# Nirmala and Banhatti-Sombor Index over Tensor and Cartesian Product of Special Class of Semigroup Graphs

Authors: Seda Oğuz Ünal
Journal: Journal of Mathematics
 (2022)

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## Abstract

The Nirmala and first Banhatti-Sombor index which is originated from Sombor index is designated byNG=∑uv∈EGdu+dv and BSO1G=∑uv∈EG1/du2+dv2, 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.


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## Body

## 1. Introduction and Preliminaries

A finite multiplicative monogenic semigroup with zero is demonstrated as(1)SM=0,x,x2,x3,…,xn,which the authors carried out in [1]. Whilst the graph ΓSM is specified by modifying the adjacent rule of vertices and sticking to the original stance. The vertices of ΓSM include all elements in SM except zero. Any two different vertices xi and xj in ΓSM that 1≤i,j≤n are adjoined in case of xi·xj = 0 with the rule xi·xj = x(i + j) = 0, if and only if i+j>n. 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 useG1 and G2 as two simple graphs. The vertex set of the tensor product and Cartesian product of G1 and G2 is indicated by VG1⊗G2=VG1×VG2 and VG1×G2=VG1×VG2, respectively. For tensor product, necessary and sufficient condition for any u=u1,u2,v=v1,v2 in VG1×VG2 to be joined is u1,v1∈EG1 and u2,v2∈EG2. For Cartesian product, necessary and sufficient condition for any u=u1,u2,v=v1,v2 in VG1×VG2 to be associated is u1=v1 and u2,v2∈EG2 or u1,v1∈EG1 and u2=v2.In this study,SM1 monogenic semigroup and SM2 monogenic semigroup will be taken as follows, respectively:(2)SM1=x1,x12,x13,…,x1m∪0,SM2=x2,x22,x23,…,x2n∪0.The vertex set of the tensor product and Cartesian product ofSM1 and SM2 is given as follows:(3)x1,x2,x12,x2,…,x1m,x2,x1,x22,x12,x22,…,x1m,x22,…,x1,x2n−1,x12,x2n−1,…,x1m,x2n−1x1,x2nx12,x2n,…,x1m,x2n.The Sombor index invented by Gutman [14] is a vertex degree-based topological index which is narrowed down as(4)SOG=∑uv∈EGdu2+dv2.Inspired by work on Sombor indices, Kulli put forward the Nirmala and first Banhatti-Sombor index of a graphG as follows:(5)NG=∑uv∈EGdu+dv,BSO1G=∑uv∈EG1du2+dv2.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 numberr, we designate by ⌊r⌋ the greatest integer ≤r and by r, the least integer ≥r. It is quite apparent that r−1<⌊r⌋≤r and r≤r<r+1. In addition to this, for a natural number n, we have(6)n2=n2,ifnis even,n+12,ifnis odd.Here, any two verticesx1i,x2j and x1a,x2b are connected to each other if and only if(7)x1ix1a∈EΓSM1⇔x1ix1a=0⇔i+a≥m+1,x2jx2b∈EΓSM2⇔x2jx2b=0⇔j+b≥n+1.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

