# An Application of Sombor Index over a Special Class of Semigroup Graph

Authors: Seda Oğuz Ünal
Journal: Journal of Mathematics
 (2021)

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## Abstract

Recently, Gutman introduced a class of novel topological invariants named Sombor index which is defined asSOG=∑uv∈EGdu2+dv2. In this study, the Sombor index of monogenic semigroup graphs, which is an important class of algebraic structures, is calculated.


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## Body

## 1. Introduction and Preliminaries

The monogenic semigroup graph is inspired by zero divisor graphs. Therefore, before moving on to the main topic, we will focus on the studies on zero divisor graphs (see [1–4]). In relation to the study of zero divisor graphs that has many authors researching commutative and noncommutative rings and how it has advanced, DeMeyer et al. [5, 6] have developed research on commutative and noncommutative semigroups related to zero divisor graphs. The authors in [7] utilised the adjacent rule of vertices while still keeping the original idea. The authors determined a finite multiplicative monogenic semigroup with 0 as follows:(1)SM=0,x,x2,x3,…,xn.By utilizing the idea defined in [5, 6], the authors obtained a new graph related to monogenic semigroups in [7]. The vertices of this graph are all nonzero elements in SM and for any two different vertices xi and xj where 1≤i,j≤n are linked to each other, if and only if i+j>n. There are many studies concerning monogenic semigroup graphs which were published by Akgüneş et al. (see for example [8–10]).In chemistry, topological indices have been around for more than half a century [11]. In newer times, they are being extensively investigated also by mathematicians. These indices are used to model structural properties of molecules and provide information of value for physical chemistry, material science, pharmacology, environmental sciences, and biology [12]. Recently, a new such graph-based topological index, called Sombor index, was put forward by Gutman [13]. Initially, the index was applied in chemistry [14–18] and soon attracted the interest of mathematicians [19–22]. Eventually, however, the Sombor index found applications also in network science and was used for modeling dynamical effects in biology, social, and technological complex systems [23]. It seems that this index became interesting also for military purposes [24]. All this happened within less than one year since the publication of the paper [13]. In view of this wide research activity on Sombor index, it may be of interest to seek for its deeper algebraic connections. In this paper, we report some results relating the Sombor index with an important class of algebraic structures, namely, with monogenic semigroups.For a graphG, its edge set and vertex set are denoted by EG and VG, respectively.Sombor index discovered by; Gutman [13] is one of the vertex-degree-based topological indices defined by(2)SOG=∑uv∈EGdu2+dv2,because the function Fx,y=x2+y2 was not utilised.Also, as a reminder, for a real numberr, we identify by r the greatest integer ≤r, and by ⌈r⌉, the least integer ≥r. It is clear that r−1<r≤r and r≤⌈r⌉<r+1. However, for a natural number n, we have(3)n2=n2,ifnis even,n−12,ifnis odd.In this paper, we focus on determining the explicit formula of Sombor index of the monogenic semigroup graph.


## 2. An Algorithm

The authors in [8] to simplify their research gave the algorithm concerning the neighborhood of vertices by utilizing the initial statement of monogenic semigroup graph. We will use this algorithm in our main theorem in the next section.In: the vertex xn is adjoining to every vertex xi11≤i1≤n−1 except itself.In−1: the vertex xn−1 is adjoining to every vertex xi22≤i2≤n−2 except itself and the vertex xn.In−2: the vertex xn−2 is adjoining to every vertex xi33≤i3≤n−3 except itself and the vertices xn and xn−1.Carrying on the algorithm this way, we get the following result, depending on whether the numbern is odd or even.Ifn is even,In/2+2: the vertex xn/2+2 is adjoining not only to the vertices xn/2−1, xn/2, and xn/2+1 but also to the vertices xn, xn−1, xn−2, …, xn/2+3.In/2+1: the vertex xn/2+1 is adjoining not only to the single vertex xn/2 but also to the vertices xn, xn−1, xn−2, …, xn/2+2.Ifn is odd,In+1/2: the vertex xn+1/2+2 is adjoining not only to the vertices xn+1/2−2, xn+1/2−1, xn+1/2, and xn+1/2+1 also adjoining to the vertices xn, xn−1, xn−2, …, xn+1/2+3.In+1/2+1: the vertex xn+1/2+1 is adjoining not only to the vertices xn+1/2−1 and xn+1/2 also adjoining to the vertices xn−1, xn−2, …, xn+1/2+2.In the lemma given below, the degrees of verticesx1,x2,…,xn∈ΓSM are denoted by d1,d2,…,dn. There are many studies on the degree series. Regarding this, you can refer to [7, 25] and references cited in these studies. In fact, in the lemma below, it is mentioned that there is an ordering between the degrees d1,d2,…,dn. You can reach the proof of this lemma from [7], as well as from the algorithm given above (see [8]).Lemma 1.
(4)d1=1,d2=2,…,dn/2=n2,dn/2+1=n2,dn/2+2=n2+1,…,dn=n−1.Remark 1.
Paying attention to Lemma1, the repeated terms are given in the following:(5)dn/2=n2=dn/2+1.
Therefore, the degree ofdn is denoted by n−1, although the number of vertices is n.


