This paper was converted on www.awesomepapers.org from LaTeX by an anonymous user.
Want to know more? Visit the Converter page.

ON THE SOMBOR INDEX OF THE TOTAL GRAPH AND THE UNIT GRAPH OF COMMUTATIVE RINGS

ABHISHEK VAIBHAV PATHAK St Joseph’s University,36, Langford Rd, Langford Gardens, Bengaluru, Karnataka 560027 [email protected] ,  ANUKUL SACHAN St Joseph’s University,36, Langford Rd, Langford Gardens, Bengaluru, Karnataka 560027 [email protected]  and  Raisa DSouza St Joseph’s University,36, Langford Rd, Langford Gardens, Bengaluru, Karnataka 560027 [email protected]
Abstract.

In this paper, we investigate the Sombor index of the total graph and unit graph of ℤn\mathbb{Z}_{n} which is denoted by TΓ​(ℤn)T_{\Gamma}(\mathbb{Z}_{n}) and G​(ℤn)G(\mathbb{Z}_{n}) respectively for n∈{2​k,pα,p​q,p2​q}n\in\{2k,p^{\alpha},pq,p^{2}q\} where pp and qq are distinct odd prime numbers such that p<qp<q. Moreover, we compute the Sombor index of any finite local ring.

Key words and phrases:
Sombor Index, Total Graph, Unit Graph, Local Rings, Topological Indices
2020 Mathematics Subject Classification:
Primary 05C50

1. Introduction

Algebra and Graph Theory are disciplines of mathematics that focus on building and investigating certain structures. In recent years, mathematicians have constructed many graphs from algebraic structures, considering different properties inherited by these structures. The properties of graphs have been studied to understand the algebraic structures better.

An edge connecting vertices uu and vv is denoted by u​vuv in a graph G.G. In the mathematical and chemical literature, many vertex–degree–based or distance based graph invariants (usually referred to as “topological indices”) have been introduced and extensively studied [1]. The general formula of a vertex-degree-based topological index is,

T​I​(G)=∑u​v∈E​(G)H​(du,dv)TI(G)=\sum_{uv\in E(G)}H(d_{u},d_{v})

where H​(x,y)H(x,y) is some real valued function with the property H​(x,y)=H​(y,x)H(x,y)=H(y,x) and dxd_{x} is the degree of vertex x.x. Similarly the general formula for a distance based topological index is,

T​I​(G)=∑{u,v}⊆V​(G)H​(d​(u,v))TI(G)=\sum_{\{u,v\}\subseteq V(G)}H(d(u,v))

where d​(u,v)d(u,v) is the distance between the vertices uu and v.v. A topological index of GG is a real number associated with G.G. It is independent of a graph’s labeling (or pictorial representation) of a graph. In 2021 mathematical chemist Ivan Gutman introduced the Sombor Index [2] of a graph as defined below

S​O​(G)=∑u​v∈E​(G)du2+dv2.SO(G)=\sum_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}.}

Further in the same year Saeid Alikhani and Nina Ghanbari [3] gave the formulae for the Sombor Index of cycle graphs, bipartite graphs, and wheel graphs. They also presented theorems concerning the aforementioned index, which are utilized to generalize the Sombor Index for various graphs, including those with ring structures. In 2022, Arif, Alper, and Necla [4] studied the Sombor index of zero-divisor graphs of ℤn\mathbb{Z}_{n}.

In this paper, we study the Sombor index of the total graph and unit graph of some commutative rings. In Section 2, we recall some basic definitions, notations, and results which will be used throughout the paper. Also, we calculate the Sombor index of the total graph and the unit graph of ℤn\mathbb{Z}_{n} in Section 3 and Section 4 respectively. In Section 5, we explore the Sombor index the total graph, and the unit graph of any finite local ring. We establish a relationship between the complete graph, kk - regular graph, and its complement in Section 6.

2. PRELIMINARIES

In this section, we recall some basic definitions, notations, and results which will be used throughout the paper. We will assume all rings are commutative with unity. Let Z​(R)Z(R) denote the set of zero-divisors and U​(R)U(R) denote the set of units of the ring R.R.

Definition 2.1.

The total graph [5] of the ring R,R, denoted TΓ​(R),T_{\Gamma}(R), is the graph obtained by setting all the elements of RR to be the vertices and distinct vertices xx and yy are adjacent if and only if x+y∈Z​(R).x+y\in Z(R).

The degree of each vertex in a total graph can easily be determined. It was given by T.Asir and Thirugnanam Tamizh Chelvam [6], among other results. We state the results below for completeness.

Theorem 2.2.

[6] Let RR be a ring. The following statements hold for the total graph of RR

  1. (1)

    If 2∉U​(R)2\notin U(R), dx=n−|U​(R)|−1d_{x}=n-\left|U(R)\right|-1, for every x∈Rx\in R

  2. (2)

    If 2∈U​(R)2\in U(R), dx=n−|U​(R)|−1d_{x}=n-\left|U(R)\right|-1, for every x∈Z​(R)x\in Z(R) and dx=n−|U​(R)|,d_{x}=n-\left|U(R)\right|, for every x∉Z​(R).x\notin Z(R).

Theorem 2.3.

[6, Theorem 2.5] The following statements hold for the total graph of ℤn:\mathbb{Z}_{n}:

  1. (1)

    If nn is even then dv=n−ϕ​(n)−1,d_{v}=n-\phi(n)-1, for every v∈ℤn.v\in\mathbb{Z}_{n}.

  2. (2)

    If nn is odd then dv=n−ϕ​(n)−1,d_{v}=n-\phi(n)-1, for every v∈Z​(ℤn)v\in Z(\mathbb{Z}_{n}) and dv=n−ϕ​(n),d_{v}=n-\phi(n), for every v∉Z​(ℤn),v\notin Z(\mathbb{Z}_{n}), where, ϕ\phi is the Euler function.

Definition 2.4.

The unit graph [7] of R,R, denoted G​(R),G(R), is the graph obtained by setting all the elements of RR to be the vertices and distinct vertices xx and yy are adjacent if and only if x+y∈U​(R).x+y\in U(R).