Ifm is even (n is even or odd),Im,n: the vertex x1m,x2n is linked to every vertex x1i,x2j1≤i≤m−1,1≤j≤n−1Im,n−1: the vertex x1m,x2n−1 is linked to every vertex x1i,x2j1≤i≤m−1,2≤j≤n,j≠n−1Im,n−2: the vertex x1m,x2n−2 is linked to every vertex x1i,x2i1≤i≤m−1,3≤j≤n,j≠n−2⋮Im,1: the vertex x1m,x21 is linked to every vertex x1i,x2j1≤i≤m−1,j=nIm−1,n: the vertex x1m−1,x2n is linked to every vertex x1i,x2j2≤i≤m−2,1≤j≤n−1⋮Im/2+1,1: the vertex x1m/2+1,x2 is linked to x1m/2,x2nIfm is odd and n is even,Im−1/2+2,1: the vertex x1m−1/2+2,x21 is linked to x1m−1/2,x2n and x1m−1/2+1,x2nIfm is odd and n is odd,Im−1/2+2,1: the vertex x1m−1/2+2,x21 is linked to x1m−1/2,x2n and x1m−1/2+1,x2nIn the following lemma, the vertice degrees(8)x1,x2,x12,x2,…,x1m,x2,x1,x22,x12,x22,…,x1m,x22,…,x1,x2n−1,x12,x2n−1,…,x1m,x2n−1,x1,x2n,x12,x2n,…,x1m,x2n∈ΓSM1⊗ΓSM2,are denoted by(9)d1,d1′,d2,d1′,…,dm,d1′,d1,d2′,d2,d2′,…,dm,d2′,…,d1,dn′,d2,dn′,…,dm,dn′.Several investigations exist on the degree series with respect to this series [1, 38].Lemma 1.
(10)d1,d1′=1,d2,d1′=2,…,dm/2,d1′=m2,dm/2+1,d1′=m2,…,dm,d1′=m−1,…,dm/2+1,dn/2′+1=m2n2.Remark 1.
Considering special observations in Lemma1, the recurrent phrases utilized that are as follows:(11)dn/2=n2=dn/2+1.
Consequently, the degree ofdn is noted by n−1, inspite of the amount of vertices is n.


