Non-singular circulant graphs and digraphs

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1 Electroic Joural of Liear Algebra Volume 26 Volume ) Article No-sigular circulat graphs ad digraphs A. K. Lal A. Satyaarayaa Reddy Follow this ad additioal works at: Recommeded Citatio Lal, A. K. ad Reddy, A. Satyaarayaa. 2013), "No-sigular circulat graphs ad digraphs", Electroic Joural of Liear Algebra, Volume 26. DOI: This Article is brought to you for free ad ope access by Wyomig Scholars Repository. It has bee accepted for iclusio i Electroic Joural of Liear Algebra by a authorized editor of Wyomig Scholars Repository. For more iformatio, please cotact scholcom@uwyo.edu.

2 NON-SINGULAR CIRCULANT GRAPHS AND DIGRAPHS A.K. LAL AND A. SATYANARAYANA REDDY Abstract. For a fixed positive iteger, ) let W be the permutatio matrix correspodig to 1 2 the permutatio. I this article, it is show that a symmetric matrix with ratioal etries is circulat if, ad oly if, it lies i the subalgebra of Q[x]/ x geerated by W +W. O the basis of this, the sigularity of graphs o -vertices is characterized algebraically. This characterizatio is the exteded to characterize the sigularity of directed circulat graphs. The kth power matrix W+W k k defies a circulat graph C. k The results above are the applied to characterize its sigularity, ad that of its complemet graph. The digraph C r,s,t is defied as that whose adjacecy matrix is circulat circa), where a is a vector with the first r-compoets equal to 1, ad the ext s,t ad r+s+t) compoets equal to zero, oe, ad zero respectively. The sigularity of this digraph graph), uder certai coditios, is also show to deped algebraically upo these parameters. A slight geeralizatio of these graphs are also studied. Key words. Graphs, Digraphs, Circulat matrices, Primitive roots. AMS subject classificatios. 05C50, 15A Itroductio ad prelimiaries. Let Q deote the set of ratioal umbers ad let a = a 0,a 1,...,a ) Q, for a fixed positive iteger. Cosider the shift operator T : Q Q, defied by Ta 0,a 1,...,a ) = a,a 0,...,a 2 ). The circulat matrix, deoted circa), associated with a is the matrix whose kth row is T k a), for k = 1,2,...,. For example, the idetity matrix, deoted I, equals circ1,0,...,0), the matrix of all 1 s, deoted J, equals circ1,1,...,1) ad W = cicr0,1,...,0) is the permutatio matrix correspodig to the permutatio ) 1 2. It is well kow that W is the geerator of the algebra Q[x]/ x 1 see, for istace, [6]), where Q[x] is the polyomial rig over Q ad x 1 deotes the pricipal ideal of Q[x] geerated by x 1. Or equivaletly, every circulat matrix of order is a polyomial i W. Lemma 1.1. [2] Let A be a matrix with ratioal etries. The A is circulat if, ad oly if, it is a polyomial over Q i W. Received by the editors o August 24, Accepted for publicatio o March 3, Hadlig Editor: Shmuel Friedlad. Departmet of Mathematics & Statistics, Idia Istitute of Techology Kapur, Kapur , Idia arlal@iitk.ac.i). Departmet of Mathematics, Shiv Nadar Uiversity, Dadri , Idia satyaarayaa.reddy@su.edu.i). 248