Analogous to Theorem 2.2, in [8] gave the degrees of vertices in G​(R).G(R).

Theorem 2.5.

Let RR be a ring. The following statements hold for the unit graph of RR

  1. (1)

    If 2∉U​(R)2\notin U(R), dx=|U​(R)|d_{x}=\left|U(R)\right|, for every x∈Rx\in R

  2. (2)

    If 2∈U​(R)2\in U(R), dx=|U​(R)|−1d_{x}=\left|U(R)\right|-1, for every x∈Z​(R)x\in Z(R) and dx=|U​(R)|,d_{x}=\left|U(R)\right|, for every x∉Z​(R).x\notin Z(R).

Theorem 2.6.

[9, Proposition 2.4] Then the following statements hold for the unit graph of ℤn:\mathbb{Z}_{n}:

  1. (1)

    If 2∉U​(ℤn),2\notin U(\mathbb{Z}_{n}), then dx=ϕ​(n)d_{x}=\phi(n) for every x∈ℤn.x\in\mathbb{Z}_{n}.

  2. (2)

    If 2∈U​(ℤn),2\in U(\mathbb{Z}_{n}), then dx=ϕ​(n)−1d_{x}=\phi(n)-1 for every x∈U​(ℤn)x\in U(\mathbb{Z}_{n}) and dx=ϕ​(n)d_{x}=\phi(n) for every x∈ℤn\U​(ℤn).x\in\mathbb{Z}_{n}\backslash U(\mathbb{Z}_{n}).

Proposition 2.7.

[10] Let RR be a finite ring, then Z​(TΓ​(R))Z(T_{\Gamma}(R)) is a complete graph if and only if RR is a local ring.

3. SOMBOR INDEX OF THE TOTAL GRAPH OF ℤn\mathbb{Z}_{n}

Recently, the total graph of the ring ℤn\mathbb{Z}_{n} has gained popularity in spectral and chemical graph theory. Many researchers have focused on this area, contributing to its advancement and exploration. Sheela Suthar and Om Prakash have examined the energy and Wiener index of the total graph TΓ(ℤnT_{\Gamma}(\mathbb{Z}_{n}) [11]. In this section, we analyze the Sombor index the total graph of ℤn\mathbb{Z}_{n}.

Theorem 3.1.

Let pp be a prime number.The following holds for the total graph of ℤn:\mathbb{Z}_{n}:

  1. (1)

    If nn is even, then S​O​(TΓ​(ℤn))=n​(n−ϕ​(n)−1)22.SO(T_{\Gamma}(\mathbb{Z}_{n}))=\frac{n(n-\phi(n)-1)^{2}}{\sqrt{2}}.

  2. (2)

    If n=pα,n=p^{\alpha}, where α≥1\alpha\geq 1 and p>2,p>2, then

    S​O​(TΓ​(ℤn))=ϕ​(n)​(n−ϕ​(n))22+(n−ϕ​(n)−1)2​(n−ϕ​(n))2.\displaystyle SO(T_{\Gamma}(\mathbb{Z}_{n}))=\frac{\phi(n)(n-\phi(n))^{2}}{\sqrt{2}}+\frac{(n-\phi(n)-1)^{2}(n-\phi(n))}{\sqrt{2}}.
Proof.
  1. (1)

    If nn is even, TΓ​(ℤn)T_{\Gamma}(\mathbb{Z}_{n}) is a n−ϕ​(n)−1n-\phi(n)-1 regular graph with nn vertices and the result follows.

  2. (2)

    For n=pα,n=p^{\alpha}, by Theorem 2.3, if x∈Z​(ℤn)x\in Z(\mathbb{Z}_{n}) then, dx=n−ϕ​(n)−1d_{x}=n-\phi(n)-1 and if x∉Z​(ℤn)x\notin Z(\mathbb{Z}_{n}) then, dx=n−ϕ​(n).d_{x}=n-\phi(n). Now, we know that for n=pαn=p^{\alpha} the set of non-units of ℤn\mathbb{Z}_{n} forms a subgroup. Hence, any two non-units of ℤn\mathbb{Z}_{n} are adjacent. Therefore, number of edges between zero-divisors is,

    (n−ϕ​(n))​(n−ϕ​(n)−1)2.\frac{(n-\phi(n))(n-\phi(n)-1)}{2}.

    By handshake lemma, the total number of edges in TΓ​(ℤn)T_{\Gamma}(\mathbb{Z}_{n}) is given by,

    ϕ​(n)​(n−ϕ​(n))+(n−ϕ​(n))​(n−ϕ​(n)−1)2.\frac{\phi(n)(n-\phi(n))+(n-\phi(n))(n-\phi(n)-1)}{2}.

    Thus, the number of edges among units will be, ϕ​(n)​(n−ϕ​(n))2.\displaystyle{\frac{\phi(n)(n-\phi(n))}{2}}. The result follows.

∎

Theorem 3.2.

Let pp and qq be two odd prime numbers with p<q.p<q. Then, the Sombor index of the graph TΓ​(ℤp​q)T_{\Gamma}(\mathbb{Z}_{pq}) is given by,

S​O​(TΓ​(ℤp​q))=2​α​(p​q−ϕ​(p​q)−1)+β​(p​q−ϕ​(p​q)−1)2+(p​q−ϕ​(p​q))2+2​(|E|−α−β)​(p​q−ϕ​(p​q))SO(T_{\Gamma}(\mathbb{Z}_{pq}))=\sqrt{2}\alpha(pq-\phi(pq)-1)+\\ \beta\sqrt{(pq-\phi(pq)-1)^{2}+(pq-\phi(pq))^{2}}+\\ \sqrt{2}(\left|E\right|-\alpha-\beta)(pq-\phi(pq))

where,

α=p​(p−1)+q​(q−1)2,β=2​(p−1)​(q−1),|E|=(p​q−1)​(p+q−1)2.\displaystyle\alpha=\frac{p(p-1)+q(q-1)}{2},\quad\beta=2(p-1)(q-1),\quad\left|E\right|=\frac{(pq-1)(p+q-1)}{2}.
Proof.

