Weakly nil clean graphs of rings

TL;DR

This paper introduces the concept of weakly nil clean graphs for rings, extending previous ideas, and investigates their graph-theoretic properties such as girth, clique number, diameter, and chromatic index for finite commutative rings.

Contribution

It generalizes the nil clean graph concept to weakly nil clean graphs and explores their properties in the context of finite commutative rings.

Findings

Determined girth, clique number, diameter, and chromatic index for weakly nil clean graphs.

Established relationships between ring properties and graph invariants.

Extended existing graph-theoretic analyses to a broader class of rings.

Abstract

In this article, the concept of nil clean graph of a ring has been generalised to weakly nil clean graph of a ring and graph theoretic properties like girth, clique number, diameter and chromatic index of the graph have been studied for a finite commutative ring.

Full text

**Weakly nil clean graphs of rings

**

Ajay Sharma, Jayanta Bhattacharyya

and Dhiren Kumar Basnet1

*Department of Mathematical Sciences, Tezpur University,

Napaam, Tezpur-784028, Assam, India.

Email: ajay123@tezu.ernet.in, jbhatta@tezu.ernet.in

and dbasnet@tezu.ernet.in*

11footnotetext: corresponding author.

**Abstract: ** In this article, the concept of nil clean graph of a ring has been generalised to weakly nil clean graph of a ring and graph theoretic properties like girth, clique number, diameter and chromatic index of the graph have been studied for a finite commutative ring.

Key words: Nil clean ring, weakly nil clean ring, weakly nil clean graph.

$2010$** Mathematics Subject Classification:** 05C75, 16N40, 16U99.

1 Introduction

Rings in this article are finite commutative rings with non zero identity. The set of idempotents, nilpotent elements of a ring $R$ and the $n\times n$ matrix ring over $R$ are denoted by $Idem(R)$, $Nil(R)$ and $M_{n}(R)$ respectively. An element $x$ of a ring $R$ is said to be a nil clean element[6, 5] if $x=n+e$, for some $e\in Idem(R)$ and $n\in Nil(R)$. The ring $R$ is said to be nil clean if each element of $R$ is nil clean. An element $x$ of ring $R$ is said to be a weakly nil clean element[4, 2] $x=n+e$ or $x=n-e$, for some $e\in Idem(R)$ and $n\in Nil(R)$. The ring $R$ is said to be weakly nil clean if each element of $R$ is weakly nil clean. The set of nil clean and weakly nil clean elements of $R$ are denoted by $NC(R)$ and $WNC(R)$ respectively.

In this article, by graph, we mean simple undirected graph. For a graph $G$, $V(G)$ denotes the set of vertices and $E(G)$ denotes the set of edges. For a positive integer $n$, P. Grimaldi[8] defined and studied various properties of the unit graph $G(\mathbb{Z}_{n})$, of the ring of integers modulo $n$, with vertex set $\mathbb{Z}_{n}$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y$ is a unit. Further in [1], authors generalized $G(\mathbb{Z}_{n})$ to unit graph $G(R)$, where $R$ is an arbitrary associative ring with non zero identity. Recently D.K. Basnet and J. Bhattacharya [3], defined nil clean graph of a finite commutative ring $R$, where the vertex set is $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y$ is nil clean and they studied various properties of nil clean graph of finite commutative ring.

In this article we have introduced weakly nil clean graph $G_{WN}(R)$ associated with a finite commutative ring $R$ motivated by the concept of Nil clean graph of a finite ring $R$ [3]. We define $G_{WN}(R)$ as $R$ is the vertex set and two vertices $x$ and $y$ are adjacent if and only if $x+y$ is a weakly nil clean element of $R$. The properties like girth, diameter, clique number and chromatic index of $G_{WN}(R)$ have been studied.