3 No-sigular Circulat Digraphs 249 For a Q, let A = circa). The Lemma 1.1 esures the existece of the polyomial γ A x) = a 0 +a 1 x+ +a x Q[x], called the represeter polyomial, with A = γ A W ). Oe also obtais the followig result about circulat matrices. Lemma 1.2. Let A = circa) be a circulat matrix. The the eigevalues of A equal γ A ζ k ) = a 0 +a 1 ζ k + +a ζ k ), for k = 0,1,...,, where ζ deotes a primitive -th root of uity. The reader is referred to [4] ad/or [1] for stadard cocepts ad results used i this article without explicit refereces. Let G = V,E) be a simple graph has o loops), directed or ot, o = V vertices, where for a fiite set S, S deotes the umber of elemets i S. Give a orderig V = {v 1,v 2,...,v } of the vertices, the adjacecy matrix of AG) = [a ij ] of G, is by defiitio the matrix whose ijth etry equals the umber of edges v i,v j ), from the vertex v i to the vertex v j. Observe that, the adjacecy matrix AG) of a graph G is symmetric, whereas the adjacecy matrix of a directed graph eed ot be symmetric. The graph G is said to be osigular if AG) is a osigular matrix ad is called circulat if AG) is a circulat matrix. For example, the adjacecy matrix of the cycle graph C o vertices equals circ0,1,0,...,1) = W +W ad γx) = x+x is its represeter polyomial. Thus, usig Lemma 1.2, the graph C is sigular if, ad oly if, is a multiple of 4. Recall that the -th cyclotomic polyomial, deoted Φ x), is a elemet of Z[x] ad equals x ζ k). Also, x = d Φ d x) here a b meas a divides gcdk,)=1 1 k b). Moreover, Φ x) is the miimal polyomial of ζ ad hece, fζ ) = 0 for some fx) Z[x] implies Φ x) divides fx) Z[x]. The ext result appears i [8]. Lemma 1.3. [8] Let p be a prime ad let be a positive iteger. The Φ p x) = { Φ x p ), if p, Φ x p ) Φ x), if p. I particular, for each positive iteger k, Φ p kx) = 1+x pk +x 2pk + +x p)pk. The followig result is a applicatio of Lemma 1.2. This result also appears i [5] also see Corollary 10 i [6]). Lemma 1.4. [5, 6] Let A = circa). The A is sigular if, ad oly if, deggcdγ A x),x )) 1. Lemma 1.4 gives us the followig useful remarks.

4 250 A.K. Lal ad A Satyaarayaa Reddy Remark Let A = circa) with a = 1,...,1,0,...,0). The A is sigular if, ad oly }{{} s times if, gcds,) > Let A = circa) with a i = 0, for 0 i k ad a k 0. The γ A x) = x k Γ A x), for some polyomial Γ A x) Q[x]. Thus, usig Lemma 1.4, A is sigular osigular) if, ad oly if, WA k is sigular osigular), for some k = 0,1,...,. That is, to study sigularity osigularity) of A, it is eough to study Γ A x). Usig Remark ad Lemma 1.4, the followig result is immediate, ad hece, the proof is omitted. Lemma 1.6. Let G be a circulat digraph of order. The G is sigular if, ad oly if, Φ d x) Γ AG) x), for some divisor d 1 of. As a immediate corollary of Lemma 1.6 ad 1.3, the followig result follows. Corollary 1.7. Let p be a prime ad let k,l be positive itegers with p k. Also, let G be a k-regular circulat graph/digraph o p l vertices. The G is osigular. Proof. Usig Lemma 1.6, it is eough to show that Φ d x) Γ AG) x), for every d p l with d 1. O the cotrary, assume that Γ AG) x) = Φ d x)gx), for some gx) Z[x]. The usig Lemma 1.3, Φ d 1) = p, for every d p l with d 1. As gx) Z[x],g1) Z ad hece k = Γ AG) 1) = Φ d 1)g1) = p g1). A cotradictio to the assumptio that p k. The remaiig part of this paper cosists of two more sectios that are maily cocered with applicatios of Lemma 1.6. Sectio 2 gives ecessary ad sufficiet coditios for a few classes of circulat graphs to be osigular ad Sectio 3 gives possible geeralizatio of the results studied i Sectio 2. Before proceedig to Sectio 2, recall that for a graph G = V,E 1 ), the complemet graph of G, deoted G c = V,E 2 ), is a graph i which u,v) E 2 wheever u,v) E 1 ad vice versa, for every u v V. Note that as G is a simple graph, G c is also simple ad AG c ) = J AG) I. Moreover, G is circulat if, ad oly if, G c is circulat. 2. Some sigular circulat graphs. We start this sectio by showig that the adjacecy matrix of a circulat graph o vertices is a polyomial i the adjacecy matrix of C. For 1 i 2, let Ci deote a graph whose vertex set is same as the vertex set of the cycle graph C ad x,y) is a edge i C i wheever the distace of x from y i C is exactly i. Also, defie the matrices A i commoly called the distace matrices

