The zero-divisor graph of a commutative ring was first introduced by Beck in [1]. He showed some conclusions about coloring. In [2], Anderson and Livingston studied a zero-divisor graph of a ring $ R $, denoted by $ \Gamma(R) $, whose vertices are nonzero divisors of $ R $, and two distinct vertices $ x $ and $ y $ are adjacent if and only if $ xy=0 $. In [3], Akbari and Habibi introduced idempotent graph, denoted by $ I(R) $, whose vertices are all nontrivial idempotent elements of a ring $ R $, and two distinct vertices $ e $ and $ f $ are adjacent if and only if $ ef=fe=0 $. Habibi [4] and Celikel introduced clean graph of a ring $ R $, denoted by $ Cl(R) $, which is a simple graph with vertices in form $ (e, u) $, where $ e $ is an idempotent and $ u $ is a unit of $ R $, and two distinct vertices $ (e, u) $ and $ (f, v) $ are adjacent if and only if $ ef=fe=0 $ or $ uv=vu=1 $. They investigated the clique number, chromatic number, independence number and domination number of the subgraph $ Cl_{2}(R) $ of $ Cl(R) $, where $ R $ is a commutative Artin ring with identity, see [4]. In [5], Boonsawang discussed the clean graph of the finite ring $ \mathbb{F}_{p^{n}}\times\mathbb{F}_{q^{m}} $, where $ p $ and $ q $ are primes, $ n $ and $ m $ are positive integers, and he gave two explicit structures of $ Cl_{2}({\mathbb{F}_{p^{n}}\times\mathbb{F}_{q^{m}}}) $. In recent years, graph theory associated with rings has attracted many researches and scholars, see for instance [6-8].

In this paper, $ R $ always denotes a finite commutative ring with identity. An element $ e $ of $ R $ is called an idempotent if $ e^{2}=e $, and $ u $ of $ R $ is called a unit, if there exists $ v $ of $ R $ such that $ uv=1 $. By $ Id(R) $, $ Id^{*}(R) $ and $ U(R) $, we mean the sets of idempotents, nonzero idempotents and units of $ R $, respectively. If an element $ a $ of $ R $ can be written by $ e+u $, where $ e\in Id(R) $, $ u\in U(R) $, then $ a $ is called clean. The ring $ R $ is clean if every element of $ R $ is clean. For any undefined notation or terminology in ring theory, we refer the reader to [9].

Let us briefly mention some other notations about graph theory which will be used in this paper. Let $ G $ denote a graph. Then the vertex set of $ G $ is referred to as $ V(G) $, its edge set as $ E(G) $. Two adjacent vertices $ a $ and $ b $ in $ G $ are denoted as $ a\sim b $. $ G $ is called connected if any two of its vertices are linked by a path in $ G $. The distance $ d_{G}(x, y) $ in $ G $ of two vertices is the length of a shortest $ x-y $ path in $ G $. The greatest distance between any two vertices in $ G $ is the diameter of $ G $, denoted by $ diam(G) $. If all the vertices of $ G $ are pairwise adjacent, then $ G $ is complete. A complete graph on $ n $ vertices is a $ K_{n} $, a $ K_{3} $ is called a triangle. The degree $ d_{G}(v)=d(v) $ of a vertex $ v $ is the number $ |E(v)| $ of edges at $ v $, the number $ \delta(G)=min\{d(v)|v\in V\} $ is the minimum degree of $ G $, the number $ \Delta(G)=max\{d(v)|v\in V\} $ is maximum degree. If all the vertices of $ G $ have the same degree $ k $, then $ G $ is $ k $-regular, or simply regular. Call a closed walk in a graph an Euler tour if it traverses every edge of the graph exactly once. A graph is Eulerian if it admits an Euler tour. The greatest integer $ r $ such that $ K_{r}\subseteq G $ is the clique number $ \omega(G) $ of $ G $. A vertex colouring of a graph $ G=(V, E) $ is a map $ C:V\rightarrow S $ with $ C(v)\neq C(w) $ whenever $ v $ and $ w $ are adjacent. The elements of the set $ S $ are called the available colours. All that interests us about $ S $ is its size: typically, we shall be asking for the smallest integer $ k $ such that $ G $ has a $ k $-colouring, a vertex colouring $ C:V\rightarrow \{1, \cdots, k\} $. This $ k $ is the (vertex-) chromatic number of $ G $, it is called by $ \chi(G) $. We refer the reader to [10] and [11] for general background on graph theory and for all undefined notions used in the text.

