2460-0245Journal of the Indonesian Mathematical SocietyJIMS2460-02452086-8952IndoMS10.22342/jims.v32i2.2009Structural Properties and Reverse Topological Indices of Order GCD Graphs of Integers Modulo RingAbdurahimAbdurahimIndonesiaabdurahim@staff.unram.ac.idRomdhiniMamika UjianitaIndonesiamamika@unram.ac.idMaharaniAndika Ellena Saufika HakimIndonesiaa.ellena.saufika@staff.unram.ac.idAstutiMuliaIndonesiamulia_astuti@unib.ac.idDepartment of MathematicsUniversity of Mataramhttps://ror.org/00fq07k50IndonesiaUniversity of Bengkuluhttps://ror.org/04w077t62IndonesiaCorresponding author: Mamika Ujianita Romdhini. Email: mamika@unram.ac.id2304202623042026322JUNE1151204202515022026Copyright (c) 2026 Journal of the Indonesian Mathematical Society2026Journal of the Indonesian Mathematical Societyhttps://creativecommons.org/licenses/by-nc-nd/4.0/This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
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.
Order GCD Graphintegers modulo ringreverse Sombor indexreverse Randic indexreverse harmonic indexFile created by JATS Editorhttps://jatseditor.comissue-created-year20261. 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 . However, this paper did not mention whether , where is an odd prime, and 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 , where is an odd prime and is an integer. Furthermore, the reverse topological indices studied include the Sombor, Randic, and Harmonic indices.
2. Preliminaries
Let be a simple graph with being the set of vertices and being the edge set. Let be the edge between vertices and and deg(u) is the number of vertices that are adjacent to the vertex . The reverse vertex degree of vertex in a graph was first introduced by Ediz and Cancan in 2016 [2]. It is denoted by and defined as , where is the maximum degree in . 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 of a graph [5] as follows:
Following the same logic as [2], the reverse indices of Randic (RR) and Harmonic (RH) of a graph are defined as follows:
Throughout this paper, the order GCD graph of the integer modulo is denoted as . The set of vertices in this graph is all elements of the ring and is denoted as . 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, or, in other words, [14].
3. Main Results
In this section, we examine the properties of the order GCD graph for and discuss the computation of its reverse-based topological indices. The mentioned 3 indices are given as follows: reverse Sombor, reverse Randic, and reverse Harmonic indices.
3.1. Properties of Order GCD Graph.
Let ring, we get the order of elements of as , ,,, , and . Now we list the gcd of every pair of elements of as seen in Table 1.
GCD of Every Pair of Elements of
-
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-
9
6
9
18
3
18
9
2
9
18
3
18
9
6
9
18
1
9
-
3
9
9
3
9
9
1
9
9
3
9
9
3
9
9
1
6
3
-
3
6
3
6
3
2
3
6
3
6
3
6
3
6
1
9
9
3
-
9
3
9
9
1
9
9
3
9
9
3
9
9
1
18
9
6
9
-
3
18
9
2
9
18
3
18
9
6
9
18
1
3
3
3
3
3
-
3
3
1
3
3
3
3
3
3
3
3
1
18
9
6
9
18
3
-
9
2
9
18
3
18
9
6
9
18
1
9
9
3
9
9
3
9
-
1
9
9
3
9
9
3
9
9
1
2
1
2
1
2
1
2
1
-
1
2
1
2
1
2
1
2
1
9
9
3
9
9
3
9
9
1
-
9
3
9
9
3
9
9
1
18
9
6
9
18
3
18
9
2
9
-
3
18
9
6
9
18
1
3
3
3
3
3
3
3
3
1
3
3
-
3
3
3
3
3
1
18
9
6
9
18
3
18
9
2
9
18
3
-
9
6
9
18
1
9
9
3
9
9
3
9
9
1
9
9
3
9
-
3
9
9
1
6
3
6
3
6
3
6
3
2
3
6
3
6
3
-
3
6
1
9
9
3
9
9
3
9
9
1
9
9
3
9
9
3
-
9
1
18
9
6
9
18
3
18
9
2
9
18
3
18
9
6
9
-
Next, the order of the product between two distinct elements of the ring is as follows in Table 2.
