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A note on co-maximal graphs of commutative rings Deepa Sinha ∗, Anita Kumari Rao South Asian University, New Delhi 110021, India Received 22 March 2018; accepted 23 March 2018 Available online xxxx
Abstract Let R be a commutative ring with unity. The co-maximal graph Γ (R) is the graph with vertex set R and two vertices a and b are adjacent if Ra + Rb = R. In this paper, we characterize rings for which the co-maximal graph Γ (R) is a line graph of some graph G and determine rings for which Γ (R) is Eulerian or Hamiltonian. We also find the diameter and girth of Γ (R). c 2018 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Finite commutative ring; Maximal ideal; Co-maximal graph; Line graph
1. Introduction In 1995, Sharma and Bhatwadekar [1] introduced a graph Γ (R) on a commutative ring R, whose vertices are elements of R and two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Further properties of these graphs were established by Maimani et al. [2] and they named this graph as co-maximal graph of R, Γ (R). The co-maximal graph Γ (Z 6 ) is shown in Fig. 1. For the preliminary notations and terminologies in abstract algebra we refer to standard text books [3,4], and for graph theory, we refer to [5–7]. Unless mentioned otherwise, all rings considered in this paper are finite and commutative with unity 1 ̸= 0. A subring A of a ring R is an ideal of R if ra, ar ∈ A, for every r ∈ R and a ∈ A. A proper ideal A of R is a maximal ideal of R if there are no other ideals contained between A and R. An element a ∈ R is called a unit of the ring R if a −1 exists. A commutative ring is quasi-local if it has only finitely many maximal ideals. In this paper, Max(R) = {M1 , M2 , . . . , Mn } denotes the set of maximal ideals of R. For a ring R, U (R) denotes the set of units of R. A subset C of the vertex set of G is called a clique if any two distinct vertices of C are adjacent. Moreover, the clique number is the maximum order of a complete subgraph of G and is denoted by ω(G). The line graph L(G) of a graph G is defined as follows: The vertices of L(G) are taken as the edges of G, and two vertices are adjacent whenever the corresponding edges of G are adjacent (i.e. they share a vertex in common). Peer review under responsibility of Kalasalingam University.
∗ Corresponding author.
E-mail addresses: deepa
[email protected] (D. Sinha),
[email protected] (A.K. Rao). https://doi.org/10.1016/j.akcej.2018.03.003 c 2018 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: D. Sinha, A.K. Rao, A note on co-maximal graphs of commutative rings, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2018.03.003.
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D. Sinha, A.K. Rao / AKCE International Journal of Graphs and Combinatorics (
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Fig. 1. Γ (Z 6 ).
Fig. 2. Graph G whose line graph is Γ (R) when R ∼ = Z 4 or
Z 2 [x] ⟨x 2 ⟩
or Z 2 × Z 2 .
Krausz [8] introduced the concept of the line graph. This concept has been independently discovered by many authors with a different name: interchange graph [9], adjoint [10], and edge-to-vertex dual [11] are a few. The purpose of this paper is to characterize rings whose co-maximal graph is a line graph of some graph G. We also identify rings for which Γ (R) and L(Γ (R)) are Hamiltonian and conclude that if R ∼ = F, where F is a field, Γ (R) and L(Γ (R)) are Hamiltonian. Also, we find the diameter and girth of Γ (R). 2. Main results Remark 2.1. Since a field F has no proper ideals it follows that Γ (F) is the complete graph of order |F| = n, where n = p k ( p is prime) and hence it is the line graph of the star K 1,n . Theorem 2.2. For a local ring R, Γ (R) is a line graph of some graph G if and only if R is isomorphic to one of the following rings Z 4 , Z⟨x2 [x] 2 ⟩ , Z 2 × Z 2 , or F, where F is a finite field. Proof. Let Γ (R) be a line graph of some graph G and R be not isomorphic to rings defined above. Then |R \ U (R)| ≥ 3. Let a, b ∈ R \ U (R) and let u ∈ U (R). Then the induced subgraph ⟨{a, b, u, 0}⟩ is isomorphic to K 1,3 and hence it follows that Γ (R) is not a line graph. Hence R is isomorphic to one of the rings given in the theorem. Conversely, suppose R is isomorphic to one of the rings Z 4 , Z⟨x2 [x] 2 ⟩ , Z 2 × Z 2 , or F, where F is a field. Then Γ (F) = K |F| = L(K 1,|F| ). In the remaining cases Γ (R) is the line graph of the graph given in Fig. 2. □ Theorem 2.3. For a ring R, if |U (R)| ≥ 2 and |Max(R)| ≥ 2 then Γ (R) is not a line graph of any graph G. Proof. Let M1 , M2 ∈ Max(R). Let a1 ∈ M1 and a2 ∈ M2 and let u 1 , u 2 ∈ U (R). Then the induced subgraph ⟨{0, u 1 , u 2 , a1 , a2 }⟩ is isomorphic to the graph given in Fig. 3 which is a forbidden subgraph of a line graph. Hence Γ (R) is not a line graph. □ Sharma and Gaur [12] have characterized ring R for which Γ (R) is Eulerian. Theorem 2.4 ([12]). Let R be a ring. Then the co-maximal graph Γ (R) is Eulerian if and only if R has odd cardinality. Please cite this article in press as: D. Sinha, A.K. Rao, A note on co-maximal graphs of commutative rings, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2018.03.003.
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Fig. 3. Forbidden graph G.
Remark 2.5. If R ∼ = F where F is a field and F ≇ Z 2 , then Γ (R) is a complete graph and hence Γ (R) and L(Γ (R)) are Hamiltonian. Proposition 2.6. Let R be a commutative ring with unity. Then { 1 if R is a field, diam(Γ (R)) = 2 otherwise. Proof. Let R ∼ = F. Then Γ (R) is a complete graph. Thus, diameter of Γ (R) is equal to one. Next, let R ≇ F. Let u, a, 0 ∈ R be a unit element, non-unit element and zero of the ring respectively. Now, the shortest path between a and 0, a − u − 0 is of length two. Also if R is a local ring, then the distance between any two non-units is equal to two. Thus, diam(Γ (R)) = 2. □ Proposition 2.7. Let R be a finite commutative ring and R ̸= Z 2 . Then g(Γ (R)) = 3 where g(Γ (R)) is the girth of Γ (R). Proof. If R is a local ring, then ⟨{0, u 1 , u 2 }⟩ where u 1 , u 2 ∈ U (R) is a triangle. If not let M1 , M2 be maximal ideals of R. Let u ∈ U (R), a1 ∈ M1 and a2 ∈ M2 . Then ⟨{u, a1 , a2 }⟩ is a triangle. Hence g(Γ (R)) = 3. □ References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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Please cite this article in press as: D. Sinha, A.K. Rao, A note on co-maximal graphs of commutative rings, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2018.03.003.