We define $ U $-clean graph of a ring R, denoted by $ U $-$ Cl(R) $, whose vertices are related to clean elements of $ R $. $ U $-$ Cl(R) $ is a graph with vertices $\{ (e, u) $: $ e\in Id^{*}(R) \}$, $ u\in U(R) $ and two distinct vertices $ (e, u) $ and $ (f, v) $ are adjacent if and only if $ ef=1 $ or $ uv=1 $. Let $ U $-$ Cl_{1}(R) $ be the subgraph of $ U $-$ Cl(R) $ induced by $ \{(1, u) $: $ u \in U(R)\} $.Let $ U $-$ Cl_{2}(R) $ be the subgraph of $ U $-$ Cl(R) $ induced by $ \{(e, u) $: $ 1\neq e\in Id^{*}(R) $, $ u\in U(R)\}$. In section $ 2 $, we give some examples for $ U $-clean graph to illustrate our definitions. We also discuss the graph structures such as connectivity, diameter, degree, regularity, etc. In section $ 3 $, we focus on the $ U $-clean graph of the finite ring $ \mathbb{Z}_{p}\times\mathbb{Z}_{q} $, where $ p $ and $ q $ are primes. Among many results in this section, we determine the degree of $ U $-$ Cl(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $, and prove $ U $-$ Cl(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $ is Eulerlian if and only if $ p=2 $, $ q=2 $. In section $ 4 $, we discuss the parameters $ (\omega(G), \chi(G)) $ of $ U $-clean graph of a finite commutative ring $ R $, and we obtain the exact value of these parameters.

Definition 2.1  Let $ R $ be a finite commutative ring with nonzero identity $ 1 $. The $ U $-clean graph, denoted by $ U $-$ Cl(R) $ of $ R $, which is a graph with vertices $ \{(e, u) $: $ e\in Id^{*}(R) $, $ u\in U(R)\} $. Two distinct vertices $ (e, u) $ and $ (f, v) $ are adjacent if and only if $ ef=1 $ or $ uv=1 $.

Let $ U $-$ Cl_{1}(R) $ be the subgraph of $ U $-$ Cl(R) $ induced by $\{ (1, u) $: $ u\in U(R)\} $. Let $ U $-$ Cl_{2}(R) $ be the subgraph of $ U $-$ Cl(R) $ induced by {$ (e, u) $: $ 1\neq e\in Id^{*}(R) $, $ u\in U(R) $}.

Definition 2.2  Let $ R $ be a finite commutative ring. Let $ U_{1}(R) $ and $ U_{2}(R) $ be the subsets of $ U(R) $, denote by $ U_{1}(R) $=$ \{ u\in U(R) $: $ u^{-1}=u \}$ and $ U_{2}(R) $=$ \{ u\in U(R) $: $ u^{-1}\neq u \}$, respectively.

Remark 2.3  $ U $-$ Cl_{1}(R) $ is a complete subgraph of $ U $-$ Cl(R) $, $ U $-$ Cl(R) $ is the union of $ U $-$ Cl_{1}(R) $ and $ U $-$ Cl_{2}(R) $. If $ R $ has no nontrivial idempotent and $ |U(R)|=n $, then $ U $-$ Cl(R) $ is a complete graph, $ K_{n} $. If $ R $ has a nontrivial idempotent and $ U_{2}(R)=\varnothing $, then there is a subgraph of $ U $-$ Cl(R) $, $ K_{|U_{1}(R)|} $.

Example 2.4  $ (1) $ $ Id^{*}(\mathbb{Z}_{2}\times\mathbb{Z}_{2})=\{(0, 1), (1, 0), (1, 1)\} $, $ U(R)=\{(1, 1)\} $. It is easy to check that $ V $($ U $-$ Cl $($ \mathbb{Z}_{2}\times\mathbb{Z}_{2} $))=$ \{((0, 1), (1, 1)), ((1, 0), (1, 1)), ((1, 1), (1, 1))\} $.