Here we mention some preliminaries about graph theory for this article. Let $G$ be a graph. The degree of the vertex $v\in G$ denoted by $deg(v)$, is the number of edges adjacent with $v$. A graph $G$ is said to be connected if for any two distinct vertices of $G$, there is a path in $G$ connecting them. Number of edges on the shortest path between vertices $x$ and $y$ is called the distance between $x$ and $y$ and is denoted by $d(x,y)$. If there is no path between $x$ and $y$ then we say $d(x,y)=\infty$. The diameter of a graph $G$, denoted by $diam(G)$, is the maximum of distances of each pair of distinct vertices in $G$. Also girth of $G$ is the length of the shortest cycle in $G$, denoted by $gr(G)$. Note that if there is no cycle in $G$ then $gr(G)=\infty$. A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A bipartite graph $G$ is a graph whose vertices can be divided into two disjoint parts $V_{1}$ and $V_{2}$, such that $V(G)=V_{1}\cup V_{2}$ and every edge in $G$ has the form $e=(x,y)\in E(G)$, where $x\in V_{1}$ and $y\in V_{2}$. Note that no two vertices both in $V_{1}$ or both in $V_{2}$ are adjacent. A complete bipartite graph is a graph where every vertex of the first part $V_{1}$ is connected to every vertex of the second part $V_{2}$, denoted by $K_{m,n}$, where $|V_{1}|=m$ and $|V_{2}|=n$. A complete bipartite graph $K_{1,n}$ is called star graph.

A clique is a subset of vertices of an undirected graph such that its induced subgraph is complete. A clique having $n$ number of vertices is called $n$-clique. The maximum clique of a graph is a clique such that there is no clique with more vertices. The clique number of a graph $G$ is denoted by $\omega(G)$ and defined by the number of vertices in the maximal clique of $G$. An edge colouring of a graph $G$ is a map $C:E(G)\rightarrow S$, where $S$ is a set of colours such that for all $e_{1},e_{2}\in E(G)$, if $e_{1}$ and $e_{2}$ are adjacent then $C(e_{1})\neq C(e_{2})$. The chromatic index of a graph $G$ is denoted by $\chi^{\prime}(G)$ and is defined as the minimum number of colours needed for a proper colouring of $G$.

2 Weakly nil clean graph

In this section we will define weakly nil clean graph of a finite commutative ring and discuss its basic properties.

Definition 2.1**.**

The weakly nil clean graph of a ring $R$ denoted by $G_{WN}(R)$ is defined by setting $R$ as the vertex set and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y$ is weakly nil clean. Here we are not considering any loop at a vertex in the graph.

From the definition, nil clean graph of a ring $R$ is always a subgraph of a weakly nil clean graph of $R$.

For illustration following are the weakly nil clean graphs of $\mathbb{Z}_{10}$ and $GF(25)$, where $\mathbb{Z}_{10}$ is the ring of integers modulo $10$ and $GF(25)$ is the finite field with $25$ elements.

Since, $GF(25)\cong\mathbb{Z}_{5}[x]/<x^{2}+x+1>=\{ax+b+<x^{2}+x+1>\,\,:\,\,a,b\in\mathbb{Z}_{5}\}$.

Define, $\alpha=x+<x^{2}+x+1>$, then we have $GF(25)=\{0,1,2,3,4,\alpha,2\alpha,3\alpha,4\alpha,1+\alpha,1+2\alpha,1+3\alpha,1+4\alpha,2+\alpha,2+2\alpha,2+3\alpha,2+4\alpha,3+\alpha,3+2\alpha,3+3\alpha,3+4\alpha,4+\alpha,4+2\alpha,4+3\alpha,4+4\alpha\}$, Observe that $WNC(GF(25))=\{0,1,4\}$.

For any prime $p$ the weakly nil clean graph of $\mathbb{Z}_{p}$ is given below. If $\frac{p-1}{2}\equiv 0(mod\,\,2)$ then the graph $G_{WN}(\mathbb{Z}_{p})$ is

If $\frac{p-1}{2}\equiv 1(mod\,\,2)$, then the graph $G_{WN}(\mathbb{Z}_{p})$ is

The following theorem is based on complete graph.