Let Z​(ℤp​q)=Sp∪SqZ(\mathbb{Z}_{pq})=S_{p}\cup S_{q} where, Sp={0,p,2​p,⋯,(q−1)​p}S_{p}=\{0,p,2p,\cdots,(q-1)p\} and Sq={0,q,2​q,⋯,(p−1)​q}.S_{q}=\{0,q,2q,\cdots,(p-1)q\}. We observe that SpS_{p} and SqS_{q} are closed under the operation ⨁p​q.\bigoplus_{pq}. The fact that SpS_{p} and SqS_{q} are closed and Sp∩Sq=0S_{p}\cap S_{q}={0}, will imply that a≁ba\nsim b\, for all a∈Sp​and​b∈Sq.\,a\in S_{p}\,\text{and}\,b\in S_{q}. We thus get complete graphs KqK_{q} and KpK_{p} for the sets SpS_{p} and SqS_{q} respectively. Thus the number of edges between zero divisors and zero divisors are

α=q​(q−1)2+p​(p−1)2.\alpha=\frac{q(q-1)}{2}+\frac{p(p-1)}{2}.

Since pp and qq are odd primes, by Theorem 2.3 dv=p​q−ϕ​(p​q)−1,d_{v}=pq-\phi(pq)-1,\, for all v∈Z​(ℤp​q).\,v\in Z(\mathbb{Z}_{pq}). Thus,

dv\displaystyle d_{v} =p​q−ϕ​(p​q)−1\displaystyle=pq-\phi(pq)-1
=p​q−(p−1)​(q−1)−1\displaystyle=pq-(p-1)(q-1)-1
=p​q−p​q+p+q−1−1\displaystyle=pq-pq+p+q-1-1
=p+q−2.\displaystyle=p+q-2.

We notice that the degree of a vertex within SpS_{p} is, dvSp=q−1d^{S_{p}}_{v}=q-1\, for all v∈Sp\,v\in S_{p} and within SqS_{q} is, dvSq=p−1d^{S_{q}}_{v}=p-1\, for allv∈Sq.\,v\in S_{q}. So the remaining adjacencies of v∈Spv\in S_{p} or Sq\,S_{q}\, in TΓ​(ℤp​q)T_{\Gamma}(\mathbb{Z}_{pq}) is with the units. The excess degree of

v∈Sp​ is ​p+q−2−q+1=p−1v\in S_{p}\,\text{ is }\,p+q-2-q+1=p-1

and of

v∈Sq​ is ​p+q−2−p+1=q−1.v\in S_{q}\,\text{ is }\,p+q-2-p+1=q-1.

Next we notice that 0≁u0\nsim u\, for allu∈U​(ℤp​q).\,u\in U(\mathbb{Z}_{pq}). Thus the number of edges between zero divisors and units are

β\displaystyle\beta =(q−1)​(p−1)+(p−1)​(q−1)\displaystyle=(q-1)(p-1)+(p-1)(q-1)
=2​(p−1)​(q−1).\displaystyle=2(p-1)(q-1).

Using Theorem 2.3 and the handshake lemma we find the total number of edges in TΓ​(ℤp​q)T_{\Gamma}(\mathbb{Z}_{pq}) as,

|E|=12​(p​q−1)​(p+q−1).\displaystyle\left|E\right|=\frac{1}{2}(pq-1)(p+q-1).

Since we know the total number of edges in TΓ​(ℤp​q)​(|E|)T_{\Gamma}(\mathbb{Z}_{pq})\,\,(\left|E\right|), the number of edges between zero divisors and zero divisors (α)(\alpha) and the number of edges between zero divisors and units (β),(\beta), we can easily get the number of edges between units and units, which will be given by |E|−α−β.\left|E\right|-\alpha-\beta.

From Theorem 2.3 we know dv=p​q−ϕ​(p​q)−1​for all​v∈Z​(ℤp​q)d_{v}=pq-\phi(pq)-1\,\text{for all}\,v\,\in Z(\mathbb{Z}_{pq}) and du=p​q−ϕ​(p​q)d_{u}=pq-\phi(pq)\, for all u∈U​(ℤp​q).\,u\,\in U(\mathbb{Z}_{pq}). The result follows. ∎

Theorem 3.3.

Let pp and qq be two odd prime numbers with p<q.p<q. Then, the Sombor index of the graph TΓ​(ℤp2​q)T_{\Gamma}(\mathbb{Z}_{p^{2}q}) is

S​O​(TΓ​(ℤp2​q))=2​α​(p2​q−ϕ​(p2​q)−1)+β​(p2​q−ϕ​(p2​q)−1)2+(p2​q−ϕ​(p2​q))2+2​(|E|−α−β)​(p2​q−ϕ​(p2​q))SO(T_{\Gamma}(\mathbb{Z}_{p^{2}q}))=\sqrt{2}\alpha(p^{2}q-\phi(p^{2}q)-1)+\\ \beta\sqrt{(p^{2}q-\phi(p^{2}q)-1)^{2}+(p^{2}q-\phi(p^{2}q))^{2}}+\\ \sqrt{2}(\left|E\right|-\alpha-\beta)(p^{2}q-\phi(p^{2}q))

where,

α\displaystyle\alpha =p​(q−1)​(p​(q−1)−1)2+p​(p−1)​(p​(p−1)−1)2\displaystyle=\frac{p(q-1)(p(q-1)-1)}{2}+\frac{p(p-1)(p(p-1)-1)}{2}
+p​(p−1)2+p2​(q−1)+p2​(p−1),\displaystyle\hskip 130.88284pt+\frac{p(p-1)}{2}+p^{2}(q-1)+p^{2}(p-1),
β\displaystyle\beta =2​p2​(p−1)​(q−1),\displaystyle=2p^{2}(p-1)(q-1),\,
|E|\displaystyle\left|E\right| =p​(p+q−1)​(p2​q−1)2.\displaystyle=\frac{p(p+q-1)(p^{2}q-1)}{2}.
Proof.

