On the reciprocal augmented Sombor index

Abdulaziz Mutlaq Alotaibi; Akbar Ali; Abdulaziz Mutlaq Alotaibi; Akbar Ali

doi:10.3934/math.2026812

AIMS Mathematics

2026, Volume 11, Issue 7: 20007-20028. doi: 10.3934/math.2026812

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Research article

On the reciprocal augmented Sombor index

Abdulaziz Mutlaq Alotaibi ¹,
Akbar Ali ^{2,3
,
,}

1.
Department of Mathematics, College of Science and Humanities in AlKharj, Prince Sattam Bin Abdulaziz University, AlKharj 11942, Saudi Arabia
2.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 81451, Saudi Arabia
3.
Jadara Research Center, Jadara University, Irbid 21110, Jordan

Received: 13 May 2026 Revised: 28 June 2026 Accepted: 02 July 2026 Published: 08 July 2026
MSC : 05C05, 05C07, 05C09, 05C35

The reciprocal augmented Sombor (RASO) index is a recently introduced vertex-degree-based graph invariant whose mathematical properties have not yet been exclusively investigated. In this paper, we present the first comprehensive study of this index. We derive sharp bounds for the RASO index in terms of fundamental graph parameters, including the order, size, minimum degree, and maximum degree of a graph. We further characterize the graphs minimizing/maximizing the RASO index among all fixed-order (ⅰ) connected graphs, (ⅱ) trees, and (ⅲ) unicyclic graphs. In addition, we examine the effect of edge addition on the RASO index and establish conditions under which this operation increases its value.
- graph invariant,
- reciprocal augmented Sombor index,
- extremal problem,
- bound,
- tree (graph),
- unicyclic graph
Citation: Abdulaziz Mutlaq Alotaibi, Akbar Ali. On the reciprocal augmented Sombor index[J]. AIMS Mathematics, 2026, 11(7): 20007-20028. doi: 10.3934/math.2026812

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Abstract

The reciprocal augmented Sombor (RASO) index is a recently introduced vertex-degree-based graph invariant whose mathematical properties have not yet been exclusively investigated. In this paper, we present the first comprehensive study of this index. We derive sharp bounds for the RASO index in terms of fundamental graph parameters, including the order, size, minimum degree, and maximum degree of a graph. We further characterize the graphs minimizing/maximizing the RASO index among all fixed-order (ⅰ) connected graphs, (ⅱ) trees, and (ⅲ) unicyclic graphs. In addition, we examine the effect of edge addition on the RASO index and establish conditions under which this operation increases its value.

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