Abstract

The Nirmala and first Banhatti-Sombor index which is originated from Sombor index is designated by and , respectively. In this work, we calculated the Nirmala and Banhatti-Sombor index over the tensor and Cartesian product of a graph of an algebraic structure by presenting two different algorithms.

1. Introduction and Preliminaries

A finite multiplicative monogenic semigroup with zero is demonstrated aswhich the authors carried out in [1]. Whilst the graph is specified by modifying the adjacent rule of vertices and sticking to the original stance. The vertices of include all elements in except zero. Any two different vertices and in that are adjoined in case of xi·xj = 0 with the rule xi·xj = x(i + j) = 0, if and only if . For detailed information about monogenic semigroup graphs, see [2–4].

Zero divisor graphs are the basis of monogenic semigroup graphs [1]. Zero divisor graphs were first conducted on commutative rings [5], and then following this study, the researchers worked on commutative and noncommutative rings [6–8]. Following the studies of zero divisor graphs on rings, in [9, 10], the authors utilized the information in commutative and noncommutative semigroups.

In the field of chemistry, studies on topological indices have been carried out for more than half a century [11]. Recently, topological indices have been thoroughly detailed in the field of mathematics [12]. Indices such as these are utilized in creating structural properties of molecules and thus equipping us with data for industrial science, applied physics, biochemistry, environmental science, and toxicology [13]. In [14], Gutman introduced a graph-based topological index named Sombor index. First, it was used in chemistry [15–20] and soon after grabbed the attention of mathematicians [21–27]. Network science used the modeling dynamical effect of biology and social technological complex systems [28]. Sombor index became popular for military use as well [29]. Since its inception in less than a year of being published, the vast interest of mathematicians researching the Sombor index has been astounding. Our research produces results based on the Nirmala and first Banhatti-Sombor index category of algebraic structures like monogenic semigroups.

Research into the extension of graphs is quintessential in applied sciences (see [30, 31]). It is with this concept that the tensor and Cartesian product have been determined. The researchers in [32, 33] estimated the Wiener index of Cartesian product of graphs, and in [34] the authors calculated the Szeged index of Cartesian product of graphs. In 2011, Yarahmadi calculated certain topological indices like the Zagreb indices and Harary and Schultz indices [35].

In this paper, we use and as two simple graphs. The vertex set of the tensor product and Cartesian product of and is indicated by and , respectively. For tensor product, necessary and sufficient condition for any in to be joined is and . For Cartesian product, necessary and sufficient condition for any in to be associated is and or and .

In this study, monogenic semigroup and monogenic semigroup will be taken as follows, respectively:

The vertex set of the tensor product and Cartesian product of and is given as follows:

The Sombor index invented by Gutman [14] is a vertex degree-based topological index which is narrowed down as

Inspired by work on Sombor indices, Kulli put forward the Nirmala and first Banhatti-Sombor index of a graph as follows:

The Nirmala index is a special case of the general sum-connectivity index introduced in [36]. For Nirmala and Banhatti-Sombor indices, see [23, 24, 37], respectively.

In addition as a note, for a real number , we designate by the greatest integer and by , the least integer . It is quite apparent that and . In addition to this, for a natural number , we have

Here, any two vertices and are connected to each other if and only if

Our research calculates results based on the Nirmala index of Cartesian and tensor product of monogenic semigroup graphs.

2. An Algorithm for Tensor Product of Monogenic Semigroup Graph

 If is even ( is even or odd), : the vertex is linked to every vertex  : the vertex is linked to every vertex  : the vertex is linked to every vertex   : the vertex is linked to every vertex  : the vertex is linked to every vertex   : the vertex is linked to  If is odd and is even, : the vertex is linked to and  If is odd and is odd, : the vertex is linked to and

In the following lemma, the vertice degreesare denoted by

Several investigations exist on the degree series with respect to this series [1, 38].

Lemma 1.

Remark 1. Considering special observations in Lemma 1, the recurrent phrases utilized that are as follows:Consequently, the degree of is noted by , inspite of the amount of vertices is .

3. Computing Nirmala Index of Tensor Product of Monogenic Semigroup Graphs

Our data acquired in this area will produce an accurate formula of the Nirmala index over tensor product of monogenic semigroup graphs by using the above algorithm.

Theorem 1. For any monogenic semigroup and , the Nirmala index over tensor product of two monogenic semigroup graph isIn the formula given above, the numbers , and , will be taken in accordance with the rules and .

Proof. Our primary focus is to methodically formulise concerning the sum of degrees. The calculation will include the tally of the total of several pieces thereafter determining each separately. Amidst our evaluations, the calculation given in Section 2 is utilized and will determine the structure of the degrees of vertices. Equalities (6), (10), and Remark 1 will allow us to evaluate further.
If and are even,Consequently, the Nirmala index is noted as the sum as follows:Whilst estimating the Nirmala index value, the minutest amount is acquired after several calculations where is odd we utilize the equation given in (6).The above equation can be given briefly with the sum symbol as follows:If similar operations applied in are applied to , we obtainIf it is continued in this way, the following equations are obtained for , , and , respectively:In this way, , ,…,,…,, and , …, are calculated one by one to obtain a general sum formula given as follows:

4. Computing Banhatti-Sombor Index of Tensor Product of Monogenic Semigroup Graphs

In this section, the exact formula of the Banhatti–Sombor index will be given over the tensor product of the monogenic semigroup graphs utilizing the algorithm given above.

Theorem 2. For any monogenic semigroup and , the Sombor index over tensor product of two monogenic semigroup graph isIn the formula given above, the numbers , and , will be taken in accordance with the rules and .

Proof. The proof is similar to the calculation of the Nirmala index of the tensor product of monogenic semigroup graphs, as in Theorem 1.

5. An Algorithm for Cartesian Product of Monogenic Semigroup Graph

 If is even ( is even or odd), : the vertex is linked to and  : the vertex is linked to and  : the vertex is linked to and   : the vertex is linked to  : the vertex is linked to and .   : the vertex is linked to  If is odd ( is even or odd), : the vertex is linked to

6. Computing Nirmala Index of Cartesian Product of Monogenic Semigroup Graphs

By using the algorithm given in Section 5, the formula of Nirmala index of Cartesian product over monogenic semigroup graphs will be calculated.

Theorem 3. For any monogenic semigroup and , the Nirmala index over Cartesian product of two monogenic semigroup graph is

In the formula given above, the numbers , and , will be taken in accordance with the rules , , and , .

Proof. Since our primary focus is to formulise concerning the total number of degrees, we need to treat the sum as the sum of the total of different blocks thereafter determining individually. Amidst our evaluations, the calculation given in Section 2 is utilized and will determine the structure of the degrees of vertices with the addition of equalities (6), (10), and Remark 1.
If and is even,