Let Z​(ℤp​q)=Sp∪Sq∪Sp​qZ(\mathbb{Z}_{pq})=S_{p}\cup S_{q}\cup S_{pq} where, Sp={0,p,2​p,⋯,(p​q−1)​p}S_{p}=\{0,p,2p,\cdots,(pq-1)p\}, Sq={0,q,2​q,⋯,(p2−1)​q}S_{q}=\{0,q,2q,\cdots,(p^{2}-1)q\} and Sp​q={0,p​q,2​p​q,⋯,(p−1)​p​q}.S_{pq}=\{0,pq,2pq,\cdots,(p-1)pq\}. Now consider Sp∗=Sp\Sp​qS^{*}_{p}=S_{p}\backslash S_{pq} and Sq∗=Sp\Sp​q.S^{*}_{q}=S_{p}\backslash S_{pq}. We observe that Sp​q,S_{pq}, Sp∗S^{*}_{p} and Sq∗S^{*}_{q} are closed under ⨁p2​q.\bigoplus_{p^{2}q}. The fact that Sp∗S^{*}_{p} and Sq∗S^{*}_{q} are closed and Sp∗∩Sq∗=ϕS^{*}_{p}\cap S^{*}_{q}=\phi, will imply that a≁ba\nsim b\, for alla∈Sp∗​and​b∈Sq∗.\,a\in S^{*}_{p}\,\text{and}\,b\in S^{*}_{q}. We thus get complete graphs Kp,K_{p}, Kp​(q−1)K_{p(q-1)} and Kp​(p−1)K_{p(p-1)} for the sets Sp​q,Sp∗,Sq∗S_{pq},\,S^{*}_{p},\,S^{*}_{q}\, respectively. We also observe that a∼ba\sim b\, for alla∈Sp∗​and​b∈Sp​q\,a\in S^{*}_{p}\,\text{and}\,b\in S_{pq} and a∼ba\sim b\, for alla∈Sq∗​and​b∈Sp​q.\,a\in S^{*}_{q}\,\text{and}\,b\in S_{pq}. We thus get complete bipartite graphs Kp​(q−1),pK_{p(q-1),p}\, between the vertices of Sp∗S^{*}_{p} and Sp​qS_{pq} and Kp​(p−1),pK_{p(p-1),p}\, between the vertices of Sq∗S^{*}_{q} and Sp​qS_{pq}. Therefore the number of edges between zero-divisors and zero-divisors are,

α=p​(q−1)​(p​(q−1)−1)2+p​(p−1)​(p​(p−1)−1)2+p​(p−1)2+p2​(q−1)+p2​(p−1).\alpha=\frac{p(q-1)(p(q-1)-1)}{2}+\frac{p(p-1)(p(p-1)-1)}{2}+\frac{p(p-1)}{2}+p^{2}(q-1)+p^{2}(p-1).

Since pp and qq are odd primes, by Theorem 2.3 dv=p2​q−ϕ​(p2​q)−1,d_{v}=p^{2}q-\phi(p^{2}q)-1,\, for all v∈Z​(ℤp2​q).\,v\in Z(\mathbb{Z}_{p^{2}q}). Thus,

dv\displaystyle d_{v} =p​(p+q−1)−1.\displaystyle=p(p+q-1)-1.

We notice that dvSp∗=p​(q−1)−1+pd^{S^{*}_{p}}_{v}=p(q-1)-1+p\, for all v∈Sp\,v\in S_{p} and dvSq∗=p​(p−1)−1+pd^{S^{*}_{q}}_{v}=p(p-1)-1+p\, for allv∈Sq.\,v\in S_{q}. So the remaining adjacencies of v∈Sp∗v\in S^{*}_{p} or Sq∗\,S^{*}_{q}\, in TΓ​(ℤp2​q)T_{\Gamma}(\mathbb{Z}_{p^{2}q}) is with the units. The excess degree of

v∈Sp∗​ is ​p​(p+q−1)−1−(p​(q−1)−1+p)=p​(p−1)v\in S^{*}_{p}\,\text{ is }\,p(p+q-1)-1-(p(q-1)-1+p)=p(p-1)

and of

v∈Sq∗​ is ​p​(p+q−1)−1−(p​(p−1)−1+p)=p​(q−1).v\in S^{*}_{q}\,\text{ is }\,p(p+q-1)-1-(p(p-1)-1+p)=p(q-1).

Also the excess degree of v∈Sp​q​i​s​ 0v\in S_{pq}\,is\,0 as degree of any element in Sp​qS_{pq} is p2​q−ϕ​(p2​q)−1.p^{2}q-\phi(p^{2}q)-1. Thus the number of edges between zero-divisors and units are,

β\displaystyle\beta =p​(p−1)​p​(q−1)+p​(q−1)​p​(p−1)\displaystyle=p(p-1)p(q-1)+p(q-1)p(p-1)
=2​p2​(p−1)​(q−1).\displaystyle=2p^{2}(p-1)(q-1).

Using Theorem 2.3 and the handshake lemma we find the total number of edges in TΓ​(ℤp2​q)T_{\Gamma}(\mathbb{Z}_{p^{2}q}) as follows,

|E|\displaystyle\left|E\right| =12[ϕ(p2q)(p2q−ϕ(p2q))+(p2q−ϕ(p2q))(p2q−ϕ(p2q)−1])\displaystyle=\frac{1}{2}[\phi(p^{2}q)(p^{2}q-\phi(p^{2}q))+(p^{2}q-\phi(p^{2}q))(p^{2}q-\phi(p^{2}q)-1])
=12​p​(p+q−1)​(p2​q−1).\displaystyle=\frac{1}{2}p(p+q-1)(p^{2}q-1).

Since we know the total number of edges in TΓ​(ℤp​q)​(|E|)T_{\Gamma}(\mathbb{Z}_{pq})\,\,(\left|E\right|), the number of edges between zero divisors and zero divisors (α)(\alpha) and the number of edges between zero divisors and units (β),(\beta), we can easily get the number of edges between units and units, which will be given by |E|−α−β.\left|E\right|-\alpha-\beta.

From Theorem 2.3 we know dv=p​q−ϕ​(p​q)−1​for all​v∈Z​(ℤp​q)d_{v}=pq-\phi(pq)-1\,\text{for all}\,v\,\in Z(\mathbb{Z}_{pq}) and du=p​q−ϕ​(p​q)d_{u}=pq-\phi(pq)\, for all u∈U​(ℤp​q).\,u\,\in U(\mathbb{Z}_{pq}). The result follows.

