ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (2014) 453-459 Commuting graphs and extremal centralizers* Alexander Guterman t Faculty of Algebra, Department of Mathematics and Mechanics, Moscow State University GSP-1, 119991 Moscow, Russia Bojan Kuzma University of Primorska, Glagoljaška 8, SI-6000 Koper, Slovenia, and IMFM, Jadranska 19, SI-1000 Ljubljana, Slovenia Received 29 September 2012, accepted 13 June 2013, published online 27 December 2013 We determine the conditions for matrix centralizers which can guarantee the connectedness of the commuting graph for the full matrix algebra Mn(F) over an arbitrary field F. It is known that if F is an algebraically closed field and n > 3, then the diameter of the commuting graph of Mn(F) is always equal to four. We construct a concrete example showing that if F is not algebraically closed, then the commuting graph of Mn (F) can be connected with the diameter at least five. Keywords: Commuting graph, matrix ring, centralizer. Math. Subj. Class.: 05C40, 15A27, 15A04 *The work is partially supported by the joint Slovene-Russian grant BI-RU/10-11-010. t The work is partially supported by RFBR grant 11-01-00794-a. * Corresponding author E-mail addresses: gregor.dolinar@fe.uni-lj.si (Gregor Dolinar), guterman@list.ru (Alexander Guterman), bojan.kuzma@famnit.upr.si (Bojan Kuzma), polona.oblak@fri.uni-lj.si (Polona Oblak) Gregor Dolinar Faculty of Electrical Engineering, University of Ljubljana Trzaska cesta 25, SI-1000 Ljubljana, Slovenia Polona Oblak i Faculty of Computer and Information Science, University of Ljubljana Tržaška cesta 25, SI-1000 Ljubljana, Slovenia Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 340 Ars Math. Contemp. 7 (2014) 337-340 1 Introduction Let n > 2 and let Mn (F) be a ring of all n x n matrices over a field F. The commuting graph r(Mn(F)) of Mn(F) is a simple graph with the vertex set consisting of all non-scalar matrices from Mn(F), and two vertices form an edge if the corresponding matrices are distinct and commute. Recently, connections between various algebraic structures and their commuting graphs were investigated, see, e.g. [1, 3, 4, 5, 8, 9, 10, 12, 15]. For example, Mohammadian [15] recently proved that a ring is isomorphic to M2(F), with F finite, if and only if the commuting graph of the ring under consideration is isomorphic to r(M2(F)). Akbari, Ghandehari, Hadian and Mohammadian conjectured in [3] that this is also true for Mn(F), where n > 2. The connectedness and diameter of the commuting graph of a matrix ring Mn(F) have been also studied extensively. If F is an algebraically closed field and n > 3, Akbari, Mohammadian, Radjavi, and Raja [4, Corollary 7] proved that the diameter of r(Mn(F)) is always equal to four. For fields which are not algebraically closed the situation is completely different, e.g. if F is the field of rational numbers, then the commuting graph is never connected (see [2, Remark 8]). A necessary and sufficient condition for which r(Mn(F)) is connected was given in [2, Theorem 6]. Namely, it was proven that r(Mn(F)) is connected if and only if every field extension of F of degree n contains at least one proper intermediate field. In the case when commuting graph of Mn (F) is connected, its diameter is known to be at most six and it is conjectured that it is at most five, see [4, Theorem 17] and [4, Conjecture 18]. In the present paper, we show that r(M9(Z2)) is connected and has the diameter at least 5, where Zm is the ring of integers modulo m. We also characterize connected commuting graphs in the language of centralizers. Observe that this characterization is different from [2, Theorem 6], where the language of field extension was used. Definition 1.1. For a matrix A e Mn(F), the centralizer of A, denoted by C(A), is the set of all matrices in Mn(F) commuting with A. Let us remark that the set of non-scalar matrices in C(A) coincides with the closed neighborhood of vertex A e r(Mn(F)). Centralizer induces a natural equivalence relation on Mn(F): Definition 1.2. Matrices A and B are C-equivalent (abbreviated A — B) if C(A) = C(B). Definition 1.3. A matrix A is C-minimal if for every X e M„(F) with C (A) D C (X) it follows that A — X. Definition 1.4. A non-scalar matrix A is C-maximal if for every non-scalar X e Mn(F) with C(A) C C(X) it follows that A - X. Let us remark that C-minimal and C-maximal matrices in Mn (F) were already classified, see Semrl [16] and recent paper [7] by the authors. Also, C-minimal and C-maximal matrices were used as a main tool by Dolinar, Kuzma, and Oblak [8] to investigate distances between vertices in the commuting graph r(Mn(F)) and they will be used to prove the results of this paper as well. We use the following notations. By Ej we denote the matrices with 1 on (i, j)-th