## 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 semigroupSM1 and SM2, the Nirmala index over tensor product of two monogenic semigroup graph ΓSM1⊗ΓSM2 is(12)NΓSM1⊗ΓSM2=∑t=m/2+1m∑r=n/2+1n∑b=1m/2∑i=1n/2t−1r−1+bi+∑t=m/2+1m∑r=n/2+1n∑b=1m/2∑i=n/2+1n/2t−1r−1+bi−1+∑t=m/2+1m∑r=n/2+1n∑b=m/2+1m∑i=1n/2t−1r−1+b−1i+∑t=m/2+1m∑r=n/2+1n∑b=m/2+1m∑i=n/2+1nt−1r−1+b−1i−1+∑t=m/2+1m∑r=1n/2∑b=1m/2∑i=n/2+1nt−1r+bi−1+∑t=m/2+1m∑r=1n/2∑b=m/2+1m∑i=n/2+1nt−1r+b−1i−1.
In the formula given above, the numberst, b and r, m will be taken in accordance with the rules t+b≥m+1 and r+i≥n+1.Proof.
Our primary focus is to methodically formuliseSOΓSM1⊗ΓSM2 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.
Ifm and n are even,(13)NΓSM1⊗ΓSM2=dm−1dn′−1+d1d1′+dm−1dn′−1+d1d2′+dm−1dn′−1+d1dn/2′+dm−1dn′−1+d1dn/2′+1−1+dm−1dn′−1+d1dn′−1−1+dm−1dn′−1+d2d1′+⋯+dm−1dn′−1+d2d2′+⋯+dm−1dn′−1+d2dn/2′+dm−1dn′−1+d2dn/2′+1−1+⋯+dm−1dn′−1+d2dn′−1−1+⋯+dm−1dn′−1+dm/2+1−1d1′+dm−1dn′−1+dm/2+1−1d2′+⋯+dm−1dn′−1+dm/2+1−1dn/2′+dm−1dn′−1+dm/2+1−1dn/2′+1−1+⋯+dm−1dn′−1+dm/2+1−1dn′+1−1+⋯+dm−1dn′−1+dm−1−1d1′+⋯+dm−1dn′−1+dm−1−1d2′+⋯+dm−1dn′−1+dm−1−1dn/2′+dm−1dn′−1+dm−1−1dn/2′+1−1+⋯+dm−1dn′−1+dm−1−1dn′−1−1+dm−1dn′−1−1+d1d2′+dm−1dn′−1−1+d1d3′+⋯+dm−1dn′−1−1+d1dn/2′+dm−1dn′−1−1+d1dn/2′+1−1+⋯+dm−1dn′−1−1+d1dn′−2−1+dm−1dn′−1−1+d2d2′+⋯+dm−1dn′−1−1+d2d3′+⋯+dm−1dn′−1−1+d2dn/2′+dm−1dn′−1−1+d2dn/2′+1−1+⋯+dmdn′−1−1+d2dn′−2−1+⋯+dm−1dn′−1−1+dm/2+1−1d2′+dm−1dn′−1−1+dm/2+1−1d3′+⋯+dm−1dn′−1−1+dm/2+1−1dn/2′+dm−1dn′−1−1+dm/2+1−1dn/2+1−1′+⋯+dm−1dn′−1−1+dm/2+1−1dn′−2−1+⋯+dm/2+1−1d1′+dm/2dn′.
Consequently, the Nirmala indexNΓSM1⊗ΓSM2 is noted as the sum as follows:(14)NΓSM1⊗ΓSM2=NΓSM1⊗ΓSM2m,n+NΓSM1⊗ΓSM2m,n−1+⋯+NΓSM1⊗ΓSM2m,1+NΓSM1⊗ΓSM2m−1,n+NΓSM1⊗ΓSM2m−1,n−1+⋯+NΓSM1⊗ΓSM2m−1,1+NΓSM1⊗ΓSM2m/2+1,1.
Whilst estimating the Nirmala index value, the minutest amount is acquired after several calculations wheren is odd we utilize the equation n/2=n+1/2 given in (6).(15)NΓSM1⊗ΓSM2m,n=m−1n−1+1⋅1+m−1n−1+1⋅2+⋯+m−1n−1+1⋅n−1+m−1n−1+2⋅1+m−1n−1+2⋅2+⋯+m−1n−1+2⋅n−1+⋯+m−1n−1+m2+1−1⋅1+m−1n−1+m2+1−1⋅2+⋯+m−1n−1+m2+1−1⋅n2+m−1n−1+m2+1−1⋅n2+1−1+⋯+m−1n−1+m2+1−1⋅n−1−1+⋯+m−1n−1+m−1−1⋅1+m−1n−1+m−1−1⋅2+⋯+m−1n−1+m−1−1⋅n2+m−1n−1+m−1−1⋅n2+1−1+⋯+m−1n−1+m−1−1⋅n−1−1.
The above equation can be given briefly with the sum symbol as follows:(16)NΓSM1⊗ΓSM2m,n=∑b=1m/2∑i=1n/2m−1n−1+bi+∑b=1m/2∑i=n/2+1n−1m−1n−1+bi−1+∑b=m/2+1m−1∑i=1n/2m−1n−1+b−1i+∑b=m/2+1m−1∑i=n/2+1n−1m−1n−1+b−1i−1.
If similar operations applied inNΓSM1⊗ΓSM2m,n are applied to NΓSM1⊗ΓSM2m,n−1, we obtain(17)NΓSM1⊗ΓSM2m,n−1=∑b=1m/2∑i=2n/2m−1n−1−1+bi+∑b=1m/2∑i=n/2+1nm−1n−1+bi−1+∑b=m/2+1m−1∑i=2n/2m−1n−1+b−1i+∑b=m/2+1m−1∑i=n/2+1n−1m−1n−1+b−1i−1,i≠n−1.
If it is continued in this way, the following equations are obtained forNΓSM1⊗ΓSM2m,n/2+1, NΓSM1⊗ΓSM2m,n/2, and NΓSM1⊗ΓSM2m,1, respectively:(18)NΓSM1⊗ΓSM2m,n/2=∑b=1m/2∑i=n/2+1nm−1n2+bi−1+∑b=m/2+1m−1∑i=n/2+1nm−1n2+b−1i−1,NΓSM1⊗ΓSM2m,1=∑b=1m/2m−112+bn−1+∑b=m/2+1m−1m−11+b−1n−1.
In this way,NΓSM1⊗ΓSM2m−1,n, NΓSM1⊗ΓSM2m−1,n−1,…,NΓSM1⊗ΓSM2m−1,1,…,NΓSM1⊗ΓSM2m/2+1,n, and NΓSM1⊗ΓSM2m/2+1,n−1, …, NΓSM1⊗ΓSM2m/2+1,1 are calculated one by one to obtain a general sum formula given as follows:(19)NΓSM1⊗ΓSM2=∑t=m/2+1m∑r=n/2+1n∑b=1m/2∑i=1n/2t−1r−1+bi+∑t=m/2+1m∑r=n/2+1n∑b=1m/2∑i=n/2+1nt−1r−1+bi−1+∑t=m/2+1m∑r=n/2+1n∑b=m/2+1m∑i=1n/2t−1r−1+b−1i+∑t=m/2+1m∑r=n/2+1n∑b=m/2+1m∑i=n/2+1nt−1r−1+b−1i−1+∑t=m/2+1m∑r=1n/2∑b=1m/2∑i=n/2+1nt−1r+bi−1+∑t=m/2+1m∑r=1n/2∑b=m/2+1m∑i=n/2+1nt−1r+b−1i−1.


