Structural Properties and Reverse Topological Indices of Order GCD Graphs of Integers Modulo Ring

Abdurahim Abdurahim; Mamika Ujianita Romdhini; Andika Ellena Saufika Hakim Maharani

doi:10.22342/jims.v32i2.2009

Abdurahim Abdurahim ⁽¹⁾ , Mamika Ujianita Romdhini ⁽²⁾ , Andika Ellena Saufika Hakim Maharani ⁽³⁾

(1) Department of Mathematics, University of Mataram, Indonesia,

(2) Department of Mathematics, University of Mataram, Indonesia,

(3) Department of Mathematics, University of Mataram, Indonesia

https://doi.org/10.22342/jims.v32i2.2009

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
April 12, 2025

Accepted
February 15, 2026

Published
April 23, 2026

Keywords:

Order GCD Graph, integers modulo ring, reverse Sombor index, reverse Randic index, reverse harmonic index

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Abstract

The order GCD graph of a ring, having the ring elements as the vertex set, and two distinct vertices are adjacent if and only if the greatest common divisor (gcd) of the order of both vertices is equal to the order of the product of these two vertices in the ring. This paper aims to analyze the reverse Sombor, Randić, and Harmonic indices of the order GCD graph where the set of vertices is integers modulo ring elements. The results provide new insights into the mathematical properties of reverse topological indices and their potential applications.

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References

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Authors

Abdurahim Abdurahim

Department of Mathematics, University of Mataram

Mamika Ujianita Romdhini

Department of Mathematics, University of Mataram

mamika@unram.ac.id (Primary Contact)

Andika Ellena Saufika Hakim Maharani

Department of Mathematics, University of Mataram

Abdurahim, A., Romdhini, M. U., & Maharani, A. E. S. H. (2026). Structural Properties and Reverse Topological Indices of Order GCD Graphs of Integers Modulo Ring. Journal of the Indonesian Mathematical Society, 32(2), 2009. https://doi.org/10.22342/jims.v32i2.2009

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1. Introduction

The reverse topological index is a modification of the conventional topological index, which is based on vertex degree. It uses the reverse vertex degree, defined as the maximum degree of the graph minus the degree of the given vertex, plus 1. The reverse topological index was first introduced by Ediz in 2015[1]. In this article, the reverse second Zagreb index of chemical trees was introduced. The following year, Ediz studied the reverse Zagreb index of the Cartesian product of graphs[2], and in 2018, he further examined the reverse first beta Zagreb index[3]. As scientific knowledge continues to advance, the reverse topological index has become an intriguing subject of research. This is evident from several publications related to the reverse topological index and its significance in chemistry[4]. These studies inspired Swamy et al. (2020) to define the reverse Sombor index[5], and the reverse Euler Sombor index has been reported in[6]. Furthermore, investigations have examined various reverse topological indices of bistar graphs and the corona product[7].

The development of graph theory is no longer limited to simple graphs defined by vertex and edge sets. Still, it has expanded to encompass various algebraic structures, such as groups, rings, and fields, as seen in [8,9] and ofers background relevant to subgroup-induced graph structures[10]. Moreover, the constructed graph is extended to various graph types, as described in [11,12]. This exploration not only enriches pure mathematics but also opens numerous applications in applied fields, including cryptography, coding theory, communication networks, and computer science. One example of a graph defined by group elements is the GCD graph. It was first formally introduced by Klotz and Sander in 2007[13], when they discussed unitary Cayley graphs, and has since become an interesting new graph to study. Six years later, they showed the relationship between the GCD graph and the graph products. In 2024, Sarkar and Patra[14]introduced a new definition: the order GCD graph associated with the integer modulo group \(\mathbb { Z } _ { n }\). However, this paper did not mention whether \(n = 2 p ^ { k }\), where \(p\) is an odd prime, and \(k \geq 2\) is an integer. This research gap warrants further examination, which is the urgency of this research.

Furthermore, in previous studies on reverse topological indices, the Order GCD graph has not been examined. There has been research on this graph[15], and in the context of the Sombor index for the group of integers modulo a prime power[16]. Therefore, this article aims to investigate several reverse topological indices of the order GCD graph derived from the ring of integers modulo \(2 p ^ { k }\), where \(p\) is an odd prime and \(k \geq 2\) is an integer. Furthermore, the reverse topological indices studied include the Sombor, Randic, and Harmonic indices.

2. Preliminaries

Let \(G\) be a simple graph with \(V ( G )\) being the set of vertices and \(E ( G )\) being the edge set. Let \(uv\) be the edge between vertices \(u\) and \(v ,\) and deg(u) is the number of vertices that are adjacent to the vertex \(u\). The reverse vertex degree of vertex \(v\) in a graph \(G\) was first introduced by Ediz and Cancan in 2016[2]. It is denoted by \(c _ { v }\) and defined as \(c _ { v } = \Delta - \deg ( v ) + 1\), where \(\Delta\) is the maximum degree in \(G\). With the definition of reverse vertex degree, several reverse topological indices have been introduced, such as the second reverse Zagreb index[1], the first reverse beta Zagreb index[2], and the reverse Sombor index[3]. Among these reverse indices, Swamy et al. in 2020 defined the reverse Sombor index \((RSO)\) of a graph \(G\)[5]as follows:

Following the same logic as[2], the reverse indices of Randic (RR) and Harmonic (RH) of a graph \(G\) are defined as follows:

Throughout this paper, the order GCD graph of the integer modulo \(2 { p } ^ { k }\) is denoted as \(\Gamma \left( \mathbb { Z } _ { 2 p ^ { k } } \right)\). The set of vertices in this graph is all elements of the \(\mathbb { Z } _ { 2 p ^ { k } }\) ring and is denoted as \(V \left( \Gamma \left( \mathbb { Z } _ { 2 p ^ { k } } \right) \right)\). Two distinct vertices \(u \not = v \in V \left( \Gamma \left( \mathbb { Z } _ { 2 p ^ { k } } \right) \right)\) are adjacent if and only if the greatest common divisor (gcd) of the order of both vertices is equal to the order of the product of these two vertices, or, in other words, \(\begin{array} { r } { \operatorname { gcd } ( o ( u ) , o ( v ) ) = o ( u \cdot v ) } \end{array}\)[14].

3. Main Results

In this section, we examine the properties of the order GCD graph for \(\mathbb { Z } _ { 2 p ^ { k } }\) and discuss the computation

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Structural Properties and Reverse Topological Indices of Order GCD Graphs of Integers Modulo Ring

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