Theorem 2.2**.**

The weakly nil clean graph $G_{WN}(R)$ is a complete graph if and only if $R$ is a weakly nil clean ring.

Proof.

Let $G_{WN}(R)$ be a complete graph. For $r\in R$, $r$ is adjacent to [math], so $r=r+0$ is weakly nil clean element in $R$. Hence $R$ is weakly nil clean ring. Converse is obvious from the definition of weakly nil clean graph. ∎

Lemma 2.3**.**

Let $R$ be a ring. If $x+Nil(R)$ and $y+Nil(R)$ are adjacent in $G_{WN}(R/Nil(R))$ then every element of $x+Nil(R)$ is adjacent to each element of $y+Nil(R)$ in the weakly nil clean graph $G_{WN}(R)$.

Proof.

Let $x+Nil(R)$ and $y+Nil(R)$ are adjacent in $G_{WN}(R/Nil(R))$. Then $(x+y)+Nil(R)=e+Nil(R)$ or $(x+y)+Nil(R)=-e+Nil(R)$, where $e\in Idem(R)$, as idempotents lift modulo $Nil(R)$. Thus $x+y=e+n$ or $x+y=-e+n$, where $n\in Nil(R)$. Now for $a\in x+Nil(R)$ and $b\in y+Nil(R)$, we have $a=x+n_{1}$ and $b=y+n_{2}$, for some $n_{1},n_{2}\in Nil(R)$. So $a+b=e+(n-n_{1}-n_{2})$ or $a+b=-e+(n-n_{1}-n_{2})$. Hence $a$ and $b$ are adjacent in $G_{WN}(R)$. ∎

The next Lemma is related to the degree of a vertex in $G_{WN}(R)$.

Lemma 2.4**.**

Let $G_{WN}(R)$ be the weakly nil clean graph of a ring $R$. For $x\in R$ we have the following:

1. (i)

If $2x$ is weakly nil clean, then $deg(x)=|WNC(R)|-1$. 2. (ii)

If $2x$ is not weakly nil clean, then $deg(x)=|WNC(R)|$.

Proof.

Proof is similar to Lemma 2.4 [3]. ∎

The result of connectedness of weakly nil clean graph of $\mathbb{Z}_{n}$ and $M_{n}(\mathbb{Z}_{n})$ is given below:

Theorem 2.5**.**

For a ring $R$. The following hold :

1. (i)

$G_{WN}(R)$* need not be connected.* 2. (ii)

Let $R=\mathbb{Z}_{n}$. For $a\in\mathbb{Z}_{n}$ there is a path from $a$ to [math]. 3. (iii)

$G_{WN}(\mathbb{Z}_{n})$* is connected.* 4. (iv)

Let $R=\mathbb{Z}_{n}$. For $A\in M_{n}(\mathbb{Z}_{n})$ there is a path from $A$ to [math], where [math] is the zero matrix of $M_{n}(\mathbb{Z}_{n})$. 5. (v)

$G_{WN}(M_{n}(\mathbb{Z}_{n}))$* is connected.*

Proof.

$(i)$ It is clear from the graph $G_{WN}(G(25))$, Figure $2$. $(ii),\,\,(iii),\,\,(iv)$ and $(v)$ are clear from Theorem 2.5 [3], as nil clean graph of a ring $R$ is a subgraph of a weakly nil clean graph of $R$. ∎

3 Invariants of weakly nil clean graph

In this section, for any ring $R$, we study about some invariants like girth, clique number, diameter and chromatic index of $G_{WN}(R)$.

3.1 Girth of $G_{WN}(R)$

Here in the given theorem we show that the girth of weakly nil clean graph of any ring is $3$.