∎

4. SOMBOR INDEX OF UNIT GRAPH OF ℤn\mathbb{Z}_{n}

An intriguing observation lies in the relationship between unit graphs and total graphs: they are complements of each other. In the preceding section, we delved into computing the Sombor index of total graphs. Now, in this section, we focus on exploring the Sombor index of unit graphs.

Theorem 4.1.

Let pp be a prime number. The following holds for unit graph of ℤn:\mathbb{Z}_{n}:

  1. (1)

    If nn is even, then S​O​(G​(ℤn))=n​(ϕ​(n))22.SO(G(\mathbb{Z}_{n}))=\frac{n(\phi(n))^{2}}{\sqrt{2}}.

  2. (2)

    If n=pα,n=p^{\alpha}, where α≥1\alpha\geq 1 and p>2p>2 then

    S​O​(G​(ℤn))=ϕ​(n)​(n−ϕ​(n))​(ϕ​(n))2+(ϕ​(n)−1)2+[ϕ​(n)​(ϕ​(n)−1)−(n−ϕ​(n))]​(ϕ​(n)−1)2.SO(G(\mathbb{Z}_{n}))=\phi(n)(n-\phi(n))\sqrt{(\phi(n))^{2}+(\phi(n)-1)^{2}}+\\ \frac{[\phi(n)(\phi(n)-1)-(n-\phi(n))](\phi(n)-1)}{\sqrt{2}.}
Proof.
  1. (1)

    For nn even, as 2∉U​(ℤn),2\notin U(\mathbb{Z}_{n}), the degree of each vertex is ϕ​(n)\phi(n) by Theorem 2.6. Thus, G​(ℤn)G(\mathbb{Z}_{n}) is a ϕ​(n)\phi(n)-regular graph with n​ϕ​(n)2\frac{n\phi(n)}{2} edges. The result follows.

  2. (2)

    For n=pα,2∈U​(ℤn).n=p^{\alpha},2\in U(\mathbb{Z}_{n}). By Theorem 2.6, if x∈U​(ℤn)x\in U(\mathbb{Z}_{n}), then dx=ϕ​(n)−1,d_{x}=\phi(n)-1, and if x∉U​(ℤn)x\notin U(\mathbb{Z}_{n}) then, dx=ϕ​(n).d_{x}=\phi(n). Now, we know that for n=pαn=p^{\alpha} the set of non-units of ℤn\mathbb{Z}_{n} forms a group under modulo addition. Hence, any two non-units of ℤn\mathbb{Z}_{n} are not adjacent. But, we know that degree of each non-unit is ϕ​(n)\phi(n), thus number of edges between units and non-units is (n−ϕ​(n))​ϕ​(n).(n-\phi(n))\phi(n). Now by Handshake Lemma, total number of edges in G​(ℤn)G(\mathbb{Z}_{n}) is given by

    ϕ​(n)​(ϕ​(n)−1)+(n−ϕ​(n))​ϕ​(n)2.\frac{\phi(n)(\phi(n)-1)+(n-\phi(n))\phi(n)}{2}.

    Thus, the number of edges among units will be the difference between the total number of edges and the number of edges among units and non-units,i.e.

    ϕ​(n)​(ϕ​(n)−1)−(n−ϕ​(n))​ϕ​(n)2=ϕ​(n)​(ϕ​(n)−1−n−ϕ​(n))2.\frac{\phi(n)(\phi(n)-1)-(n-\phi(n))\phi(n)}{2}=\frac{\phi(n)(\phi(n)-1-n-\phi(n))}{2}.

    The result follows.

∎

Theorem 4.2.

Let pp and qq be two odd prime numbers with p<q.p<q. Then, Sombor index of graph G​(ℤp​q)G(\mathbb{Z}_{pq}) is,

S​O​(G​(ℤp​q))=2​α​ϕ​(p​q)+β​(ϕ​(p​q))2+(ϕ​(p​q)−1)2+2​(|E|−α−β)​(ϕ​(p​q)−1)SO(G(\mathbb{Z}_{pq}))=\sqrt{2}\alpha\phi(pq)+\beta\sqrt{(\phi(pq))^{2}+(\phi(pq)-1)^{2}}+\sqrt{2}(\left|E\right|-\alpha-\beta)(\phi(pq)-1)

where, α=(p−1)​(q−1),β=(p−1)​(q−1)​(p+q−3),|E|=(p​q−1)​ϕ​(p​q)2.\alpha=(p-1)(q-1),\quad\beta=(p-1)(q-1)(p+q-3),\quad\left|E\right|=\displaystyle{\frac{(pq-1)\phi(pq)}{2}}.

Proof.

Let Z​(ℤp​q)=Sp∪SqZ(\mathbb{Z}_{pq})=S_{p}\cup S_{q} where Sp={0,p,2​p,⋯,(q−1)​p}S_{p}=\{0,p,2p,\cdots,(q-1)p\} and Sq={0,q,2​q,⋯,(p−1)​q}.S_{q}=\{0,q,2q,\cdots,(p-1)q\}. We can observe that the sets SpS_{p} and SqS_{q} forms a closed set under the operation ⨁p​q.\bigoplus_{pq}. Since the unit graph is the complement of the total graph and in the total graph, SpS_{p} and SqS_{q} induce complete graphs, thus there will be no edges inside SpS_{p} and SqS_{q}. But we will get a complete bipartite graph with the partite sets SpS_{p} and SqS_{q}.

Also observe that 0≁v0\nsim v\, for allv∈Z​(ℤp​q)\,v\,\in Z(\mathbb{Z}_{pq}). Thus the sets SpS_{p} and SqS_{q} gives us K(p−1),(q−1).K_{(p-1),(q-1)}. So that the number of edges between zero divisors and zero divisors is,

α=(p−1)​(q−1).\alpha=(p-1)(q-1).

