian Institute of Education, Fuzhou 350025, China
2.
Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian 350108, China
3.
Fujian Science & Technology Innovation Laboratory for Optoelectronic Information of China, Fuzhou, Fujian 350108, China
4.
Center for Applied Mathematics of Fujian Province, Fuzhou, Fujian 350108, China

Received: 02 June 2026 Revised: 01 September 2026 Accepted: 15 September 2026 Published: 24 September 2026
MSC : 05C09, 05C35, 05C92

The augmented Sombor (ASO) index is a degree-based topological descriptor for connected graphs with at least three vertices, defined by
$ \text{ASO}(G) = \sum\limits_{uv \in E(G)} \sqrt{\frac{d(u)^2 + d(v)^2}{d(u) + d(v) - 2}}. $
This paper determines the sharp upper bound for the ASO index over the class of unicyclic graphs with a fixed order. Our approach is based on two local transformations, called leaf contracting and leaf transferring, together with monotonicity properties and auxiliary estimates for the edge-contribution function. We prove that these operations strictly increase the ASO index whenever the graph does not already have the required extremal structure. Consequently, every maximizing unicyclic graph of order $ n $ must consist of a triangle whose remaining $ n-3 $ vertices are pendant vertices adjacent to one common vertex. For every $ n\geq 3 $, we also derive an explicit closed formula for the resulting maximum. The argument covers the small orders directly and establishes uniqueness for all larger orders through the transformation lemmas. The result completes the upper-bound problem for the ASO index on unicyclic graphs and provides a structural benchmark for degree concentration in this graph class.
- augmented Sombor index,
- unicyclic graph,
- pendant vertex,
- graphical index,
- extremal graph,
- chemical graph theory
Citation: Qinfei Tang, Qinghai Liu. Maximal augmented sombor index in unicyclic graphs[J]. AIMS Mathematics, 2026, 11(9): 31538-31551. doi: 10.3934/math.20261244

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Abstract

The augmented Sombor (ASO) index is a degree-based topological descriptor for connected graphs with at least three vertices, defined by

$ \text{ASO}(G) = \sum\limits_{uv \in E(G)} \sqrt{\frac{d(u)^2 + d(v)^2}{d(u) + d(v) - 2}}. $

This paper determines the sharp upper bound for the ASO index over the class of unicyclic graphs with a fixed order. Our approach is based on two local transformations, called leaf contracting and leaf transferring, together with monotonicity properties and auxiliary estimates for the edge-contribution function. We prove that these operations strictly increase the ASO index whenever the graph does not already have the required extremal structure. Consequently, every maximizing unicyclic graph of order $ n $ must consist of a triangle whose remaining $ n-3 $ vertices are pendant vertices adjacent to one common vertex. For every $ n\geq 3 $, we also derive an explicit closed formula for the resulting maximum. The argument covers the small orders directly and establishes uniqueness for all larger orders through the transformation lemmas. The result completes the upper-bound problem for the ASO index on unicyclic graphs and provides a structural benchmark for degree concentration in this graph class.

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