Theorem 3.1**.**

For a ring $R$ with $|R|\geq 3$, girth of a weakly nil clean graph $G_{WN}(R)$ is 3.

Proof.

For a ring $R$, clearly $\{0,1,-1\}\subseteq WNC(R)$, so the weakly nil graph of $R$ contains the given subgraph

So the girth of a ring $R$ is always $3$. ∎

Corollary 3.2**.**

The weakly nil clean graph of a ring $R$ can not be bipartite.

Corollary 3.3**.**

Weakly nil clean graph of any ring $R$ can not be a star graph.

3.2 Clique number of $G_{WN}(R)$

Here we study about the clique number of weakly nil clean graph of any finite field and graph $G_{WN}(\mathbb{Z}_{2p})$.

Theorem 3.4**.**

The clique number of the field $\mathbb{Z}_{p}$ is $3$, where $p$ is prime and $p\geq 3$.

Proof.

If $p=3$ then clearly $\omega(G_{WN}(\mathbb{Z}_{3}))=3$. Let $p>3$. The weakly nil clean graph of $\mathbb{Z}_{p}$ is connected, as nil clean graph of $\mathbb{Z}_{n}$ is connected and girth of $G_{WN}(\mathbb{Z}_{p})$ is $3$. If possible, assume that $\omega(G_{WN}(\mathbb{Z}_{p}))>3$. Then there exist four vertices say $\{v_{1},v_{2},v_{3},v_{4}\}$ having $deg(v_{i})\geq 3$ and the subgraph induced by $\{v_{1},v_{2},v_{3},v_{4}\}$ is complete. Since $G_{WN}(\mathbb{Z}_{p})$ is connected and $p\geq 5$, so there exists $v_{j}\in\{v_{1},v_{2},v_{3},v_{4}\}$ such that $deg(v_{j})>3$, which is a contradiction as $WNC(\mathbb{Z}_{p})=\{0,1,-1\}$. ∎

It is clear from Figure $5$ [3], that the weakly nil clean graph of $GF(p^{n})$ is partitioned into disjoint subgraphs, where one subgraph contains $p$ number of vertices and rest other subgraph contains each of $2p$ number of vertices.

Theorem 3.5**.**

The clique number of weakly nil clean graph of any finite field is $3$.

Proof.

Let, $GF(p^{n})$ be a field of order $p^{n}$. Let $\{G_{1},G_{2},G_{3},\cdot\cdot\cdot,G_{m}\}$ be the disjoint connected subgraphs of $G_{WN}(GF(p^{n}))$, where $G_{1}$ contains $p$ number of vertex and $G_{i}$ contains $2p$ number of vertex, for $2\leq i\leq m$. Clearly

is a subgraph of $G_{WN}(GF(p^{n}))$. If possible let, clique number of $G_{WN}(GF(p^{n}))$ is 4. Then there exist a $4$-$clique$ which lies in $G_{i}$, for some $i$ with $1\leq i\leq m$. By similar procedure of Theorem 3.4, we get a contradiction, as required. ∎

In graph theory, the neighbourhood of a vertex $v$ of a graph $G$ is denoted by $N(v)$ and defined to be the set of all vertices adjacent to $v$. Note that $v\notin N(v)$.

Lemma 3.6**.**

For $\mathbb{Z}_{2p}$, where $p\geq 5$ is a prime number, the following hold:

1. (i)

$N(a)\cap N(-a)=\phi$, for $a\in\mathbb{Z}_{2p}$ and $\,\,a,-a\notin\{0,1,p,p+1,\frac{p+1}{2},\frac{p-1}{2}\}$. 2. (ii)

$N(a)\cap N(b)=\phi$, for $a,b\in\mathbb{Z}_{2p}$ with $a+b=1$ and $a\notin\{0,1,p,p+1,\frac{p-1}{2},\frac{p+1}{2},\frac{p+3}{2},\frac{3p-1}{2},\frac{3p+1}{2},\frac{3p+3}{2}\}$. 3. (iii)

$N(a)\cap N(b)=\phi$, for $a,b\in\mathbb{Z}_{2p}$ with $a+b=-1$ and $a\notin\{0,-1,p,p-1,\frac{p-1}{2},\frac{p+1}{2},\frac{3p-3}{2},\frac{p-3}{2},\frac{3p-1}{2},\frac{3p+1}{2}\}$.