Since pp and qq are odd primes, then by Theorem 2.6 dv=ϕ​(p​q)d_{v}=\phi(pq)\, for allv∈Z​(ℤp​q).\,v\in Z(\mathbb{Z}_{pq}). Thus,

dv\displaystyle d_{v} =ϕ​(p​q)=(p−1)​(q−1)\displaystyle=\phi(pq)=(p-1)(q-1)

Notice that for all v∈Z​(ℤp​q)​dv=ϕ​(p​q)=(p−1)​(q−1).\,v\in Z(\mathbb{Z}_{pq})\,d_{v}=\phi(pq)=(p-1)(q-1). Also, for all v∈Sp\{0}​dv=(p−1).\,v\in S_{p}\backslash\{0\}\,d_{v}=(p-1). and for all v∈Sq\{0}​dv=(q−1).\,v\in S_{q}\backslash\{0\}\,d_{v}=(q-1). Thus the remaining degrees of,

v∈Sp\{0}​ is ​(p−1)​(q−1)−(p−1)=(p−1)​(q−2)v\in S_{p}\backslash\{0\}\,\text{ is }\,(p-1)(q-1)-(p-1)=(p-1)(q-2)

and of

v∈Sq\{0}​ is ​(p−1)​(q−1)−(q−1)=(p−2)​(q−1).v\in S_{q}\backslash\{0\}\,\text{ is }\,(p-1)(q-1)-(q-1)=(p-2)(q-1).

Also 0∼u0\sim u\, for all u∈U​(ℤp​q)\,u\,\in U(\mathbb{Z}_{pq}), which implies d0=ϕ​(p​q)=(p−1)​(q−1).d_{0}=\phi(pq)=(p-1)(q-1). Therefore the number of edges between zero divisors and units are,

β\displaystyle\beta =(q−1)​(p−1)​(q−2)+(p−1)​(p−2)​(q−1)+(p−1)​(q−1)\displaystyle=(q-1)(p-1)(q-2)+(p-1)(p-2)(q-1)+(p-1)(q-1)
=(p−1)​(q−1)​(p+q−3)\displaystyle=(p-1)(q-1)(p+q-3)

Using Theorem 2.6 and handshake lemma we can find the total number of edges in G​(ℤp​q)G(\mathbb{Z}_{pq}) as follows,

|E|\displaystyle\left|E\right| =12[ϕ(pq)(pq−ϕ(pq))+(ϕ(pq)−1)(ϕ(pq)])\displaystyle=\frac{1}{2}[\phi(pq)(pq-\phi(pq))+(\phi(pq)-1)(\phi(pq)])
=12​((p−1)​(q−1))​(p​q−1).\displaystyle=\frac{1}{2}((p-1)(q-1))(pq-1).

Now since we know the total number of edges in G​(ℤp​q)G(\mathbb{Z}_{pq}) (|E|)(\left|E\right|), number of edges between zero divisors and zero divisors (α)(\alpha) and the number of edges between zero divisors and units (β)(\beta), we can easily get the number of edges between units and units, which will be given by |E|−α−β.\left|E\right|-\alpha-\beta.

From Theorem 2.6 we know dv=ϕ​(p​q)d_{v}=\phi(pq)\, for all v∈Z​(ℤp​q)\,v\,\in Z(\mathbb{Z}_{pq}) and du=ϕ​(p​q)−1d_{u}=\phi(pq)-1\, for all u∈U​(ℤp​q).\,u\,\in U(\mathbb{Z}_{pq}). The result follows. ∎

Theorem 4.3.

Let pp and qq be two odd prime numbers with p<q.p<q. Then, the Sombor index of the graph G​(ℤp2​q)G(\mathbb{Z}_{p^{2}q}) is,

S​O​(G​(ℤp2​q))=2​α​ϕ​(p2​q)+β​(ϕ​(p2​q))2+(ϕ​(p2​q)−1)2+2​(|E|−α−β)​(ϕ​(p2​q)−1)SO(G(\mathbb{Z}_{p^{2}q}))=\sqrt{2}\alpha\phi(p^{2}q)+\beta\sqrt{(\phi(p^{2}q))^{2}+(\phi(p^{2}q)-1)^{2}}+\sqrt{2}(\left|E\right|-\alpha-\beta)(\phi(p^{2}q)-1)

where, α=p2​(p−1)​(q−1),β=p2​(p−1)​(q−1)​(p+q−3),|E|=p2​(p−1)​(q−1)​(p2​q−1)2.\alpha=p^{2}(p-1)(q-1),\quad\beta=p^{2}(p-1)(q-1)(p+q-3),\quad\left|E\right|=\displaystyle{\frac{p^{2}(p-1)(q-1)(p^{2}q-1)}{2}.}

Proof.

Let Z​(ℤp2​q)=Sp∪Sq∪Sp​qZ(\mathbb{Z}_{p^{2}q})=S_{p}\cup S_{q}\cup S_{pq} where, Sp={0,p,2​p,⋯,(p​q−1)​p}S_{p}=\{0,p,2p,\cdots,(pq-1)p\}, Sq={0,q,2​q,⋯,(p2−1)​q}S_{q}=\{0,q,2q,\cdots,(p^{2}-1)q\} and Sp​q={0,p​q,2​p​q,⋯,(p−1)​p​q}.S_{pq}=\{0,pq,2pq,\cdots,(p-1)pq\}. Now consider Sp∗=Sp\Sp​qS^{*}_{p}=S_{p}\backslash S_{pq} and Sq∗=Sp\Sp​q.S^{*}_{q}=S_{p}\backslash S_{pq}. Since the unit graph is the complement of the total graph and in the total graph, Sp∗S^{*}_{p}, Sq∗S^{*}_{q} and Sp​qS_{pq} induce complete graphs, thus here there will be no edges inside Sp∗S^{*}_{p}, Sq∗S^{*}_{q} and Sp​q.S_{pq}. Also, in the total graph we get complete bipartite graphs induced by Sp∗S^{*}_{p} & Sp​qS_{pq} and Sq∗S^{*}_{q} & Sp​q.S_{pq}. Thus here there will be no edges between Sp∗S^{*}_{p} & Sp​qS_{pq} and Sq∗S^{*}_{q} & Sp​q.S_{pq}. So the only edges between zero-divisors and zero-divisors will be between Sp∗S^{*}_{p} and Sq∗S^{*}_{q}, which will give us a complete bipartite graph. Also observe that 0≁v0\nsim v\, for allv∈Z​(ℤp2​q)\,v\,\in Z(\mathbb{Z}_{p^{2}q}).Thus the number of edges between zero divisors and zero divisors are,

α=p2​(p−1)​(q−1).\alpha=p^{2}(p-1)(q-1).