5 No-sigular Circulat Digraphs 251 of C ) by A i = W i +W i ad A 2 = The ote that A i = AC i ), for 1 i 2. { W 2, if is eve, W 2 +W 2, if is odd. The idetity x k +x k ) = x+x )x k +x 1 k ) x k 2 +x 2 k ) eables us to readily establish, by mathematical iductio, that x k +x k is a moic polyomial i x+x of degree k. Cosequetly, A i s, for 1 i 2, are polyomials of degree i, i the adjacecy matrix of C over Q. Thus, every circulat symmetric matrix with ratioal etries lies i the subalgebra of Q[x]/ x geerated by W +W for eve, ote that 2W 2 = W 2 +W 2 ). Theorem 2.1. Let B = circq), with q Q. The B is a symmetric if, ad oly if, B is a polyomial i W +W with ratioal coefficiets. Proof. Oe part is immediate, as every polyomial i W +W circulat matrix. i=0 Coversely, let B = circq) be symmetric. The B = q i W i. But B = B t implies q i = q i, for 1 i. Thus, B = i=0 is a symmetric q i W i ad B t = 2 i=0 q i A i. Observe that for 1 i < 2, γ A i x) = x i 1+x 2i) ad γ A 2 x) = x 2, if is eve, ad γ A 2 x) = x 2 1+x), if is odd. Usig these, the ext result gives a ecessary ad sufficiet coditio for the graphs C, i for 1 i 2, to be sigular. Propositio 2.2. Let 3 ad 1 i 2. The the graph Ci is sigular if, ad oly if, is a multiple of 4 ad gcdi, 2 ) 4. Proof. Note that Γ x) = 1 + x if is odd ad Γ A A x) = 1 if is eve. 2 2 Hece, A 2 is osigular for all. For 1 i < 2, Γ A i x) = 1+x 2i ad hece A i is sigular if, ad oly if, Γ Ai ζ) k = 0, for some k,1 k 1. That is, oe eeds ζ) k 2i =. This holds if, ad oly if, 4 divides ad gcdi, 2 ) 4. As a immediate cosequece of Propositio 2.2, the graph C is sigular if, ad oly if, 4. The ext result gives a ecessary ad sufficiet coditio for the complemet graph C i ) c, of C i, to be sigular. Propositio 2.3. Let 4. The the graph 1. C 2 ) c is sigular if, ad oly if, is eve or 3 mod 6). 2. C i ) c, for 1 i < 2, is sigular if, ad oly if, 3 ad gcdi,) 3.