## 3. Calculating Sombor Index ofΓSM

In this section, we will obtain an exact formula of Sombor index over monogenic semigroup graph.Theorem 1.
For any monogenic semigroupSMas given in (1), the Sombor index of the graphΓSMis(6)SOΓSM=∑k=1n/2−1∑i=kn−k−1n−k2+i2+∑k=1n/2n−k2+n22,ifnis even,∑k=1n−1/2∑i=kn−k−1n−k2+i2+∑k=1n−1/2n−k2+n22,ifnis odd.Proof.
Since our aim is to formulateSOΓSM in terms of the total number of degrees, we need to treat the sum as the sum of different blocks and then calculate each separately. During our calculations, we will use the algorithm given in Section 2 here, as it offers a very systematic way of calculating the degrees of vertices. We will also make use of equations (3) and (4) and Remark 1
Ifn is odd,(7)SOΓSM=dn2+d12+dn2+d22+dn2+d32+⋯+dn2+dn−22+dn2+dn−12++dn−12+d22+dn−12+d32+⋯+dn−12+dn−22++⋯++dn+1/2+22+dn+1/2−22+dn+1/2+22+dn+1/2−12+dn+1/2+22+dn+1/22+dn+1/2+22+dn+1/2+12+dn+1/2+12+dn+1/2−12+dn+1/2+12+dn+1/22.
As a result, the Sombor index ofΓSM is written as the sum below:(8)SOΓSM=∑ij∈EGdi2+dj2=SOn+SOn−1+⋯+SOn+1/2+2+SOn+1/2+1.
When calculating the Sombor index sum, we will write the smallest degree at the end of the line, so we will get a second total and this will provide us with ease of operation. By the way, while making these calculations, we use the equationn/2=n−1/2 given in (2) for the case where n is odd.(9)SOn=n−12+12+n−12+22+n−12+32+⋯+n−12+n22+⋯++n−12+n−22+n−12+n22=∑i=1n−2n−i2+i2+n−12+n−122.
If similar operations applied inSOn are applied in SOn−1, we obtain(10)SOn−1=∑i=2n−3n−i2+i2+n−22+n−122,SOn+1/2+2=n+322+n−322+n+322+n−122+n+322+n−122+n+322+n+122,and finally,(11)SOn+1/2+1=n+122+n−122+n+122+n−122.
Hence,(12)SOn+SOn−1+⋯+SOn+1/2+2+SOn+1/2+1=∑k=1n−1/2∑i=kn−k−1n−k2+i2+∑k=1n−1/2n−k2+n22.
If we follow similar steps as ifn is odd, we will get the following sum if n is even:(13)SOn+SOn−1+⋯+SOn/2+2+SOn/2+1=∑k=1n/2−1∑i=kn−k−1n−k2+i2+∑k=1n/2n−k2+n22.Corollary 1.
In [26, 27], the authors exhibited that the Sombor index can be an integer in several graph structures. In monogenic semigroup graphs, it is seen that it is not possible for the Sombor index to take an integer value according to the formula given in Theorem 1.We will give the following examples to reinforce Theorem1.Example 1.
Consider the monogenic semigroupSM6 given below and calculate the Sombor index of ΓSM6 graph by applying the rule given in Theorem 1:(14)SM6=x,x2,x3,x4,x5,x6∪0.
Monogenic semigroup graphs, which are defined with inspiration from zero divisor graphs, also contain the 0 element. Because the vertices ofxi and xj, which are taken arbitrarily in the monogenic semigroup, can be connected with each other, that is, the necessary and sufficient condition for the condition of xixj=0 is to be i+j>n. In line with this information, the SM6 graph is given in Figure 1.(15)SOΓSM6=∑k=12∑i=k5−k6−k2+i2+∑k=136−k2+32=52+12+52+22+52+32+52+42+42+22+42+32+52+32+42+32+32+32.Figure 1
SM6 monogenic semigroup graph.In the example below, the Sombor index of the corresponding hydrogen-suppressed molecular graph, which is equivalent toΓSM4 monogenic semigroup graph, is calculated.Example 2.
The Sombor index of the monogenic semigroupSM4 given below is calculated by applying Theorem 1.(16)SM4=x,x2,x3,x4∪0.
TheSM4 graph is given in Figure 2.(17)SOΓSM4=∑k=1∑i=124−k2+i2+∑k=124−k2+22=32+12+32+22+32+22+22+22.
As can be seen, the Sombor index of a monogenic semigroup graph can be calculated very easily with the given formula in Theorem1.Figure 2
SM4 monogenic semigroup graph (corresponding hydrogen-suppressed molecular graph).



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