(2) For the ring $ \mathbb{Z}_{2}\times\mathbb{Z}_{3} $, $ V $($ U $-$ Cl(\mathbb{Z}_{2}\times\mathbb{Z}_{3})) $={((0, 1), (1, 1)), ((0, 1), (1, 2)), ((1, 0), (1, 1)), ((1, 0), (1, 2)), ((1, 1), (1, 1)), ((1, 1), (1, 2))}.

(3) Note that $ V $($ U $-$ Cl $($ \mathbb{Z}_{3}\times\mathbb{Z}_{3} $))=$ \{((0, 1), (1, 1)), ((0, 1), (1, 2)), ((0, 1), (2, 2)), ((0, 1), (2, 1)), ((1, 0), (1, 1)), ((1, 0), (1, 2)), ((1, 0), (2, 1)), ((1, 0), (2, 2)), ((1, 1), (1, 1)), ((1, 1), (1, 2)), ((1, 1), (2, 1)), ((1, 1), (2, 2))\} $.

Theorem 2.5   Let $ R $ be a finite commutative ring. Then $ U $-$ Cl(R) $ is connected and $ diam $($ U $-$ Cl(R) $)$ \leq3 $.

Proof   Let $ x=(e, u) $ and $ y=(f, v) $ be two vertices of $ U $-$ Cl(R) $. It is clear that $ (e, u)\sim(1, u^{-1})\sim(1, v^{-1})\sim(f, v) $, thus $ U $-$ Cl(R) $ is connected.

We discuss the following two cases.

Case 1. $ e=f $. If $ e=f=1 $, then $ (1, u)\sim(1, v) $ is a path; if $ e, f\neq1 $, then $ (e, u)\sim(1, u^{-1})\sim(1, v^{-1})\sim(f, v) $, thus $ d_{G}(x, y)=3 $.

Case 2. $ e\neq f $. If $ e $ or $ f $ is equal to $ 1 $, we assume that $ e=1 $ without loss of generality. It follows that $ (1, u)\sim(1, v^{-1})\sim(f, v) $ is a path of length $ 2 $; if $ e, f\neq1 $, then $ (e, u)\sim(1, u^{-1})\sim(1, v^{-1})\sim(f, v) $, thus $ d_{G}(x, y)=3 $.

Hence, $ diam $($ U $-$ Cl(R) $)$ \leq3 $.

This completes the proof of Theorem 2.5.

Lemma 2.6   Let $ R $ be a commutative ring. If $ R $ has two different nontrivial idempotents or three different units, then $ U $-$ Cl(R) $ has a triangle.

Proof    If $ e, f\in Id^{*}(R)\backslash\{1\} $ and $ e\neq f $, then there is a cycle $ (e, 1)\sim(1, 1)\sim(f, 1)\sim(e, 1) $. If $ u, v, w\in U(R) $ and they are different, then there is a triangle $ (1, u)\sim(1, v)\sim(1, w)\sim(1, u) $.

Thus the result follows.

Lemma 2.7   Let $ R $ be a commutative ring with nontrivial idempotents, and $ x=(e, u) $ be a vertex of $ U $-$ Cl(R) $.

$ (1) $ If $ e=1 $, then

$ \begin{align*} d(x)=|Id^{*}(R)|+|U(R)|-2.\end{align*} $

(2.1)

$ (2) $ If $ e\neq1 $, then

$ \begin{align*} d(x)=\left\{ \begin{aligned} |Id^{*}(R)|-1 & & {\text{if}\; \; u\in U_{1}(R), }\\ |Id^{*}(R)|\; \; \; \; \; \; \; & & {\text{if}\; \; u\in U_{2}(R).} \end{aligned} \right. \end{align*} $

(2.2)

Proof   $ (1) $ If $ u\in U_{1}(R) $, then $ (1, u)\sim (e_{1}, u) $, $ e_{1}\in{Id^{*}(R)\backslash\{1\}} $. $ (1, u)\sim (1, u_{1}) $, $ u_{1}\in{U(R)\backslash\{u\}} $. Thus

$ \begin{equation*} d(x)=|Id^{*}(R)|+|U(R)|-2. \end{equation*} $

If $ u\in U_{2}(R) $, then $ (1, u)\sim (e_{1}, u^{-1}) $, $ e_{1}\in {Id^{*}(R)\backslash\{1\}} $. $ (1, u)\sim (1, u_{1}) $, $ u_{1}\in{U(R)\backslash\{u\}} $. Thus