## 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 semigroupSM1 and SM2, the Sombor index over tensor product of two monogenic semigroup graph ΓSM1⊗ΓSM2 is(20)BSO1ΓSM1⊗ΓSM2=∑t=m/2+1m∑r=n/2+1n∑b=1m/2∑i=1n/21t−1r−12+bi2+∑t=m/2+1m∑r=n/2+1n∑b=1m/2∑i=n/2+1n1t−1r−12+bi−12+∑t=m/2+1m∑r=n/2+1n∑b=m/2+1m∑i=1n/21t−1r−12+b−1i2+∑t=m/2+1m∑r=n/2+1n∑b=m/2+1m∑i=n/2+1n1t−1r−12+b−1i−12+∑t=m/2+1m∑r=1n/2∑b=1m/2∑i=n/2+1n1t−1r2+bi−12+∑t=m/2+1m∑r=1n/2∑b=m/2+1m∑i=n/2+1n1t−1r2+b−1i−12.
In the formula given above, the numberst, b and r, m will be taken in accordance with the rules t+b≥m+1 and r+i≥n+1.Proof.
The proof is similar to the calculation of the Nirmala index of the tensor product of monogenic semigroup graphs, as in Theorem1.


## 5. An Algorithm for Cartesian Product of Monogenic Semigroup Graph

Ifm is even (n is even or odd),Im,n: the vertex x1m,x2n is linked to x1i,x2n and x1m,x2j1≤i≤m−1,1≤j≤n−1Im,n−1: the vertex x1m,x2n−1 is linked to x1i,x2n−1 and x1m,x2j1≤i≤m−1,2≤j≤n−2Im,n−2: the vertex x1mx2n−2 is linked to x1i,x2n−2 and x1m,x2j1≤i≤m−1,3≤j≤n−3⋮Im,1: the vertex x1m,x21 is linked to x1i,x211≤i≤m−1Im−1,n: the vertex x1m−1,x2n is linked to x1i,x2n and 2≤i≤m−2,1≤j≤n−1. x1m−1,x2j⋮Im/2+1,1: the vertex x1m/2+1,x21 is linked to x1m/2,x21Ifm is odd (n is even or odd),Im−1/2+2,1: the vertex x1m−1/2+2,x21 is linked to x1i,x21m−1/2≤i≤m−1/2+1