Proof.

$(i)$ If $a\notin\{0,1,p+1,p,\frac{p+1}{2},\frac{p-1}{2}\}$, then the possible values of $a$ are the elements of $S=\{2,3,\cdot\cdot\cdot,\frac{p-3}{2},p-2,p-3,\cdot\cdot\cdot,\frac{p+3}{2}\}$ and from Figure $7$, it is clear that $N(a)\cap N(-a)=\phi$, for $a\in S$.

$(ii)$ The possible pairs $(a,b)\in\mathbb{Z}_{2p}\times\mathbb{Z}_{2p}$ satisfying the condition $a+b=1$ and $a\notin\{0,1,p,p+1,\frac{p-1}{2},\frac{p+1}{2},\frac{p+3}{2},\frac{3p-1}{2},\frac{3p+1}{2},\frac{3p+3}{2}\}$ are $S=\{(-1,2),(-2,3),\cdot\cdot\cdot,(\frac{3p+5}{2},\frac{p-3}{2})\}\cup\{(p-1,p+2),(p-2,p+3),\cdot\cdot\cdot,(\frac{p+5}{2},\frac{3p-3}{2})\}$. For $(a,b)\in S$, from Figure $7$ of weakly nil clean graph of $\mathbb{Z}_{2p}$, we conclude that $N(a)\cap N(b)=\phi$. Similarly $(iii)$ follows from Figure $7$.

∎

Theorem 3.7**.**

There are exactly five $4$-cliques of $G_{WN}(\mathbb{Z}_{2p})$, where $p\geq 5$ is a prime number.

Proof.

It is clear that $WNC(\mathbb{Z}_{2p})=\{0,1,-1,p,p-1,p+1\}$. Let $C=\{a_{1},a_{2},a_{3},a_{4}\}$ be a $4$-clique of $G_{WN}(\mathbb{Z}_{2p})$. So $a_{i}+a_{j}\in WNC(\mathbb{Z}_{2p})$, for any $1\leq i,j\leq 4$. Without loss of generality we consider $a_{1}$ and $a_{2}$.

Case I: If $a_{1}+a_{2}=0$ then $a_{1}=-a_{2}$. If $0\in C$ then $a_{i}\in WNC(\mathbb{Z}_{2p})$, for $1\leq i\leq 4$ and it gives only two $4$-cliques $viz.$ $\{1,-1,p,0\}$ and $\{p-1,p+1,p,0\}$. If $0\notin C$ and $a_{1}\in WNC(\mathbb{Z}_{2p})$ then $a_{1}=1$ or $p+1$. If $a_{1}=1$ then $a_{2}=-1$ and $a_{3}=p$ as $0\notin C$, this must imply $a_{4}=0$, which is a contradiction. Similarly if $a_{1}=p+1$ then also we get a contradiction. If $0\notin C$ and $a_{1}\notin WNC(\mathbb{Z}_{2p})$ then from Lemma 3.6 $(i)$, $a_{1}=\frac{p+1}{2}$ or $\frac{p-1}{2}$. If $a_{1}=\frac{p+1}{2}$ then $a_{2}=\frac{3p-1}{2}$ and it gives the $4$-clique $\{\frac{p+1}{2},\frac{p-1}{2},\frac{3p+1}{2},\frac{3p-1}{2}\}$. Similarly if $a_{2}=\frac{p-1}{2}$ then also it gives the same $4$-clique $\{\frac{p+1}{2},\frac{p-1}{2},\frac{3p+1}{2},\frac{3p-1}{2}\}$.