Since pp and qq are odd primes, then by Theorem 2.6 dv=ϕ​(p2​q)d_{v}=\phi(p^{2}q)\, for allv∈Z​(ℤp2​q).\,v\in Z(\mathbb{Z}_{p^{2}q}). Thus,

dv\displaystyle d_{v} =ϕ​(p2​q)=p​(p−1)​(q−1).\displaystyle=\phi(p^{2}q)=p(p-1)(q-1).

Notice that for all v∈Z​(ℤp2​q)​dv=ϕ​(p2​q)=p​(p−1)​(q−1).\,v\in Z(\mathbb{Z}_{p^{2}q})\,d_{v}=\phi(p^{2}q)=p(p-1)(q-1). Also, for all v∈Sp∗​dv=p​(q−1).\,v\in S_{p}^{*}\,d_{v}=p(q-1). and for all v∈Sq∗​dv=p​(p−1).\,v\in S_{q}^{*}\,d_{v}=p(p-1). Thus the remaining degrees of

v∈Sp∗​is​p​(p−1)​(q−1)−p​(q−1)=p​(p−2)​(q−1)v\in S_{p}^{*}\,\text{is}\,p(p-1)(q-1)-p(q-1)=p(p-2)(q-1)

and of

v∈Sq∗​is​p​(p−1)​(q−1)−p​(p−1)=p​(p−1)​(q−2).v\in S_{q}^{*}\,\text{is}\,p(p-1)(q-1)-p(p-1)=p(p-1)(q-2).

Also each element in Sp​qS_{pq} is adjacent to every unit. Therefore the number of edges between zero divisors and units are,

β\displaystyle\beta =p2​(q−1)​(p−1)​(p−2)+p2​(p−1)​(q−2)​(q−1)+p2​(p−1)​(q−1)\displaystyle=p^{2}(q-1)(p-1)(p-2)+p^{2}(p-1)(q-2)(q-1)+p^{2}(p-1)(q-1)
=p2​(p−1)​(q−1)​(p+q−3)\displaystyle=p^{2}(p-1)(q-1)(p+q-3)

Using Theorem 2.6 and the handshake lemma we can find the total number of edges in G​(ℤp2​q)G(\mathbb{Z}_{p^{2}q}) as follows,

|E|\displaystyle\left|E\right| =12[ϕ(p2q)(p2q−ϕ(p2q))+(ϕ(p2q)−1)(ϕ(p2q)])\displaystyle=\frac{1}{2}[\phi(p^{2}q)(p^{2}q-\phi(p^{2}q))+(\phi(p^{2}q)-1)(\phi(p^{2}q)])
=12​(p2​(p−1)​(q−1))​(p2​q−1).\displaystyle=\frac{1}{2}(p^{2}(p-1)(q-1))(p^{2}q-1).

Now since we know the total number of edges in G​(ℤp2​q)G(\mathbb{Z}_{p^{2}q}) (|E|)(\left|E\right|), the number of edges between zero divisors and zero divisors (α)(\alpha) and the number of edges between zero divisors and units (β)(\beta), we can easily get the number of edges between units and units, which will be given by |E|−α−β.\left|E\right|-\alpha-\beta.

From Theorem 2.6 we know dv=ϕ​(p2​q)d_{v}=\phi(p^{2}q)\, for all v∈Z​(ℤp2​q)\,v\,\in Z(\mathbb{Z}_{p^{2}q}) and du=ϕ​(p2​q)−1d_{u}=\phi(p^{2}q)-1\, for all u∈U​(ℤp2​q).\,u\,\in U(\mathbb{Z}_{p^{2}q}). The result follows. ∎

5. SOMBOR INDEX OF LOCAL RINGS

It is easy to calculate the Sombor index of the total graph and the unit graph of the rings ℤpα,\mathbb{Z}_{p^{\alpha}}, where pp is a prime number and α\alpha is a natural number. In this section, we generalize the computation of the Sombor index of the total graph and the unit graph of any finite local ring.

Theorem 5.1.

Let RR be a local ring

  1. (1)

    If 2∉U​(R)2\notin U(R) then,

    SO(TΓ(R)=n​(n−|U​(R)|−1)22SO(T_{\Gamma}(R)=\frac{n(n-\left|U(R)\right|-1)^{2}}{\sqrt{2}}
  2. (2)

    If 2∈U​(R)2\in U(R) then,

    S​O​(TΓ​(R))=(n−|U​(R)|)​(n−|U​(R)|−1)22+(|U​(R)|)​(n−|U​(R)|)22.SO(T_{\Gamma}(R))=\frac{(n-\left|U(R)\right|)(n-\left|U(R)\right|-1)^{2}}{\sqrt{2}}+\frac{(\left|U(R)\right|)(n-\left|U(R)\right|)^{2}}{\sqrt{2}}.
Proof.
  1. (1)

    If 2∉U​(R)2\notin U(R), then by Theorem 2.2 we know dx=n−|U​(R)|−1d_{x}=n-\left|U(R)\right|-1, for every x∈R.x\in R. Thus by the handshake lemma we get the total number of edges as

    |E|=n​(n−|U​(R)|−1)2\left|E\right|=\frac{n(n-\left|U(R)\right|-1)}{2}

    and the result follows.