6 252 A.K. Lal ad A Satyaarayaa Reddy Proof. NotethatA = AC 2 ) c ) = J A 2 I adγ A x) = x x 1+x 2 ), for eve. Thus, usig Lemma 1.2, the graph C 2 ) c is sigular, wheever is eve. Ad for odd, γ A x) = x x 1+x 2 +x 2 +1 ). Cosequetly, C 2 ) c is sigular if, ad oly if, 1+ζ k) 2 +ζ k) 2 +1 = 0, for some k,1 k. Or equivaletly, ζ k 2 is a primitive 3-rd root of uity. Thus, 3 mod 6). Now assume that 1 i < 2. The ACi )c ) = J A i I, ad hece, γ AC i ) c )x) = x x 1 + xi + x i ). Thus, C i)c is sigular if, ad oly if, = 0, for some k, 1 k. That is, ζ ki uity. Thus, C) i c is sigular if, ad oly if, 3 ad gcdi,) 3. 1+ζ ki +ζ ki is a primitive 3-rd root of As a immediate cosequece of Propositio 2.3, it follows that the complemet graph C c, of C, is sigular if, ad oly if, 3. We ow obtai ecessary ad sufficiet coditios for osigularity of circulat graphs that appeared i [9]. For the sake of clarity, his otatios have bee modified. Fix a positive iteger 3 ad let 1 r < 2. The first class of circulat graphs, deoted C r), has the same vertex set as the vertex set of the cycle C ad x,y) is a edge wheever the legth of the shortest path from x to y i C is at most r. Note that C 2 ) is the complete graph. The secod class of graphs, deoted C,r) is a graph o vertices ad its adjacecy matrix is the sum of the adjacecy matrices of C r) ad C, where 1 r <. A ecessary ad sufficiet coditio for the graph C,r) to be sigular appears later i Theorem 2.7. A similar result o C r) appears as Theorem 2.2 of [9]. A separate proof is give for the sake of completeess. Theorem 2.4. [9] Let 3 ad let 1 r < 2. The the graph Cr) sigular if, ad oly if, oe of the followig coditios hold: is 1. gcd,r) > 1, 2. gcd,r) = 1, is eve ad gcdr +1,) divides 2. Proof. Note that A = AC r) ) = circ0,1,...,1 0,...,0 1,...,1) ad hece }{{}}{{}}{{} r 2r r x 1)Γ A x) = x r 1)1 + x r ). Therefore, C r) is sigular if, ad oly if, Γ A ζ) d = 0, for some d,1 d 1. Or equivaletly either ζ) d r 1 = 0 or 1+ζ d) r = 0. But ζ d )r = 0 wheever gcdr,) > 1. If gcdr,) = 1 the 1+ζ d ) r = 0 ad this implies that is eve ad gcdr +1,) divides 2.

7 No-sigular Circulat Digraphs 253 The followig result follows from Remark Corollary 2.5. Let 3 ad let 1 r < 2. The the graph Cr) ) c is osigular if, ad oly if, gcd,2r+1) = 1. Proof. Note that A = AC r) ) c ) = circ0,...,0 1,...,1 0,...,0) ad hece }{{}}{{}}{{} r+1 2r r x)γ A x) = x 2r. Thus, usig Remark 1.5, C r) ) c is sigular if, ad oly if, gcd2r+1,) > 1. Before proceedig with the ext result that gives a ecessary ad sufficiet coditio for the graph C,r) to be sigular, we state a result that appears as Propositio 1 i Kursha & Odlyzko [7] Lemma 2.6. [7] Let m ad be positive itegers with m ad let ζ be a primitive root of uity. The there exists a uit u Z[ζ ] depedet o m, ad ζ such that pu, if m = pα, p a prime, α > 0; 1 ζ p α)u, if m Φ m ζ ) = = p α, p a prime, α > 0; p m; 1 ζ pα +1) p u, if m = p α, p a prime, α > 0; p m; u, otherwise. Theorem 2.7. Let ad r be positive itegers such that the circulat graph C,r) is well defied. The the circulat graph C,r) is sigular if, ad oly if, gcd,2r+1) 3. Proof. Note that A = AC,r)) = circ0,1,...,1,0,...,0,1,0,...,0,1,...,1). }{{}}{{}}{{}}{{} r r r r Cosequetly, γ A x) = x+x 2 + +x r +x +x r + +x = xγ A x) ad x)γ A x) = x r +x x)+x r x r ) = x r 1 x 2r )+x ) x x ) = x )x +1) x r x 2r ). Now, let us assume that gcd,2r+1) = d 3. The ζ /d ζ /d ) = ζ ) /d = 1 = ζ ) 2r+1)/d = Hece, the circulat graph C, r) is sigular. ζ /d )Γ A ζ /d ) = 0 as ) 2r+1 = ζ /d ) 2r. Coversely, let us assume that the graph C, r) is sigular. This implies that there exists a eigevalue of C,r) that equals zero. That is, there exists a k, 1 k 1, such that γ A ζ k ) = 0. We will ow show that if gcd,2r + 1) = 1