Case II: Let $a_{1}+a_{2}=1$. Here $a_{i}+a_{j}\in\{-1,p,p-1,p+1\}$ except $a_{1}+a_{2}$, where $1\leq i,j\leq 4$. Here two cases arise

1. (i)

If $a_{i}\in WNC(\mathbb{Z}_{2p})$ then the possible pair for $(a_{1},a_{2})$ are $(0,1)$ and $(p,p+1)$. If $a_{1}=0$ and $a_{2}=1$ then it gives $a_{3}=\{-1,p,p-1\}$. Since $p+(p-1)$ and $(-1)+p\in WNC(\mathbb{Z}_{2p})$, so we get the $4$-cliques $\{0,1,-1,p\}$ and $\{0,1,p,p-1\}$. Similarly if $a_{1}=p$ and $a_{2}=p+1$ then it gives two $4$-cliques $\{p,p+1,p-1,0\}$ and $\{p,p+1,-1,0\}$. 2. (ii)

Let $a_{i}\notin WNC(\mathbb{Z}_{2p})$, for some $1\leq i\leq 4$. Without loss of generality assume that $i=1$. By Lemma 3.6, $a_{1}\in\{\frac{p-1}{2},\frac{p+1}{2},\frac{3p-1}{2}\}$. If $a_{1}=\frac{p+1}{2}$ then it gives the $4$-clique namely $\{\frac{p-1}{2},\frac{p+1}{2},\frac{3p-1}{2},\frac{3p+1}{2}\}$. Again if $a_{1}=\frac{p-1}{2}$ or $\frac{3p-1}{2}$ then we will get a contradiction that $\{a_{1},a_{2},a_{3},a_{4}\}$ is a $4$-clique.

Case III: If $a_{1}+a_{2}=-1$, then similar to Case II, we conclude that the total possibilities of $4$-cliques are $\{0,-1,1,p\}$, $\{p,p-1,1,0\}$, $\{p,p-1,p+1,0\}$, $\{0,-1,p,p+1\}$ and $\{\frac{p-1}{2},\frac{3p-1}{2},\frac{p+1}{2},\frac{3p+1}{2}\}$.

Case IV: Let $a_{i}+a_{j}\in\{p,p-1,p+1\}$. Assume $a_{1}$ is fixed then the other three vertices of the $4$-clique are $p-a_{1},\,\,p+1-a_{1}$ and $p-1-a_{1}$, so $(p-a_{1})+(p+1-a_{1})=2p-2a_{1}+1\in\{p,p-1,p+1\}$. If $2p-2a_{1}+1=p$ then $a_{1}=\frac{p+1}{2}$ and $p+1-a_{1}=\frac{p+1}{2}$, a contradiction. Also $2p-2a_{1}+1\neq p-1$ and $2p-2a_{1}+1\neq p+1$. Hence in this case no such $4$-clique exists.

From the above $4$ cases we conclude that the only $4$-cliques of the weakly nil clean graph of $\mathbb{Z}_{2p}$ are $\{0,1,-1,p\},\{0,1,p,p-1\},\{0,-1,p,p+1\},\{0,p,p-1,p+1\}$ and $\{\frac{p-1}{2},\frac{p+1}{2},\frac{3p-1}{2},\frac{3p+1}{2}\}$

∎

Theorem 3.8**.**

The clique number of $G_{WN}(\mathbb{Z}_{2p})$ is $4$.

Proof.

If possible let $\{a,b,c,d,e\}$ be a $5$-clique. Then clearly it contains any one of the $4$-cliques mentioned in Theorem 3.7.