$ \begin{equation*} d(x)=|Id^{*}(R)|+|U(R)|-2. \end{equation*} $

$ (2) $ If $ u\in U_{1}(R) $, then $ (e, u)\sim (e_{1}, u) $, $ e_{1}\in{Id^{*}(R)\backslash\{e\}} $. Thus

$ \begin{equation*} d(x)=|Id^{*}(R)|-1. \end{equation*} $

If $ u\in U_{2}(R) $, then $ (e, u)\sim (e_{1}, u^{-1}) $, $ e_{1}\in Id^{*}(R) $. Thus

$ \begin{equation*} d(x)=|Id^{*}(R)|. \end{equation*} $

Thus we are done.

Theorem 2.8  Let $ R $ be a commutative ring. Then $ U $-$ Cl(R) $ is regular if and only if $ |U(R)|=1 $ or $ R $ has no nontrivial idempotent.

proof  If $ |U(R)|={1} $, then

$V(U-C l(R))=\left\{(e, 1): e \in I d^*(R)\right\}, U-C l(R) \cong K_{\left|I d^*(R)\right|} .$

If $ R $ has no nontrivial idempotent, then

$V(U-C l(R))=\{(1, u): u \in U(R)\}, U-C l(R) \cong K_{|U(R)|} .$

Conversely, let $ U $-$ Cl(R) $ be regular. Assume that $ |U(R)|\geq2 $ and $ e $ is a nontrivial idempotent. Let $ x=(e, 1) $, $ y=(1, 1) $. Then by (2.1), (2.2), we have $ d(x)=|Id^{*}(R)|-1 $ and $ d(y)=|Id^{*}(R)|+|U(R)|-2 $, thus $ d(x)<d(y) $. Hence $ U $-$ Cl(R) $ is not regular, a contradiction.

This completes the proof of Theorem 2.8.

Proposition 2.9   Let $ R $ be a finite commutative ring. Then the following statements hold:

(1)

$ \Delta(U-C l(R))=|U(R)|+\left|I d^*(R)\right|-2 . $

(2.3)

(2)

$ \delta(U-C l(R))= \begin{cases}\left|I d^*(R)\right|-1 & \text { if } R \text { has nontrivial idempotents } \\ |U(R)|-1 & \text { if } R \text { has no nontrivial idempotent }\end{cases} $

(2.4)

Example 2.10  $ (1) $ Let $ \mathbb{Z}_{12} $ be the ring of integers modulo 12. Note that $ Id^{*}(R)=\{1, 4, 9\} $, $ U(R)=\{1, 5, 7, 11\} $. By (2.3), (2.4), we have

$ \Delta\left(U-C l\left(\mathbb{Z}_{12}\right)\right)=5, \delta\left(U-C l\left(\mathbb{Z}_{12}\right)\right)=2,$

and the $ U-C l\left(\mathbb{Z}_{12}\right)$.

$ (2) $ Let $ R $ be $ \mathbb{Z}_{3}[x]/(x^{2}+1) $. Then $ U $-$ Cl(R) $ is regular, because the degree of any vertex in $ U $-$ Cl(R) $ is 7. We give $ U $-$ Cl(R) $ as follows.

Let $ p $ and $ q $ be primes such that $ p\leq q $. In this part, we show the degree of each vertex of $ U $-$ Cl(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $. It is obvious that $ Id^{*}(\mathbb{Z}_{p}\times\mathbb{Z}_{q})=\{(0, 1), (1, 0), (1, 1)\} $ and $ |U(\mathbb{Z}_{p}\times\mathbb{Z}_{q})|=(p-1)(q-1) $. From Theorem 2.5, $ U $-$ Cl(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $ is connected.