position and 0 elsewhere. By 0k and /fc we denote the k x k zero matrix and the k x k G. Dolinar et al.: Commuting graphs and extremal centralizers 455 identity matrix, respectively. When it is clear from the context, we omit the subscript. For a given scalar A G F, define Jk(A) = XI + J2i-i Ej(i+1) to be an elementary upper-triangular Jordan matrix. We denote Jk = Jk (0). A matrix A g Mn (F) is an idempotent if A2 = A, it is a nilpotent if there exists an integer k > 1 such that Ak = 0. Non-zero matrix A with A2 =0 is called a square-zero matrix. For a monic polynomial m g F[x] we let C(m) = n—1 E(i+1)j - J2n=i mi-1Ein g Mn(F) be a companion matrix of m, where m(x) = m0+m1x+^ • •+mn-1xn—1 +xn. Givenamatrix A, letF[A] = {p(A)| p g F[x]} be the unital algebra generated by A. 2 Connectedness of commuting graphs In this section we provide a characterization of matrices for which r(Mn(F)) is connected in the language of extremal centralizers. We need the following result on C-maximal matrices from our recent paper [7]. Proposition 2.1. [7, Theorem 3.2] Let F be an arbitrary field. The following statements are equivalent for a non-scalar matrix A g Mn(F). (i) A is C-maximal. (ii) A belongs to one of the following three classes: (a) A is C-equivalent to an idempotent, (b) A is C-equivalent to a square-zero matrix, (c) A is similar to C © • • • © C, where C is a companion matrix of an irreducible polynomial, such that there is no proper intermediate field between F and F[C ]. Theorem 2.2. Let n > 2 and let F be an arbitrary field. A commuting graph r(Mn(F)) is not connected if and only if there exists a matrix in Mn(F) which is simultaneously C-minimal and C-maximal. Proof. First, let n = 2. Then, r(M2(F)) is never connected because every non-scalar matrix in C(E11) is diagonal, hence C-equivalent to E11 and thus E11 and E12 are not connected in r(M2 (F)) (see also Akbari and Raja [5, Remark 8]). Moreover, E11 is always a C-minimal and C -maximal matrix, so the theorem is true in the case n = 2 for every field F. Second, let n > 3. We will prove each direction of the equivalence in the theorem separately. (i). Suppose A is a C-minimal and C-maximal matrix. According to Proposition 2.1 we have to consider three cases separately. Case 1. Let A be C-equivalent to an idempotent. Then we may assume that A = Ir © 0n-r for some r g {2,..., n - 1}. Thus C(A) = Mr(F) © Mn-r(F). Recall that Jr g Mr (F) is a nilpotent upper-triangular Jordan cell, so C(Ir + Jr) = {a0Ir + a1 Jr + • • • + ar—1 Jr-1| ai G F} = F[Jr]. It easily follows that C ((Ir + Jr ) © 0n-r ) = F[Jr ] © Mn-r (F). Hence C((Ir + Jr) © 0n-r) C C(A), so A is not C-minimal, a contradiction. 340 Ars Math. Contemp. 7 (2014) 337-340 Case 2. Let A be C-equivalent to a non-scalar square-zero matrix. Then we may assume /0r 0 Ir\ A = I 0 0n-2r 0 I = for some integer r, 1 < r < n. However, C(Jn) £ \ o 0 0ry C(A), hence A is not C-minimal, a contradiction. Case 3. Let A = C © • • • © C, where C is a companion matrix of some irreducible monic polynomial m G F[x], such that there is no proper intermediate field between F and F[C]. We will prove that actually A = C. Suppose F is an infinite field. Then A being C-minimal implies that A is non-derogatory (see [7, Lemma 2.7]). Therefore A = C, i.e. A contains only one summand. Hence F[C] is a field extension of F of degree n. Recall that C is a companion matrix of an irreducible polynomial, such that there is no proper intermediate field between F and F[C] thus it follows by [2, Theorem 6] that T(M„(F)) is not connected. Suppose F = GF (pk) is a finite field and suppose that A = C © • • • © C contains more than one summand. Since F[C] is a field extension of F with degree d = deg m, it is isomorphic to K = GF(pkd). Let yc G K correspond to matrix C under this isomorphism. Since A contains more than one summand, d is a proper divisor of n. So, K = GF(pkd) is a proper intermediate field between F and GF(pkn). It is known that the multiplicative group of GF(pkn) is cyclic, so let £ G GF(pkn) be its generator. Then F[£] = GF(pkn) and minimal polynomial f G F[x] for £ is irreducible over F of degree n. For matrix X = C(f) G Mn(F), F[X] is a field isomorphic to GF(pkn). Since GF(pkd) C GF(pkn), some polynomial in X is isomorphic to yc g GF(pkd), and hence also to C. Consequently, by Skolem-Noether theorem, p(X) is similar to a matrix A = C © • • • © C for some p G F[x]. By applying a suitable similarity to X we can assume thatp(X) = A, hence C(X) C c(a). Since A is C-minimal, C(X) = C(A), so X is a polynomial in A by the centralizer Theorem (see [13, p. 113, Corollary 1] and also [17, p. 106, Theorem 2]). Therefore the rational form of X (see [11, Chapter 3] for details) has at least as many cells as the rational form of A. A contradiction to the fact that