Case I: Suppose $\{a,0,1,-1,p\}$ be a $5$-clique. Then $a+0\in WNC(\mathbb{Z}_{2p})=\{0,1,-1,p,p-1,p+1\}$. If $a+0=0,1,-1$ or $p$ then $a=0,1,-1$ or$p$, a contradiction. If $a+0=p+1$ then $a=p+1$ implies $(p+1)+1\in WNC(\mathbb{Z}_{2p})$, a contradiction. Similarly if $a+0=p-1$ then we get $p-2\in WNC(R)$, a contradiction. Hence there is no possibility of $5$-clique containing the $4$-clique $\{0,1,-1,p\}$.

Case II: If the $5$-clique contains $4$-clique $\{0,1,p,p-1\}$ or $\{0,p,p+1,-1\}$ or $\{p,p+1,p-1,0\}$, then similar as Case I, we get a contradiction.

Case III: Suppose $\{a,\frac{p-1}{2},\frac{p+1}{2},\frac{3p-1}{2},\frac{3p+1}{2}\}$ be a $5$-clique. Then $a+\frac{p+1}{2}\in WNC(\mathbb{Z}_{2p})$. If $a+\frac{p+1}{2}=0$ then $a=\frac{3p-1}{2}$, a contradiction. If $a+\frac{p+1}{2}=1$ then $a=\frac{3p+1}{2}$, a contradiction. If $a+\frac{p+1}{2}=-1$ then $a=\frac{3p-3}{2}$ implies $\frac{3p-3}{2}+\frac{3p-1}{2}=p-2\in WNC(\mathbb{Z}_{2p})$, a contradiction. If $a+\frac{p+1}{2}=p$ then $a=\frac{p-1}{2}$, a contradiction. If $a+\frac{p+1}{2}=p+1$ then $a=\frac{p+1}{2}$, a contradiction and if $a+\frac{p+1}{2}=p-1$ then $a=\frac{p-3}{2}$ implies $\frac{p-3}{2}+\frac{p-1}{2}=p-2\in WNC(\mathbb{Z}_{2p})$, a contradiction.

From the above cases and Theorem 3.7 we conclude that $5$-clique does not exist in $G_{WN}(\mathbb{Z}_{2p})$. Hence the clique number of $G_{WN}(\mathbb{Z}_{2p})$ is $4$. ∎

3.3 Diameter

Lemma 3.9**.**

$R$* is weakly nil clean ring if and only if $diam(G_{WN}(R))=1$.*

Here we define an element $x\in R$ weakly nil clean element of type $1$ if $x=n+e$, and is of type $2$ if $x=n-e$, where $n\in Nil(R)$ and $e\in Idem(R)$.

Theorem 3.10**.**

Let $R$ and $S$ are two weakly nil clean rings but not nil clean rings then $diam(G_{WN}(R\times S))$ is either $2$ or $3$.

Proof.

Clearly $R\times S$ is not weakly nil clean ring by Theorem 2.3 [2] and $diam(R\times S)\geq 2$ by Lemma 3.9. Let $(x_{1},x_{2}),(y_{1},y_{2})\in R\times S$, so $x_{i}=n_{i}+e_{i}$ or $x_{i}=n_{i}-e_{i}$ and $y_{i}=m_{i}+f_{i}$ or $y_{i}=m_{i}-f_{i}$, where $n_{i}\in Nil(R)$, $e_{i}\in Idem(R)$, $m_{i}\in Nil(S)$, $f_{i}\in Idem(S)$ and $1\leq i\leq 2$.

Case I: If all $x_{i}$ and $y_{i}$ are weakly nil clean element of same type then $(x_{1},x_{2})\leftrightarrow(0,0)\leftrightarrow(y_{1},y_{2})$ is a path from $(x_{1},x_{2})$ to $(y_{1},y_{2})$ in $G_{WN}(R\times S)$.

Case II: If any two from $x_{i}$ and $y_{i}$ are weakly nil clean elements of different types then without loss of generality assume $x_{1}=n_{1}-e_{1}$ and all are weakly nil clean elements of type 1. Clearly $(x_{1},x_{2})\leftrightarrow(e_{1},0)\leftrightarrow(0,0)\leftrightarrow(y_{1},y_{2})$ is a path from $(x_{1},x_{2})$ to $(y_{1},y_{2})$.