We show an explicit structure of $ U $-$ Cl(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $ by considering $ U_{2}(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $. If $ U_{2}(\mathbb{Z}_{p}\times\mathbb{Z}_{q})=\varnothing $, we have shown as Example 2.4. If $ U_{2}(\mathbb{Z}_{p}\times\mathbb{Z}_{q})\neq\varnothing $, we consider $ U $-$ Cl(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $ by partitioning the vertex set of $ U $-$ Cl(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $:

$ \begin{align*} \left\{ \begin{aligned} A_{1}=\{((0, 1), (x, y))|(x, y)^{-1}=(x, y)\}, \\ A_{2}=\{((1, 0), (x, y))|(x, y)^{-1}=(x, y)\}, \\ A_{3}=\{((1, 1), (x, y))|(x, y)^{-1}=(x, y)\}, \\ A_{4}=\{((0, 1), (x, y))|(x, y)^{-1}\neq(x, y)\}, \\ A_{5}=\{((1, 0), (x, y))|(x, y)^{-1}\neq(x, y)\}, \\ A_{6}=\{((1, 1), (x, y))|(x, y)^{-1}\neq(x, y)\}.\\ \end{aligned} \right. \end{align*} $

(3.1)

Proposition 3.1  Let $ R $ be $ \mathbb{Z}_{p} \times \mathbb{Z}_q $. Then

$ \begin{equation*} |U_{1}(R)\bigl| =\left\{ \begin{array}{rcl} 1 & & {\text{if}\; \; p=q=2, }\\ 2 & \; \; & {\text{if}\; \; p=2 \; \; \text{and}\; \; q\geq 3, }\\ 4 & & {\text{if}\; \; p\geq3 \; \; \text{and}\; \; q\geq 3, } \end{array} \right. \end{equation*} $

and

$ \begin{equation*} |U_{2}(R)\bigl| =\left\{ \begin{array}{rcl} (p-1)(q-1)-1 & & {\text{if}\; \; p=q=2, }\\ (p-1)(q-1)-2 & & {\text{if}\; \; p=2 \; \; \text{and}\; \; q\geq 3, }\\ (p-1)(q-1)-4 & & {\text{if}\; \; p\geq3 \; \; \text{and}\; \; q\geq 3.} \end{array} \right. \end{equation*} $

Proposition 3.2  Let $ R $ be $ \mathbb{Z}_{p} \times \mathbb{Z}_q $, and $ A_{i} $ be in (3.1). Then the following statements hold:

$ (1) $ $ \bigcup_{i=1}^{6}A_{i} $=$ V $($ U $-$ Cl $($ R $)) and $ A_{i}\cap A_{j}=\varnothing $, $ 1\leq i <j\leq 6 $.

$ (2) $ For $ i\in\{1, 2, 3\} $,

$ \begin{align*} |A_{i}| = \left\{ \begin{array}{lll} 1 & & {\text{if}\; \; p=q=2, }\\ 2 & & {\text{if}\; \; p=2 \; \; \text{and}\; \; q\geq 3, }\\ 4 & & {\text{if}\; \; p\geq3 \; \; \text{and}\; \; q\geq 3.} \end{array} \right. \end{align*} $

(3.2)

For $ j\in\{4, 5, 6\} $,

$ \begin{align*} |A_{j}| =\left\{ \begin{array}{lll} 0 & & {\text{if}\; \; p=q=2, }\\ q-3 & & {\text{if}\; \; p=2 \; \; \text{and}\; \; q\geq 3, }\\ (p-1)(q-1)-4 & & {\text{if}\; \; p\geq3 \; \; \text{and}\; \; q\geq 3, } \end{array} \right. \end{align*} $

(3.3)

Proposition 3.3  Let $ p, \; q $ be primes with $ p+q>6 $. For $ A_{i} $ in (3.1), the following statements hold:

$ (1) $ No two distinct vertices in $ A_{i} $ are adjacent, $ i\in\{1, 2\} $.

$ (2) $ For each vertex in $ A_{i} $, there exists a unique distinct vertex in $ A_{i} $ such that they are adjacent, $ i\in\{4, 5\} $.

$ (3) $ The subgraph induced by $ A_{i} $ is complete, $ i\in\{3, 6\} $.

$ (4) $ The subgraph induced by $ A_{3}\cup A_{6} $ is complete.

$ (5) $ The subgraph induced by $ A_{i}\cup A_{j} $ is bipartite graph, $ (i, j)\in\{(1, 2), (4, 5)\} $.

$ (6) $ Each vertex in $ A_{i} $ is not adjacent to any vertex in $ A_{j} $, $ (i, j)\in\{(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5)\} $.

Proof  $ (1) $ It suffices to prove the case $ i=1 $. Let $ ((0, 1), (x, y)) $ and $ ((0, 1), (a, b)) $ be two distinct vertices in $ A_{1} $. Since $ (0, 1)\cdot(0, 1)\neq(1, 1) $ and $ (x, y)\cdot(a, b)\neq(1, 1) $, we get that $ ((0, 1), (x, y)) $ and $ ((0, 1), (a, b)) $ are not adjacent.