## 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 semigroupSM1 and SM2, the Nirmala index over Cartesian product of two monogenic semigroup graph ΓSM1×ΓSM2 is(21)NΓSM1⊗ΓSM2=∑t=m/2+1m∑b=1m/2∑r=n/2+1nt−1+r−1+b+r−1+∑t=m/2+1m∑b=1m/2∑r=1n/2t−1+r+b+r+∑t=m/2+1m∑r=n/2+1n∑i=1n/2t−1+r−1+t−1+i+∑t=m/2+1m∑i=n/2+1n∑r=n/2+1nt−1+r−1+t−1+i−1+∑t=m/2+1m∑r=1n/2∑b=m/2+1m−1t−1+r+b−1+r+∑t=1m/2∑r=n/2+1n∑i=n/2+1nt+r−1+t+i−1.In the formula given above, the numberst, b and r, m will be taken in accordance with the rules t+b≥m+1, t≻b, and r+i≥n+1, r≻i.Proof.
Since our primary focus is to formuliseNΓSM1×ΓSM2 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.
Ifm and n is even,(22)NΓSM1×ΓSM2=dm−1+dn′−1+d1+dn′−1+dm−1+dn′−1+d2+dn′−1+⋯+dm−1+dn′−1+dm/2+dn′−1+dm−1+dn′−1+dm/2+1−1+dn′−1+⋯+dm−1+dn′−1+dm−1−1+dn′−1+dm−1+dn′−1+dm−1+d1′+dm−1+dn′−1+dm−1−1+dn/2′+dm−1+dn′−1+dm−1+dn/2+1′−1+⋯+dm−1+dn′−1+dm−1+dn−1′−1+dm−1+dn−1′−1+d1+dn−1′−1+dm−1+dn−1′−1+d2+dn−1′−1+⋯+dm−1+dn−1′−1+dm/2+dn−1′−1+dm−1+dn−1′−1+dm/2+1−1+dn−1′−1+⋯+dm−1+dn−1′−1+dm−1−1+dn−1′−1+dm−1+dn−1′−1+dm−1+d2′+⋯+dm−1+dn−1′−1+dm−1+dn−2′−1+⋯+dm−1+dn/2′+d1+dn/2′+dm−1+dn/2′+d2+dn/2′+⋯+dm−1+dn/2′+dm−1−1dn−1′−1+dm−1+dn′−1+dm−1−1+d1′+dm−1+dn′−1+dm−1−1+d2′+⋯+dm−1−1+dn′−1+dm−1−1+dn−1′−1+dm−1−1+dn−1′−1+d2+dn−1′−1+dm−1−1+dn−1′−1+d3+dn−1′−1+⋯+dm−1−1+dn−1′−1+dm/2+1−1+dn−1′−1+⋯+dm−1−1+dn−1′−1+dm−1−1+d2′+⋯+dm−1−1+dn−1′−1+dm−1−1+dn/2′+⋯+dm−1−1+dn/2′+d2+dn/2′+dm−1−1+dn/2′+d3+dn/2′+⋯+dm−1−1+dn/2′+dm−2−1+dn/2′+dm−1−1+d1′+d2+d1′+dm−1−1+d1′+d3+d1′+⋯+dm−1−1+d1′+dm−2−1+d1′+dm/2+1−1+dn′−1+dm/2+dn′−1+dm/2+1−1+dn′−1+dm/2+1−1+d1′+dm/2+1−1+dn′−1+dm/2+1−1+d2′+⋯+dm/2+1−1+dn′−1+dm/2+1−1+dn−1′−1+dm/2+1−1+dn−1′−1+dm/2+dn−1′−1+dm/2+1−1+dn−1′−1+dm/2+1−1+d2′+⋯+dm/2+1−1+dn−1′−1+dm/2+1−1+dn−2′−1+dm/2+1−1+dn/2′+dm/2+dn/2′+⋯+dm/2+1−1+d1′+dm/2+d1′+dm/2+dn′−1+dm/2+d1′+⋯+dm/2+dn′−1+dm/2+dn/2′+⋯+dm/2+dn′−1+dm/2+dn−1′−1+dm/2+dn−1′−1+dm/2+d2′+⋯+dm/2+dn−1′−1+dm/2+dn−2′−1+d2+dn′−1+d2+d1′+d2+dn′−1+d2+d2′+⋯+d2+dn′−1+d2+dn−1′−1+d2+dn−1′−1+d2+dn−2′−1+⋯+d2+dn−1′−1+d2+dn/2′+d1+dn′−1+d1+d1′+d1+dn′−1+d1+d2′+⋯+d1+dn′−1+d1+dn−1′−1+d1+dn−1′−1+d1+d2′+⋯+d1+dn−1′−1+d1+dn−2′−1.