  2. (2)

    If 2∈U​(R)2\in U(R), then by Theorem 2.7 Z​(TΓ​(R))≅K|Z​(R)|Z(T_{\Gamma}(R))\cong K_{\left|Z(R)\right|}, thus the number of edges between zero divisors and zero divisors are given by

    |E1|=|Z​(R)|​(|Z​(R)|−1)2=(n−|U​(R)|)​(n−|U​(R)|−1)2\left|E_{1}\right|=\frac{\left|Z(R)\right|(\left|Z(R)\right|-1)}{2}=\frac{(n-\left|U(R)\right|)(n-\left|U(R)\right|-1)}{2}

    By Theorem 2.2 when 2∈U​(R)2\in U(R) we know that dx=n−|U​(R)|−1d_{x}=n-\left|U(R)\right|-1, for every
    x∈Z​(R).x\in Z(R). Also |Z​(R)|=n−|U​(R)|\left|Z(R)\right|=n-\left|U(R)\right| and Z​(TΓ​(R))≅K|Z​(R)|Z(T_{\Gamma}(R))\cong K_{\left|Z(R)\right|} which implies
    dx=n−|U​(R)|−1d_{x}=n-\left|U(R)\right|-1, for every x∈Z​(R)x\in Z(R) i.e. all the degrees of zero divisors are exhausted. Thus there will be no edges between zero divisors and units.
    From Theorem 2.2 when 2∈U​(R)2\in U(R) we know dx=n−|U​(R)|d_{x}=n-\left|U(R)\right|, for every x∉Z​(R).x\notin Z(R). Thus the number of edges between units and units is given by,

    |E2|=(n−|U​(R)|)​(n−|U​(R)|−1)2.\left|E_{2}\right|=\frac{(n-\left|U(R)\right|)(n-\left|U(R)\right|-1)}{2}.

    Again the result follows.

∎

Theorem 5.2.

Let RR be a local ring

  1. (1)

    If 2∉U​(R)2\notin U(R) then,

    SO(G(R)=n​(|U​(R)|)22SO(G(R)=\frac{n(\left|U(R)\right|)^{2}}{\sqrt{2}}
  2. (2)

    If 2∈U​(R)2\in U(R) then,

    S​O​(G​(R))=|U​(R)|​(n−|U​(R)|)​|U​(R)|2+(n−|U​(R)|)2SO(G(R))={\left|U(R)\right|\left(n-\left|U(R)\right|\right)\sqrt{\left|U(R)\right|^{2}+(n-\left|U(R)\right|)^{2}}}
Proof.

The proof follows similar to Theorem 5.1. ∎

6. SOMBOR INDEX OF kk- REGULAR GRAPH AND ITS COMPLEMENT

In this section, we give the relationship between the Sombor index of a complete graph, kk-regular graph, and its complement.

Theorem 6.1.

Let GG be a k−k- regular graph of order nn and G¯\bar{G} be its complement. Then,

S​O​(Kn)=(S​O​(G)+S​O​(G¯))2SO(K_{n})=\left(\sqrt{SO(G)}+\sqrt{SO(\bar{G})}\right)^{2}
Proof.

As the graph GG is a k−k- regular graph, it is clear that the graph G¯\bar{G} is a (n−k−1)−(n-k-1)- regular graph. Therefore,

S​O​(G)=n​k22a​n​dS​O​(G¯)=n​(n−k−1)22.SO(G)=\frac{nk^{2}}{\sqrt{2}}\hskip 28.45274ptand\hskip 28.45274ptSO(\bar{G})=\frac{n(n-k-1)^{2}}{\sqrt{2}}.

Now, we know that,

S​O​(Kn)\displaystyle SO(K_{n}) =n​(n−1)22\displaystyle=\frac{n(n-1)^{2}}{\sqrt{2}}
=n​(k+(n−k−1))22\displaystyle=\frac{n(k+(n-k-1))^{2}}{\sqrt{2}}
=S​O​(G)+2​n​(2​S​O​(G)n)​(2​S​O​(G¯)n)+S​O​(G¯)\displaystyle=SO(G)+\sqrt{2}n\sqrt{\left(\frac{\sqrt{2}SO(G)}{n}\right)}\sqrt{\left(\frac{\sqrt{2}SO(\bar{G})}{n}\right)}+SO(\bar{G})
=(S​O​(G)+S​O​(G¯))2\displaystyle=\left(\sqrt{SO(G)}+\sqrt{SO(\bar{G})}\right)^{2}

∎

References

  • [1] R. Todeschini and V. Consonni, Molecular Descriptors for Chemoinformatics. Wiley–VCH, 2009.
  • [2] I. Gutman, “Geometric approach of degree based topological indicies: Sombor indices,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 86, no. 1, pp. 11–16, 2021.
  • [3] N. Ghanbari and S. Alikhani, “Sombor index of certain graphs,” Iranian Journal of Mathematical Chemistry, vol. 12, no. 1, pp. 27–37, 2021.
  • [4] A. Gursoy, A. Ulker, and N. K. Gursoy, “Somber index of zero - divisor graphs of commutative rings,” Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, vol. 30, no. 2, pp. 231–257, 2022.
  • [5] D. F. Anderson and A. Badawi, “The total graph of a commutative ring,” Journal of Algebra, vol. 320, no. 7, pp. 2706–2719, 2008.
  • [6] T.Asir and T. T. Chelvam, “Necessary and sufficient conditions for unit graphs to be hamiltonian,” A note on total graph of Zn,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 14, no. 1, pp. 1–7, 2010.
  • [7] N. Ashrafi, H. R. Maimani, and M. R. Pournaki, “Unit graphs associated with rings,” Communications in Algebra, vol. 38, no. 8, pp. 2851–2871, 2010.
  • [8] N. Ashrafi, H. Maimani, M. R. Pournaki, and S. Yassemi, “Unit graphs associated with rings,” Communications in Algebra - COMMUN ALGEBRA, vol. 38, pp. 2851–2871, 2010.
  • [9] N. Ashrafi, H. R. Maimani, and M. R. Pournaki, “Unit graphs associated with rings,” Communications in Algebra, vol. 38, no. 8, pp. 2851–2871, 2010.
  • [10] R. P. Grimaldi, “Graphs from rings,” Proceedings of the 20th SoutheasternConference on Combinatorics, Graph Theory, and Computing, 1990.
  • [11] S. Suthar and O. Prakash, “Energy and wiener index of total graph over ring zn,” Electronic Notes in Discrete Mathematics, vol. 63, pp. 485–495, 2017.