Order of the Product Between Two Distinct Elements of
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
0
-
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-
9
6
9
18
3
18
9
2
9
18
3
18
9
6
9
18
2
1
9
-
3
9
9
3
9
9
1
9
9
3
9
9
3
9
9
3
1
6
3
-
3
6
1
6
3
2
3
6
1
6
3
2
3
6
4
1
9
9
3
-
9
3
9
9
1
9
9
3
9
9
3
9
9
5
1
18
9
6
9
-
3
18
9
2
9
18
3
18
9
6
9
18
6
1
3
1
3
3
3
-
3
3
1
3
3
1
3
3
1
3
3
7
1
18
9
6
9
18
3
-
9
2
9
18
3
18
9
6
9
18
8
1
9
9
3
9
9
3
9
-
1
9
9
3
9
9
3
9
9
9
1
2
1
2
1
2
1
2
1
-
1
2
1
2
1
2
1
2
10
1
9
9
3
9
9
3
9
9
1
-
9
3
9
9
3
9
9
11
1
18
9
6
9
18
3
18
9
2
9
-
3
18
9
6
9
18
12
1
3
3
1
3
3
1
3
3
1
3
3
-
3
3
1
3
3
13
1
18
9
6
9
18
3
18
9
2
9
18
3
-
9
6
9
18
14
1
9
9
3
9
9
3
9
9
1
9
9
3
9
-
3
9
9
15
1
6
3
2
3
6
1
6
3
2
3
6
1
6
3
-
3
6
16
1
9
9
3
9
9
3
9
9
1
9
9
3
9
9
3
-
9
17
1
18
9
6
9
18
3
18
9
2
9
18
3
18
9
6
9
-
According to Table 1 and Table 2, we obtain as shown in Figure 1a. By the same argument, we also get the order GCD graph as seen in Figure 1b.
Order GCD Graph
Figure 1.
Taking into account the illustrations in Figure 1a and Figure 1b it can be seen that consists of two types of degree of vertex. However, the pattern of vertex degrees difers for odd and even prime numbers . Theorems 3.1 and 3.2 support this distinction, as presented below.Theorem 3.1.Let be the order GCD graph of the integer modulo ring ， where is an odd prime and is an integer. Suppose that , then
where.
Proof. Let . We consider three cases as follows:
Case 1: For , let and with .
We get and which implies . Meanwhile, yields . Hence, . Therefore, and are not adjacent.
Case 2: When , then we have and . Consequently, and . Since , in the same manner, we obtain Thus which means and are adjacent.
Case 3: Let and . According to the argument in the first and second cases, we obtain . On the other hand, for . Therefore, it sufices to prove that which means and are adjacent.
Based on the three cases above, it can be concluded that any vertex is adjacent to all vertices in . Meanwhile, any two vertices and in are not adjacent. Furthermore, the number of elements in is given by . Since each vertex is adjacent to all vertices in , its degree is given by . On the other hand, since each vertex is only adjacent to vertices outside , its degree is equal to minus , which results in .
Theorem 3.2.Let be the order GCD graph of integers modulo ring for is an even prime and is an integer. Suppose that , then
where
Proof. The proof follows a similar approach to the proof of Theorem 3.1.
The following theorem presents the reverse vertex degrees of the GCD graph for odd prime in Theorem 3.3 and even prime in Theorem 3.4.
Theorem 3.3. Let be the order GCD graph of integers modulo ring for odd prime and is an integer, then the reverse vertex degree of a vertex in is
where
Proof. Based on Theorem 3.1, we obtain . Furthermore, by the definition of the reverse degree, we have and for and , respectively.
Theorem 3.4.Let be the order GCD graph of integers modulo ring for even prime and is an integer, then the reverse vertex degree of a vertex in is
where.
Proof. It is proven based on Theorem 3.2 and the definition of the reverse degree.
Furthermore, Theorems 3.1 and 3.2 imply the following theorem about the computation of the number of edges in .
Theorem 3.5.Let be the order GCD graph of integers modulo ring for odd prime and be an integer. Then the number of edges in Γ is
Proof. Let be defined as . Based on Theorem 3.1, the degrees of the vertices are given by and for and . Furthermore, the number of vertices in is . This means that the number of vertices with degree is . Consequently, the number of vertices with degree is given by the total number of vertices minus , Since the number of edges in is equal to half of the sum of all vertex degrees [17], we obtain the following result based on Theorem 3.1:
Theorem 3.6.Let be the order GCD graph of integers modulo ring for even prime and be an integer. Then the number of edges in is
Proof. Let . Using the same reasoning as in the proof of Theorem 3.5, we obtain and . Based on Theorem 3.2, we derive the following result:
Observe the red vertices in Figure 1 and Figure 1b. If the edges incident to these vertices are removed, a complete subgraph is formed for and a complete subgraph is formed for . This can be seen in Figure 2.
Complete Subgraph of Order GCD Graph
Figure 2.
However, if the edges forming the complete graph are removed from the graph pin Figure 1a, the resulting subgraph is a complete bipartite subgraph as shown in Figure 3a. The partition of the set of vertices and with all the members of each set which and , where . Similarly, for the graph in Figure 1b, the resulting subgraph is the complete bipartite graph , as illustrated in Figure 3b.
Complete Bipartite Subgraph of Z2·23
Figure 3.
Therefore, a theorem regarding the subgraphs of can be established. This is formally stated in Theorem 3.7 and Theorem 3.8 , as follows.
Theorem 3.7. Let be the order GCD graph of ring of integers modulo where an odd prime and be an integer, then a complete subgraph and a complete bipartite subgraph are formed.
