In a recent paper C. Miguel proved that the diameter of the commuting graph of the matrix ring Mn(R) is equal to 4 if either n = 3 or n = 5. But the case n = 4 remained open, since the diameter could be 4 or 5. In this work we close the problem showing that also in this case the diameter is 4

Grau, J.M.

Oller-Marcén, A.M.

Tasis, C.

English

Repositorio Universidad de Zaragoza

Bull. Iranian Math. Soc.Vol. 43 (2017), No. 1, pp. 217–221Online ISSN: 1735-8515ON THE DIAMETER OF THE COMMUTING GRAPH OFTHE FULL MATRIX RING OVER THE REAL NUMBERSJ. M. GRAU, A. M. OLLER-MARCE´N∗ AND C. TASIS(Communicated by Ebadollah S. Mahmoodian)Abstract. In a recent paper C. Miguel proved that the diameter of thecommuting graph of the matrix ring Mn(R) is equal to 4 if either n = 3or n ≥ 5. But the case n = 4 remained open, since the diameter could be4 or 5. In this work we close the problem showing that also in this casethe diameter is 4.Keywords: Commuting graph, diameter, idempotent matrix.MSC(2010): Primary: 05C50; Secondary: 15A27.1. IntroductionFor a ring R, the commuting graph of R, denoted by Γ(R), is a simpleundirected graph whose vertices are all non-central elements of R, and twodistinct vertices a and b are adjacent if and only if ab = ba. The commutinggraph was introduced in [1] and has been extensively studied in recent yearsby several authors [2–7,12,13].In a graph G, a path P is a sequence of distinct vertices (v1, . . . , vk) suchthat every two consecutive vertices are adjacent. The number k − 1 is calledthe length of P. For two vertices u and v in a graph G, the distance betweenu and v, denoted by d(u, v), is the length of the shortest path between u andv, if such a path exists. Otherwise, we define d(u, v) = ∞. The diameter of agraph G is defineddiam(G) = sup{d(u, v) : u and v are vertices of G}.A graph G is called connected if there exists a path between every two distinctvertices of G, equivalently, diam(G) <∞.Most research has been conducted regarding the diameter of commutinggraphs of certain classes of rings [3, 7–10]. Here, we deal with the full matrixrings over fields. Let F be an arbitrary field. We known that Γ(M2(F)) is neverArticle electronically published on February 22, 2017.Received: 9 June 2015, Accepted: 3 November 2015.∗Corresponding author.c⃝2017 Iranian Mathematical Society217Diameter commuting graph matrix ring over R 218connected. It was proved in [4] that Γ(Mn(F)) is connected if and only if everyfield extension of F of degree n contains a proper intermediate field. Moreover,it was shown in [3] that if Γ(Mn(F)) is connected, then 4 ≤ diam(Γ(Mn(F))) ≤ 6and it is conjectured that diam(Γ(Mn(F))) ≤ 5. Let Q and R be the fields ofrational and real numbers, respectively. We know from [3, 4] that Γ(Mn(Q))is disconnected for any n ≥ 2 and diam(Γ(Mn(F))) = 4 for every algebraicallyclosed field F and n ≥ 3. Quite recently, C. Miguel [11] has verified thisconjecture for R, proving the following result.Theorem 1.1. Let n ≥ 3 be any integer. Then, diam(Γ(Mn(R))) = 4 forn ̸= 4 and 4 ≤ diam(Γ(M4(R))) ≤ 5.Unfortunately, this result left open the question wether diam(Γ(M4(R))) is4 or 5. In this paper we solve this open problem. Namely we will prove thefollowing result.Theorem 1.2. The diameter of Γ(M4(R)) is equal to 4.2. On the diameter of Γ(Mn(R)Before we proceed, let us introduce some notation. If a, b ∈ R, we define thematrix Aa,b asAa,b :=(a b−b a).Now, given two matrices X,Y ∈Mn(R), we defineX ⊕ Y :=(X 00 Y)∈ M2n(R).Finally, two matrices A,B ∈ Mn(R) are called similar and are written as A ∼ Bif there exists an invertible matrix P such that P−1AP = B.The proof of Theorem 1.1 in [11] relies on the possible forms of the Jordancanonical form of a real matrix. In particular, it is proved that the distancebetween two matrices A,B ∈ M4(R) is at most 4 unless we are in the situationwhere A and B have no real eigenvalues and only one of them is diagonalizableover C. In other words, the case whenA ∼(Aa,b 00 Ac,d), B ∼(As,t I20 As,t).The following result will provide us the main tool to prove that the distancebetween A and B is at most 4 also in the previous setting. It is true for anydivision ring D. In what follows, given a matrix A, LA and RA will denote theleft and right multiplication by A, respectively.Proposition 2.1. Let A,B ∈ Mn(D) matrices such that A2 = A and B2 = 0.Then, there exists a non-scalar matrix commuting with both A and B.219 Grau, Oller-Marce´n and TasisProof. Since A2 = A; i.e., A(I − A) = (I − A)A = 0, then one of nullity A ornullity (I − A) is at least n/2. Since I − A is also idempotent and a matrixcommutes with A if and only if it commutes with I − A we can assume thatnullity A ≥ n/2. Moreover, since B2 = 0, it follows that nullity B ≥ n/2.Now, if KerLA ∩KerLB ̸= {0} and KerRA ∩KerRB ̸= {0} we can apply [3,Lemma 4] and the result follows. Hence, we assume that KerLA∩KerLB = {0},since in the case KerRA ∩ KerRB = {0} we can consider the transposes of Aand B instead of A and B, respectively. Note that, in these conditions, n = 2rand the nullities of A and B are equal to r.Let B1 and B2 be bases for KerLA and KerLB , respectively, and considerB = B1 ∪ B2 a basis for Dn. Since A is idempotent, it follows that Dn =KerLA ⊕ ImLA.We want to find the representation matrix of LA in the basis B. To do so,if v ∈ B2, we write v = a + a′ with a ∈ KerLA and a′ ∈ ImLA. If a′ = Aa′′for some a′′ ∈ Dn, then Av = Aa + Aa′ = 0 + A(Aa′′) = Aa′′ = a′ = −a + v.Since Av = 0 for every v ∈ B1, we get that the representation matrix of LA inthe basis B is of the form (0 A′0 Ir),with A′ ∈ Mr(D).Now, we want to find the representation matrix of LB in the basis B. Clearly,Bv = 0 for every v ∈ B2. Let w ∈ B1. Then, Bw = w1 + w2 with w1 ∈ KerLAand so w2 ∈ KerLB . Hence, 0 = B2w = Bw1 and w1 ∈ KerLA ∩KerLB = {0}.Thus, the representation matrix of LB in the basis B is of the form(0 0B′ 0),with B′ ∈ Mr(D).As a consequence of the previous work we can find a regular matrix P suchthat:PAP−1 =(0 A′0 Ir), PBP−1 =(0 0B′ 0).Now, if A′B′ ̸= B′A′, then P−1(A′B′ ⊕B′A′)P is a non-scalar matrix com-muting with A and B. If A′ and B′ commute, we can find a non-scalar matrixS ∈ Mr(D) commuting with both A′ and B′. Therefore P−1(S⊕S)P commuteswith both A and B and the proof is complete. !We are now in the condition to prove the main result of the paper.Theorem 2.2. The diameter of Γ(M4(R)) is four.Proof. In [11] it was proved that d(A,B) ≤ 4 for every A,B ∈ M4(R), unlessA ∼(Aa,b 00 Ac,d)and B ∼(As,t I20 As,t),Diameter commuting graph matrix ring over R 220for some real numbers a, b, c, d, s, t. Hence, we only focus on this case. AssumethatA = P−1(Aa,b 00 Ac,d)P and B = Q−1(As,t I20 As,t)Q,for some invertible matrices P and Q. LetM = P−1(0 00 I2)P and N = Q−1(0 I20 0)Q.It is straightforwardly checked that M2 = M , N2 = 0, AM = MA, andBN = NB. Furthermore, Proposition 2.1 implies that there exists a non-scalar matrix X that commutes both with M and N .Thus, we have found a path (A,M,X,N,B) of length 4 connecting A andB and the result follows. !Corollary 2.3. For every n ≥ 3, diam(Γ(M4(R))) = 4.AcknowledgementsWe want to thank the anonymous referee for his useful comments that helpedto improve the paper, in particular its introduction.References[1] S. Akbari, M. Ghandehari, M. Hadian and A. Mohammadian, On commuting graphs ofsemisimple rings, Linear Algebra Appl. 390 (2004) 345–355.[2] S. Akbari, D. Kiani and F. Ramezani, Commuting graphs of group algebras, Comm.Algebra 38 (2010), no. 9, 3532–3538.[3] S. Akbari, A. Mohammadian, H. Radjavi and P. Raja, On the diameters of commutinggraphs, Linear Algebra Appl. 418 (2006), no. 1, 161–176.[4] S. Akbari, H. Bidkhori and A. Mohammadian, Commuting graphs of matrix algebras,Comm. Algebra 36 (2008), no. 11, 4020–4031.[5] S. Akbari and P. Raja, Commuting graphs of some subsets in simple rings, LinearAlgebra Appl. 416 (2006) 1038–1047.[6] J. Araujo, M. Kinyon and J. Konieczny, Minimal paths in the commuting graphs ofsemigroups, European J. Combin. 32 (2011), no. 2, 178–197.[7] G. Dolinar, B. Kuzma and P. Oblak, On maximal distances in a commuting graph,Electron. J. Linear Algebra 23 (2012) 243–256.[8] D. Dolzˇan, D. Kokol Bukovsˇek and P. Oblak, Diameters of commuting graphs of matricesover semirings, Semigroup Forum 84 (2012), no. 2, 365–373.[9] D. Dolzˇan and P. Oblak, Commuting graphs of matrices over semirings, Linear AlgebraAppl. 435 (2011), no. 7, 1657–1665.[10] M. Giudici and A. Pope, The diameters of commuting graphs of linear groups and matrixrings over the integers modulo m, Australas. J. Combin. 48 (2010) 221–230.[11] C. Miguel, A note on a conjecture about commuting graphs, Linear Algebra Appl. 438(2013), no. 12, 4750–4756.[12] A. Mohammadian, On commuting graphs of finite matrix rings, Comm. Algebra 38(2010), no. 3, 988–994.[13] G. R. Omidi and E. Vatandoost, On the commuting graph of rings, J. Algebra Appl. 10(2011), no. 3, 521–527.221 Grau, Oller-Marce´n and Tasis(Jose Maria Grau) Departamento de Matema´ticas, Universidad de Oviedo. AvenidaCalvo Sotelo s/n, 33007 Oviedo, Spain.E-mail address: grau@uniovi.es(Antonio M. Oller-Marce´n) Centro Universitario de la Defensa de Zaragoza. Car-retera de Huesca s/n, 50090 Zaragoza, Spain.E-mail address: oller@unizar.es(Carmen Tasis) Departamento de Matema´ticas, Universidad de Oviedo. AvenidaCalvo Sotelo s/n, 33007 Oviedo, Spain.E-mail address: ctasis@uniovi.es

On the diameter of the commuting graph of the full matrix ring over the real numbers

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