A note on co-maximal graphs of commutative rings

3 downloads 0 Views 226KB Size Report
[4] David S. Dummit, Richard M. Foote, Abstract Algebra, John Wiley and Sons, Inc., 2004. [5] Gray Chartrand, Ping Zhang, Introduction to Graph Theory, ...
Available online at www.sciencedirect.com

ScienceDirect AKCE International Journal of Graphs and Combinatorics (

)

– www.elsevier.com/locate/akcej

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.

2

D. Sinha, A.K. Rao / AKCE International Journal of Graphs and Combinatorics (

)



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.

D. Sinha, A.K. Rao / AKCE International Journal of Graphs and Combinatorics (

)



3

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]

P.K. Sharma, S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176 (1995) 124–127. H.R. Maimani, M. Salimi, A. Sattari, S. Yassemi, Comaximal graph of commutative rings, J. Algebra 319 (4) (2008) 1801–1808. Joseph A. Gallian, Contemporary Abstract Algebra, Narosa Publication House, 1999. David S. Dummit, Richard M. Foote, Abstract Algebra, John Wiley and Sons, Inc., 2004. Gray Chartrand, Ping Zhang, Introduction to Graph Theory, McGraw Hill Education Private Ltd, 2006. F. Harary, Graph Theory, Addison-Wesley Publ. Comp., Reading, Massachusetts, 1969. D.B. West, Introduction to Graph Theory, Prentice-Hall of India Pvt. Ltd, 1996. J. Krausz, Demonstration nouvelle d’ume Theoreme de Whitney sur les Reseaux, Mat. Fiz. Lapok 50 (1943) 75–85. O. Ore, Theory of Graphs, Vol. 83, American Mathematical Society, Providence, 1962, pp. 19–21. V. Menon, The isomorphism between graphs and their adjoint graphs, Canad. Math. Bull. 8 (1965) 7–15. S. Seshu, M.B. Reed, Linear Graphs and Electrical Networks, Addison-Wesley, 1961, p. 296. A. Sharma, A. Gaur, Line graphs associated to the maximal graph, J. Algebra Relat. Top. 3 (2015) 1–11.

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.