Consequently, the Nirmala index ofNΓSM1×ΓSM2 is stated as the sum as follows:(23)NΓSM1×ΓSM2=NΓSM1×ΓSM2m,n+NΓSM1×ΓSM2m,n−1+⋯+NΓSM1×ΓSM2m,1+NΓSM1×ΓSM2m−1,n+NΓSM1×ΓSM2m−1,n−1+⋯+NΓSM1×ΓSM2m−1,1+NΓSM1×ΓSM21,n+⋯+NΓSM1×ΓSM21,n/2+1.
Whilst estimating the Nirmala index value, the minutest amount is acquired after several calculations wheren is odd; we utilize the equation n/2=n+1/2 given in (6).(24)NΓSM1×ΓSM2m,n=m−1+n−1+1+n−1+m−1+n−1+2−2+n−1+⋯+m−1+n−12+m2+n−12+m−1+n−1+m2+1−1+n−1+⋯+m−1+n−12+m−1−1+n−12+m−1+n−12+m−1+12+⋯+m−1+n−12+m−1+n22+m−1+n−1+m−1+n2+1−1.
The above equation can be given briefly with the sum symbol as follows:(25)NΓSM1×ΓSM2m,n=∑b=1m/2m−1+n−1+b+n−1+∑b=m/2+1m−1m−1+n−1+b−1+n−1+∑i=1n/2m−1+n−1+m−1+i+∑i=n/2+1n−1m−1+n−1+m−1+i−1.
If similar operations applied inNΓSM1×ΓSM2m,n are applied in NΓSM1×ΓSM2m,n−1, we obtain(26)NΓSM1×ΓSM2m,n−1=∑b=1m/2m−1+n−1+b+n−1−1+∑m/2+1m−1m−1+n−1−2+b−1+n−1−1+∑m/2+1m−1∑i=2n/2m−1+n−1+b−1+i+∑i=1n/2m−1+n−1−1+m−1+i+∑i=n/2+1n−2m−1+n−1−1+m−1+i−1.
If it is continued in this way, the following equations are obtained forNΓSM1×ΓSM2m,n/2+1, NΓSM1×ΓSM2m,n/2, and NΓSM1×ΓSM2m,1, respectively.(27)NΓSM1×ΓSM2m,n/2=∑b=1m/2m−1+n2−n22+b−b+n2−n22,NΓSM1×ΓSM2m,1=∑b=1m/2m−1+1+b+1+∑b=m/2+1m−1m−1+1+b−1+n−1.
In this way,NΓSM1×ΓSM2m−1,n, NΓSM1×ΓSM2m−1,n−1, …, NΓSM1×ΓSM2m−1,1, …, NΓSM1×ΓSM2m/2+1,n, NΓSM1×ΓSM2m/2+1,n−1, …, NΓSM1×ΓSM2m/2+1,1 are calculated one by one to obtain a general sum formula as follows:(28)NΓSM1×ΓSM2=∑t=m/2+1m∑b=1m/2∑r=n/2+1nt−1+r−1+b+r−1+∑t=m/2+1m∑b=1m/2∑r=1n/2t−1+r+b+r+∑t=m/2+1m∑r=n/2+1n∑b=m/2+1mt−1+r−1+b−1+r−1+∑t=m/2+1m∑r=n/2+1n∑i=1n/2t−1+r−1+t−1+i+∑t=m/2+1m∑i=n/2+1n∑r=n/2+1nt−1+r−1+t−1+i−1+∑t=m/2+1m∑r=1n/2∑b=m/2+1m−1t−1+r+b−1+r∑t=1m/2∑r=n/2+1n∑i=1n/2t+r−1+t+i∑t=1m/2∑r=n/2+1n∑i=n/2+1nt+r−1+t+i−1.