Case III: If $x_{1}$ and $x_{2}$ are weakly nil clean elements of different types and $y_{1}$ and $y_{2}$ are also weakly nil clean elements of different types. Assume that $x_{1}$ and $y_{2}$ are of type 1 and $x_{2}$ and $y_{1}$ are of type 2, then $(x_{1},x_{2})\leftrightarrow(0,e_{2})\leftrightarrow(f_{1},0)\leftrightarrow(y_{1},y_{2})$ is a path from $(x_{1},x_{2})$ to $(y_{1},y_{2})$. Similarly for any pair satisfying Case III we get a path of length $3$ from $(x_{1},x_{2})$ to $(y_{1},y_{2})$.

Hence from the above three cases we conclude that the diameter of $G_{WN}(R\times S)$ is either $2$ or $3$.

∎

The weakly nil clean graph of $\mathbb{Z}_{2p}$ is given below for the next theorem on diameter of $G_{WN}(\mathbb{Z}_{2p})$.

Theorem 3.11**.**

The following hold for the ring $\mathbb{Z}_{n}:$

1. (i)

If $n=2^{k}3^{l}$, for some integer $k\geq 0$ and $l\geq 0$, then $diam(G_{WN}(\mathbb{Z}_{n}))=1$. 2. (ii)

For a prime $p$, $diam(G_{WN}(\mathbb{Z}_{p}))=\frac{p-1}{2}$. 3. (iii)

If $n=2p$, where $p\geq 5$ is an odd prime, then $diam(G_{WN}(\mathbb{Z}_{n}))=\frac{p-1}{2}$. 4. (iv)

For a finite field $F$ having order $p^{k}$, $diam(G_{WN}(F))=\infty$ if and only if $k>1$.

Proof.

$(i)$ is follows from Theorem $2.9$ [2] and Lemma $3.9$. $(ii)$ and $(iii)$ can be shown easily from the weakly nil clean graph of $\mathbb{Z}_{p}$ in Figure $3$, $4$ and weakly nil clean graph of $\mathbb{Z}_{2p}$ in Figure $7$. $(iv)$ follows from Figure $5$ [3]. ∎

3.4 Chromatic index

Let $\vartriangle$ be the maximum vertex degree of $G$, then Vizing’s theorem [7] gives $\vartriangle\leq\chi^{\prime}(G)\leq\vartriangle+1$. Vizing’s theorem divides the graphs into two classes according to their chromatic index; graphs satisfying $\chi^{\prime}(G)=\vartriangle$ are called graphs of class $1$, those with $\chi^{\prime}(G)=\vartriangle+1$ are graphs of class $2$.

Following theorem shows that $G_{WN}(R)$ is of class $1$.

Theorem 3.12**.**

Let $R$ be a ring then the weakly nil clean graph of $R$ is of class $1$.

Proof.

We colour the edge $ab$ by the colour $a+b$. By this colouring, every two distinct edges $ab$ and $ac$ have different colours and $C=\{a+b\,\,|\,\,ab\mbox{ is an edge in }G_{WN}(R)\}$ is the set of colours. Therefore the weakly nil clean graph has a $|C|$-edge colouring and so $\chi^{\prime}(G_{WN}(R))\leq|C|$. But $C\subset WNC(R)$ and $\chi^{\prime}(G_{WN}(R))\leq|C|\leq|WNC(R)|$. By Lemma 2.4, $\vartriangle\leq|WNC(R)|$ and so by Vizing’s theorem, we have $\chi^{\prime}(G_{WN}(R))\geq\vartriangle=|WNC(R)|$. Therefore $\chi^{\prime}(G_{WN}(R))=|WNC(R)|=\vartriangle,$ i.e., $G_{WN}(R)$ is of class $1$. ∎