X is similar to C(f). It follows that A = C and we conclude as in the infinite case that r(Mn(F)) is not connected. (ii). Suppose the commuting graph T(M„(F)) is not connected. By [4, Theorem 11] any two non-scalar idempotents are connected and thus there exists a non-scalar matrix A which is not connected to any non-scalar idempotent. We may assume that A is already in its rational form. Then A consists of a single cell because otherwise a matrix Ai © A2 would be connected to an idempotent I © 0. Hence A = C (ma) for some irreducible polynomial m and positive integer a. If a > 2, then A would be connected to a non-scalar square-zero matrix B = m(A)a-i. It is easy to see that B commutes with a rank-one matrix which further commutes with an idempotent, a contradiction. So, a = 1 and thus A is non-derogatory, hence C-minimal as it was proved in [7, Theorem 2.6]. If A = C(m) is not C-maximal, then there exists a proper intermediate field K between F and F[A] by Proposition 2.1. We can assume K = F[X] is a simple extension for some X G F[A]. The minimal polynomial of X has smaller degree than the minimal polynomial of A, otherwise F[X] = F[A]. Hence the rational form of X contains more than one cell and therefore X, and thus also A is connected to a non-scalar idempotent, a contradiction. □ G. Dolinar et al.: Commuting graphs and extremal centralizers 457 3 Commuting graph with diameter greater than four Recall that the diameter of a commuting graph r(Mn(F)), where F is algebraically closed and n > 3, is equal to four [4]. Below we provide an example showing that if F is not algebraically closed, then the diameter of r(Mn(F)) can be indeed greater than 4. Theorem 3.1. The graph r(M9(Z2)) is connected with diameter at least 5. Proof. Note that Z2 permits only one field extension of degree n = 9, the Galois field GF(29) which contains GF(23) as the only proper intermediate field. So, by [2, Theorem 6] the commuting graph of M9(Z2) is connected. To see that its diameter is at least 5, consider a polynomial m(A) = A9 + A8 + A4 + A2 + 1 G Z2[Aj. It is easy to see that this polynomial is irreducible. Let A = C(m) G M9(Z2) be its companion matrix. Since A has a cyclic vector, C(A) = Z2[A] by a well known Frobenius result on dimension of centralizer (see for example [2, Corollary 1]), andjhis is a field extension of Z2 [14, Theorem 4.14, p. 472] of index n = 9. Actually, C(A) is isomorphic to GF(29) by the uniqueness of field extensions for finite fields. In the sequel we identify these two fields. Since the field extension Z2 c GF(29) contains only GF(23) as a proper intermediate field, we see that each X g C(A) \ GF(23) satisfies Z2X] = Z2A = C(A) and in particular X and A are polynomials in each other so they are C-equivalent. Moreover, each non-scalar Y g GF(23) satisfies Z2 [Y] = GF(23), because no proper intermediate fields exist between Z2 and its overfield GF(23), and in particular, C(Yi) = C(Y2) for any two non-scalar Y, Y G GF(23) c GF(29) = C(A). There exists a polynomial p so thatY = p(A) G GF (23)\{0,1}. As the field GF (23) contains no idempotents other than 0 and 1 we see that the rational canonical form of Y consists only of cells which correspond to some powers of the same irreducible polynomial. Likewise, the field contains no non-zero nilpotents, so each cell of Y corresponds to the same irreducible polynomial. Moreover, GF (23) has no subfields other than Z2, so Z2[y] = GF (23) and hence the minimal polynomial of Y G GF (23) has degree [GF (23) : Z2] = 3. This polynomial is relatively prime to its derivative, so in a splitting field, Y has three distinct eigenvalues. It easily follows that Y is similar to a matrix C © C © C, with C being a 3 x 3 companion matrix of some irreducible polynomial of degree 3. Let S1 be an invertible matrix such that Y = S-1 (C © C © C)S1 and define Clearly, p(A) = Si Y S -1 A = S1A4S-1. C © C © C and it follows that C (p(A)) Z2 [C] Z2[C] Z2 [C]' Z2 [C ] Z2[C] Z2 [C ] Z2 [C ] Z2[C] Z2 [C ] (3.1) Since Z2 [Y] = GF(23) we obtain Z2 [C] Consider a 3 3 block matrix GF(23). N = E13 0 0 ' 0 0 e13 E32 0 0 E13, E13, E32 G M3(Z2). 340 Ars Math. Contemp. 7 (2014) 337-340 It is immediate that N3 = 0, so I + N is invertible. Define B = (I + N)A(1 + N )-1. We will show that d(A, B) > 5. Suppose there exists a path A — V — Z — W — B of length 4. Note that V e GF(23) C C(A). Otherwise, if V e C(A) \ GF(23), then C(V) = C(A) and such V has exactly the same neighbours as A. Since B = (I + N)A(1 + N)-1, it follows W = (I + N)U(I + N)-1 for some U e GF(23) C C(A) = (I + N)-1 C(B)(1 + N). Recall that any two non-scalar elements in GF(23) have the same centralizer. So in particular we might take U = V = p(A) = C © C © C where polynomial p was defined before. 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