## 7. Computing Banhatti-Sombor Index of Cartesian Product of Monogenic Semigroup Graphs

In this section, by utilizing the algorithm given in Section 5, the formula of Banhatti-Sombor index of Cartesian product over monogenic semigroup graphs will be calculated.Theorem 4.
For any monogenic semigroupSM1 and SM2, the first Banhatti-Sombor index over Cartesian product of two monogenic semigroup graph ΓSM1×ΓSM2 is(29)BSO1ΓSM1×ΓSM2=∑t=m/2+1m∑b=1m/2∑r=n/2+1n1t−1+r−12+b+r−12+∑t=m/2+1m∑b=1m/2∑r=1n/21t−1+r2+b+r2+∑t=m/2+1m∑r=n/2+1n∑b=m/2+1m1t−1+r−12+b−1+r−12+∑t=m/2+1m∑r=n/2+1n∑i=1n/21t−1+r−12+t−1+i2+∑t=m/2+1m∑i=n/2+1n∑r=n/2+1n1t−1+r−12+t−1+i−12+∑t=m/2+1m∑r=1n/2∑b=m/2+1m−11t−1+r2+b−1+r2+∑t=1m/2∑r=n/2+1n∑i=1n/21t+r−12+t+i2+∑t=1m/2∑r=n/2+1n∑i=n/2+1n1t+r−12+t+i−12.
In the formula given above, the numberst, b and r, m will be taken in accordance with the rules t+b≥m+1, t≻b, and r+i≥n+1, r≻i.Proof.
The proof is similar to the calculation of the Nirmala index of the Cartesian product of monogenic semigroup graphs, as in Theorem3.
The examples given below are showing the calculation of the Nirmala and Banhatti-Sombor index of the tensor product and Cartesian product ofΓSM6 and ΓSM4 to support the main theorems.Example 1.
LetSM6 and SM4 be a monogenic semigroup given as follows:(30)SM6=x,x2,x3,x4,x5,x6∪0,SM4=x,x2,x3,x4∪0.
Now, we will calculate the Nirmala and Banhatti-Sombor index ofΓSM6⊗ΓSM4 graph by using the technique given in Theorems 1 and 2, respectively. In Figures 1 and 2, the monogenic semigroup graphs of (SM6) and SM4 are given, respectively. In Figure 3, the tensor product of the SM6 and SM4 monogenic semigroup graphs is given.
For the Nirmala index ofΓSM1⊗ΓSM2, we write the formula which is given in Theorem 1.(31)NΓSM1⊗ΓSM2=∑t=46∑r=34∑b=13∑i=12t−1r−1+bi+∑t=46∑r=34∑b=13∑i=34t−1r−1+bi−1+∑t=46∑r=34∑b=46∑i=12t−1r−1+b−1i+∑t=46∑r=34∑b=46∑i=34t−1r−1+b−1i−1+∑t=46∑r=12∑b=13∑i=34t−1r+bi−1+∑t=46∑r=12∑b=46∑i=34t−1r+b−1i−1.
By using the formula given above, we obtain(32)NΓSM1⊗ΓSM2=4+6+24+9+5+3+5+6+25+9+25+9+25+12+26+6+28+4+68+6+58+9+29+3+69+6+210+2+210+3+210+4+610+6+210+8+410+9+10+12+12+2+212+3+212+4+412+6+15+1+315+2+15+3+315+4+315+6+215+8.
For the first Banhatti-Sombor index ofΓSM1⊗ΓSM2, we write the formula which is given in Theorem 2.(33)BSO1ΓSM1⊗ΓSM2=∑t=46∑r=34∑b=13∑i=12t−1r−12+bi2+∑t=46∑r=34∑b=13∑i=34t−1r−12+bi−12+∑t=46∑r=34∑b=46∑i=12t−1r−12+b−1i2+∑t=46∑r=34∑b=46∑i=34t−1r−12+b−1i−12+∑t=46∑r=12∑b=13∑i=34t−1r2+bi−12+∑t=46∑r=12∑b=46∑i=34t−1r2+b−1i−12.