$ (2) $ Suppose that $ ((0, 1), (x, y))\in A_{4} $. $ (x, y)^{-1}\neq(x, y) $, $ ((0, 1), (x, y)^{-1})\in A_{4} $. We know that $ ((0, 1), (x, y)) $ and $ ((0, 1), (x, y)^{-1}) $ are adjacent, because $ (x, y)\cdot(x, y)^{-1}=(1, 1) $. Next, let $ ((0, 1), (a, b))\in A_{4} $ such that $ (a, b)\neq(x, y)^{-1} $. Since $ (0, 1)\cdot(0, 1)\neq(1, 1) $ and $ (x, y)\cdot(a, b)\neq(1, 1) $, we have $ ((0, 1), (x, y)) $ is not adjacent to $ ((0, 1), (a, b)) $. The case $ i=5 $ can be proved similarly.

$ (3) $ Since $ (1, 1)\cdot(1, 1)=(1, 1) $, there is an edge between $ ((1, 1), (x, y)) $ and $ ((1, 1), (a, b)) $.

$ (4) $ The proof is similar to $ (3) $.

$ (5) $ The proof splits into the following cases:

Case 1. $ (i, j)=(1, 2) $. The subgraph induced by $ A_{1} \cup A_{2} $ is shown as follows:

Case 2. $ (i, j)=(4, 5) $. The subgraph induced by $ A_{4} \cup A_{5} $ is shown as follows:

Note that $ \{u_{1}, u_{2}, \cdots, u_{n}\}\in U_{1}(R) $ and $ \{v_{1}, v_{2}, \cdots, v_{n}, v_{1}^{-1}, v_{2}^{-1}, \cdots, v_{n}^{-1}\}\in U_{2}(R) $.

$ (6) $ Let $ ((0, 1), (x, y))\in A_{1} $ and $ ((0, 1), (a, b))\in A_{4} $. Then $ (x, y)^{-1}=(x, y) $ and $ (a, b)^{-1}\neq(a, b) $. It follows that $ (x, y)\neq(a, b) $. So because of $ (0, 1)\cdot(0, 1)\neq(1, 1) $ and $ (x, y)\cdot(a, b)\neq(1, 1) $, we see that$ ((0, 1), (x, y)) $ is not adjacent to $ ((0, 1), (a, b)) $.

Thus we are done.

Example 3.4  Let $ p=2 $, $ q=5 $. Then $ |U_{1}(\mathbb{Z}_{2}\times\mathbb{Z}_{5})|=2 $, $ |U_{2}(\mathbb{Z}_{2}\times\mathbb{Z}_{5})|=2 $, $ |A_{1}|=|A_{2}|=|A_{3}|=2 $ and $ |A_{4}|=|A_{5}|=|A_{6}|=2 $, $ U $-$ Cl(\mathbb{Z}_{2}\times\mathbb{Z}_{5}) $ is shown as follows:

Lemma 3.5  [10, Theorem 1.8.1] A connected graph is Eulerian if and only if every vertex has even degree.

Lemma 3.6   Let $ (x, y) $ be a vertex in $ U $-$ Cl(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $. If $ p+q>6 $, then

$ \begin{align*} d(x, y)= \left\{ \begin{array}{lll} 2 &{\text{if}\; \; (x, y)\in A_{1}}, \\ 2 &{\text{if}\; \; (x, y)\in A_{2}}, \\ 1+(p-1)(q-1) &{\text{if}\; \; (x, y)\in A_{3}}, \\ 3 &{\text{if}\; \; (x, y)\in A_{4}}, \\ 3 &{\text{if}\; \; (x, y)\in A_{5}}, \\ 1+(p-1)(q-1) &{\text{if}\; \; (x, y)\in A_{6}}. \end{array} \right. \end{align*} $

(3.4)

Proof   Case 1. Let $ (x, y) $ be the vertex in $ A_{1} $. By Proposition 3.3, each vertex in $ A_{1} $ is adjacent to a unique vertex in $ A_{2} $ and $ A_{3} $, respectively. Then $ d(x, y)=2 $.

Case 2. It is similar to Case 1.