Proof. Let . From the explanation in Theorem 3.1, it is clear that two distinct vertices that are not in , that is, or equivalently are adjacent. Therefore, the subgraph induced by the set of vertex forms a complete graph. Furthermore, the number of vertices in . Thus, one of the subgraphs of is the complete subgraph .
Observe that if the edges forming the complete subgraph are removed from , it is evident that any vertex is adjacent to any vertex . According to Theorem 3.1, any two distinct vertices are not adjacent. Therefore, a complete bipartite subgraph is formed with two partition sets: dan . Thus, we obtain the complete bipartite subgraph
Theorem 3.8.Let be the order GCD graph of ring of integers modulo where an even prime and be an integer, then a complete subgraph and a complete bipartite subgraph are formed.
Proof. Let , then , Furthermore, we obtain . Using the same reasoning as in the proof of Theorem 3.7, we derive the complete subgraph and the complete bipartite subgraph .
The number of edges in the subgraphs formed from Theorem 3.7 and Theorem 3.8 is stated, respectively, in Theorem 3.9 and Theorem 3.10 below.
Theorem 3.9.Let be the order GCD graph of ring of integers modulo where is an odd prime and is an integer. The number of edges in the complete subgraph and the complete bipartite subgraph are respectively
and
Proof. Since is a complete subgraph, the number of vertices in the subgraph is , and each vertex has a degree of . Furthermore, since the number of edges in a graph is equal to half of the sum of the degrees of all vertices [17], we obtain
The complete bipartite graph is partitioned into two vertex sets, namely and . Moreover, based on the explanations in the proofs of Theorems 3.1 and 3.7, we obtain and . In this case, can be interpreted as the degree of any vertex , and similarly, can be interpreted as the degree of any vertex . Therefore, the number of edges in the complete bipartite graph is given by
Theorem 3.10. Let be the order GCD graph of the ring of integers modulo where an even prime and be an integer. The number of edges in the complete subgraph and the complete bipartite subgraph are respectively
and
Proof. Using a similar explanation as in Theorem 3.10 and the information from Theorem 3.10, we obtain
and
3.2. Reverse-Based Topological Indices of Order GCD Graph.
This part investigates the reverse-based topological indices of . In particular, we first derive the reverse Sombor index, which is presented in Theorem 3.11.
Theorem 3.11. Let be the order GCD graph of the ring of integers modulo , where an odd prime and be an integer, then the reverse Sombor index is
Proof. Let where and . From Theorem 3.3, we obtain and . The number of edges connecting vertices and is equal to the number of edges in the complete bipartite subgraph conforming Theorem 3.9), which is . Next, the number of edges connecting vertices and is equal to the number of edges in the complete subgraph by Theorem 3.9), which is .
Thus, the general formula for the Reverse Sombor index of is given by
By substituting the values obtained earlier, then
and the proof is completed.
The following theorem presents the reverse Randic index, thereby complementing the previously discussed results.
Theorem 3.12.Let be the order GCD graph ofthe ring ofintegers modulo where an odd prime and be an integer, then the reverse Randic index is
Proof. With the same argument as in the proof of Theorem 3.11, we have
Thus, the general formula for the Reverse Randi´c index of is
Next, we present the reverse harmonic index, which further extends the sequence of reverse-based topological indices discussed above.
Theorem 3.13.Let be the order GCD graph ofthe ring ofintegers modulo where an odd prime and be an integer, then the reverse harmonic index is
Proof. Under the same assumptions as in the proof of Theorem 3.11, we obtain
Thus, the general formula for the reverse Harmonic index of is
In the order GCD graph , see Figure 1a, it is evident that . In other words, the other vertices have a degree of 17. Based on the definition of the reverse degree, we obtain while the other vertices have a reverse degree of 1. The number of edges adjacent to vertices 3, 6, 12, and 15 is 56. Next, according to Theorem 3.5, we find that . Thus, the number of edges adjacent to vertices other than 3, 6, 12, and 15 is 91. Therefore, based on the definition of the Reverse Sombor Index, we get
Next, by applying Theorem 3.11, we have
Using the same argument as in the calculation of the reverse Sombor index above, and based on the definition of the reverse Randic index, we have the following
According to 3.12, we can state that
The reverse harmonic Index of based on its definition is given by
Moreover, by Theorem 3.13, it can be written as
Regardless of whether the reverse index values are computed using the definition or the theorem, the results are the same.
4. Conclusion
In this article, we have described the properties of the order GCD graph of and the formulation of reverse-based indices referred to as reverse Sombor, Randic, and Harmonic indices. We have calculated these indices for two cases: odd and even primes p. Future work may extend this study to finite abelian p-groups and to fuzzy subgroups of finite abelian groups, as discussed in recent literature. These frameworks ofer promising tools for developing more general subgroup-based and fuzzy graph models, thereby enhancing the scope and relevance of the results within algebraic graph theory.
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