By using the formula given above, we obtain(34)BSO1ΓSM1⊗ΓSM2=142+62+2142+92+152+32+152+62+2152+92+2152+92+2152+122+2162+62+2182+42+6182+62+5182+92+2192+32+6192+62+21102+22+21102+32+21102+42+61102+62+21102+82+41102+92+1102+122+1122+22+21122+32+21122+42+41122+62+1152+12+31152+22+1152+32+31152+42+31152+62+21152+82.Figure 1
SM6 monogenic semigroup graph.Figure 2
SM4 monogenic semigroup graph.Figure 3
Tensor product ofSM6 and SM4 monogenic semigroup graphs.Example 2.
Cartesian product of the graphsΓSM1 and ΓSM2 is given below. By using the formula given in Theorems 3 and 4, we will calculate the Nirmala and first Banhatti-Sombor index of ΓSM1×ΓSM2, respectively. Cartesian product of SM6 and SM4 monogenic semigroup graphs is given in Figure 4.
For the Nirmala index over Cartesian product ofSM6 and SM4 monogenic semigroup graphs, we use the following formula given in Theorem 3:(35)NΓSM1×ΓSM2=∑t=46∑b=13∑r=34t−1r−1+b+r−1+∑t=46∑b=13∑r=12t−1+r+b+r+∑t=46∑r=34∑b=46t−1+r−1+b−1+r−1+∑t=46∑r=34∑i=12t−1+r−1+t−1+i+∑t=46∑i=34∑r=34t−1+r−1+t−1+i−1+∑t=46∑r=12∑b=45t−1+r+b−1+r+∑t=13∑r=34∑i=12t+r−1+t+i+∑t=13∑r=34∑i=34t+r−1+t+i−1.
By utilizing the formula given above, we have(36)NΓSM1×ΓSM2=3+3+4+2+24+3+24+4+25+3+45+4+45+5+6+2+26+3+66+4+96+5+27+3+27+4+67+5+67+6+7+7+8+4+8+5+38+6+38+7.
For the first Banhatti-Sombor index over Cartesian product ofSM6 and SM4 monogenic semigroup graphs, we use the following formula given in Theorem 4:(37)SOΓSM1×ΓSM2=∑t=46∑b=13∑r=341t−1r−12+b+r−12+∑t=46∑b=13∑r=121t−1+r2+b+r2+∑t=46∑r=34∑b=461t−1+r−12+b−1+r−12+∑t=46∑r=34∑i=121t−1+r−12+t−1+i2+∑t=46∑i=34∑r=341t−1+r−12+t−1+i−12+∑t=46∑r=12∑b=451t−1+r2+b−1+r2+∑t=13∑r=34∑i=121t+r−12+t+i2+∑t=13∑r=34∑i=341t+r−12+t+i−12.
By utilizing the formula given above, we have(38)SOΓSM1×ΓSM2=132+32+142+22+1242+32+1242+42+1252+32+1452+42+1452+52+162+22+1262+32+1662+42+1962+52+1272+32+1272+42+1672+52+1672+62+172+72+182+42+182+52+1382+62+1382+72.
As will be understood, the Nirmala and Banhatti-Sombor index of a graph of tensor and Cartesian product of two monogenic semigroups is found easily by considering the exact formula obtained in the main theorems.Figure 4
Cartesian product ofSM6 and SM4 monogenic semigroup graphs.


## 8. Conclusion

In this work, we connected the theory of two Sombor-type indices with abstract algebra. In particular, we computed these over a graph of monogenic semigroups. A method is provided (Theorems1–4), by means of which the tensor and Cartesian products of a monogenic semigroup graph of the respective indices can be calculated. Also, pertinent algorithms are designed.



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