Case 3. Let $ (x, y) $ be a vertex in $ A_{3} $. By Proposition 3.3, the subgraph induced by $ A_{3}\cup A_{6} $ is a complete graph. Each vertex in $ A_{3} $ is adjacent to a unique vertex in $ A_{1} $ and $ A_{2} $, respectively. Then

$ \begin{equation*} d(x, y)=|A_{3}|+|A_{6}|+1. \end{equation*} $

Let $ p=q=2 $. Then by (3.2), (3.3), we obtain

$ \begin{equation*} d(x, y)=1+(p-1)(q-1). \end{equation*} $

Let $ p=2 $ and $ q\geq 3 $. Then by (3.2), (3.3), we have

$ \begin{equation*} d(x, y)=1+(p-1)(q-1). \end{equation*} $

Let $ p\geq 3 $ and $ q\geq 3 $. Then by (3.2), (3.3), we get

$ \begin{equation*} d(x, y)=1+(p-1)(q-1). \end{equation*} $

Case 4. Let $ (x, y) $ be the vertex in $ A_{4} $. By Proposition 3.3, each vertex in $ A_{4} $ is adjacent to a unique vertex in $ A_{5} $ and $ A_{6} $, respectively. For each vertex in $ A_{4} $, there exists a unique vertex in $ A_{4} $, which is adjacent. Then $ d(x, y)=3 $.

Case 5. It is similar to Case 4.

Case 6. It is similar to Case 3.

This completes the proof of Lemma 3.6.

Theorem 3.7  Let $ p $, $ q $ be primes. Then $ U $-$ Cl(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $ is Eulerian if and only if $ p=2 $, $ q=2 $.

Proof  On the one hand, we consider the case $ U_{2}(\mathbb{Z}_{p}\times\mathbb{Z}_{q})=\varnothing $.

Case 1. $ p=q=2 $. By Figure 1, $ U $-$ Cl(\mathbb{Z}_{2}\times\mathbb{Z}_{2}) $ is an Eulerian.

Case 2. $ p=2 $, $ q=3 $. By Figure 2, $ U $-$ Cl(\mathbb{Z}_{2}\times\mathbb{Z}_{3}) $ has a vertex of odd degree.

Case 3. $ p=3 $, $ q=3 $. By Figure 3, $ U $-$ Cl(\mathbb{Z}_{3}\times\mathbb{Z}_{3}) $ has a vertex of odd degree.

On the other hand, we consider the case $ p+q>6 $. If $ v_{1}\in A_{4} $, then $ d(v_{1})=3 $ by (3.4). Thus $ U $-$ Cl(\mathbb{Z}_{p}\times\mathbb{Z}_{q}) $ has a vertex of odd degree.

This completes the proof of Theorem 3.7.

In next section, we will give some parameters of $ U $-clean graph of every finite commutative ring $ R $. Moreover, the exact value of these parameters is given.

For any finite commutative ring $ R $ with $ Id^{*}(R)=\{1\} $. It follows that

$ V(U-C l(R))=\{(1, u) \mid u \in U(R)\}, U-C l(R)=K_{|U(R)|},$

thus

$ \omega(U-C l(R))=\chi(U-C l(R))=|U(R)| . $

Theorem 4.1  Let $ R $ be a finite commutative ring and $ R_{1}, \cdots, R_{n} $ be rings with $ Id^{*}(R_{i})=\{1\} $. If $ R\cong R_{1}\times \cdots \times R_{n} $, then

$ \omega(U-C l(R))=\chi(U-C l(R))=max \left\{|U(R)|, 2^n-1\right\} .$

Proof   Assume that $ 1_{R} $ is identity of $ R $. We divide the set $ V $ as follows:

$ \begin{equation*} \left. \begin{array}{lll} V_{1}=\{(e, u)|e\in Id^{*}(R), u\in U_{1}(R)\}, \\[7pt] V_{2}=\{(e, u)|(1_{R}\neq e\in Id^{*}(R), u\in U_{2}(R)\}, \\[7pt] V_{3}=\{(1_{R}, u)|u\in U_{2}(R)\}. \end{array} \right. \end{equation*} $

Clearly, the subgraphs induced by $ \{(1_{R}, u)|u\in U_{1}(R)\}\in V_{1} $ and $ \{(e, u_{1})|e\in Id^{*}(R), u_{1}\in U_{1}(R)\}\in V_{1} $ are both complete. By $ K_{max\{|U_{1}(R)|, \; 2^{n}-1\}} $, $ K_{2} $, $ K_{|U_{2}(R)|} $, we mean the largest complete subgraphs of $ V_{1} $, $ V_{2} $ and $ V_{3} $, respectively. Each vertex in $ V_{1} $ is not adjacent to any vertex in $ V_{2} $, because $ e\neq 1_{R} $ and $ u_{1}u_{2}\neq 1_{R} $, where $ u_{1}\in U_{1}(R) $, $ u_{2}\in U_{2}(R) $. There is no vertex in $ V_{3} $ that is adjacent to both vertices in $ \{(e^{'}, u), (e^{''}, u^{-1})\} $, where $ e^{'}, e^{''}\neq 1_{R} $ and $ u\in U_{2}(R) $. Then each vertex in $ V_{3} $ is adjacent to all vertices in $ \{(1_{R}, u)|u\in U_{1}(R)\}\subseteq V_{1} $. Thus

$\omega(U-C l(R))=max \left\{|U(R)|, 2^n-1\right\} .$

Note that

$\omega(U-C l(R)) \leq \chi(U-C l(R)) .$

Next, we consider the following two cases:

Case 1. $ |U(R)|\geq 2^{n}-1 $. Let $ V_{11}=\{(1_{R}, u)|u\in U_{1}(R)\} $. We color the vertices of $ V_{11} $ by $ 1, \cdots, |U_{1}(R)| $ and $ V_{3} $ by $ |U_{1}(R)|+1, \cdots, |U(R)| $, respectively. We show the following figure to provide a more intuitive experience of the coloring situation:

Note that $ u_{1}, \cdots, u_{|U_{1}(R)|}\in U_{1}(R), u_{|U_{1}(R)|+1}, \cdots, u_{|U(R)|}\in U_{2}(R) $ and $ e_{1}, \cdots, e_{2^{n}-2}\in Id^{*}(R) $ are not equal to $ 1_{R} $. Clearly,

$\chi(U-C l(R)) \leq \omega(U-C l(R)), $

and thus,

$\omega(U-C l(R))=\chi(U-C l(R))=|U(R)| .$

Case 2. $ |U(R)|< 2^{n}-1 $. Let $ V_{e1}=\{(e_{i}, u_{1})|e_{i}\in Id^{*}(R), u_{1}\in U_{1}(R)\} $. Then we color the vertices of $ V_{e1} $ by $ 1, \cdots, 2^{n}-1 $. Similar to Case $ 1 $, we have

$\chi(U-C l(R)) \leq \omega(U-C l(R)), $

and thus,

$\omega(U-C l(R))=\chi(U-C l(R))=2^n-1 .$

Hence

$\omega(U-C l(R))=\chi(U-C l(R))=max \left\{|U(R)|, 2^n-1\right\} .$

This completes the proof of Theorem 4.1.

In [9, Theorem 8.7], it was proved that every commutative Artin ring is uniquely a finite direct product of Artin local rings. Then we have the following result.

Corollary 4.2  Let R be a commutative Artin ring with n distinct maximal ideals. Then we have

$ \begin{equation*} \omega(U -Cl(R))=\chi(U -Cl(R))= \left\{ \begin{array}{lll} |U(R)| & & {\text{if}\; \; n=1, }\\ max\{|U(R)|, \; 2^{n}-1\} & & {\text{if}\; \; n\geq 2.} \end{array} \right. \end{equation*} $

Proof   It is clear that $ R \cong R_1 \times \cdots \times R_n$, where R_i is a local ring for each $i \in\{1, \cdots, n\}$.The proof is directly from Theorem 4.1.

Corollary 4.3  Let m, n be positive integers, and p, q be primes with p ≤ q. Then

$\omega\left(U-C l\left(\mathbb{Z}_{p^m} \times \mathbb{Z}_{q^n}\right)\right)=\chi\left(U-C l\left(\mathbb{Z}_{p^m} \times \mathbb{Z}_{q^n}\right)\right)= \begin{cases}3 & \text { if } p^m+q^n \leq 5, \\ \left(p^m-p^{m-1}\right)\left(q^n-q^{n-1}\right) & \text { if } p^m+q^n>5 .\end{cases}$