8 254 A.K. Lal ad A Satyaarayaa Reddy the the expressio x)γ A x) evaluated at x = ζ k ca ever equal zero, for ay k, 1 k, ad this will complete the proof of the result. We eed to cosider two cases depedig o whether k is odd or k is eve. Let k = 2m, 1 m <. The, usig gcd,2r + 1) = 1, ζ 2m 1)Γ Aζ 2m) equals ) [ ζ 2m r ζ ) ] 2m 2r+1) 0. [ ζ 2m+1 2.1) Now, let k = 2m+1, 0 m. The ζ 2m+1 )Γ A ζ 2m+1 ) equals ) ][ ζ ) 2m+1 ) +1 ] ζ 2m+1 ) r [ ζ ) 2m+1 2r ] = 2 [ ζ 2m+1) = ζ2m+1 +1 ζ 2m+1)r+1) = ζ2m+1 ζ 2m+1)r+1) [ ] ζ 2m+1)r+1) 2ζ 2m+1)r 2ζ 2m+1)r + [ + ζ2m+1)2r+1) ζ 2m+1) l 2r+1),l 1 1 ζ 2m+1)2r+1) ] Φ l ζ 2m+1. ) Note that, ζ 2m+1 is a d-th primitive root of uity, for some d dividig. As gcd2r+ 1,) = 1, gcd2r +1,d) = 1 ad hece usig Lemma 2.6, Φ l ζ 2m+1 ) is a l 2r+1),l 1 uit i Z[ζ d ]. That is, Φ l ζ 2m+1 ) = 1. Hece, i equatio 2.1), the term l 2r+1),l 1 i the parethesis caot be zero. Thus, the result for the odd case as well. NotethattheecessarypartofTheorem2.7wasstatedadprovedasTheorem2.1 i [9]. We will ow try to uderstad the complemet graph C,r) c of C,r). Propositio 2.8. Let ad r be positive itegers such that the circulat graph C,r) is well defied. The C,r) c is osigular if, ad oly if, 1. ad r have the same parity, 2. gcd,r+1) = 1, ad 3. the highest power of 2 dividig is strictly smaller tha the highest power of 2 dividig r. Proof. Note that A = AC,r) c ) = circ0,...,0 }{{} r+1 1,...,1 }{{} r ] 0 1,...,1 }{{} r 0,...,0). }{{} r Thus, x 1)Γ A x) = 1 + x r )x r 1). Let us ow assume that the graph C,r) c is osigular. That is, Γ A ζ k ) 0, for ay k = 1,2,...,. Note that if ad r have opposite parity the gcd, r 1) = d 2 ad hece Γ A ζ /d ) = 0. Also, if ad r have the same parity ad gcd,r+1) = d > 2 the r is odd ad gcd, r) = gcd, r) = gcd,r+1) = d.

9 Hece, i this case agai, Γ A ζ /d ) = 0. No-sigular Circulat Digraphs 255 Now, the oly case that we eed to check is ad r have the same parity, gcd,r + 1) = 1 ad the highest power of 2 dividig is greater tha or equal to the highest power of 2 dividig r. Asadrhavethesameparityadgcd,r+1) = 1, wegetgcd, r) = 1 ad thus ζ k ) r 1 0, for ay k = 1,2,..., 1. Therefore, we eed to fid a coditio o k so that 1 + ζ k ) r = 0. But this is true if, ad oly if, gcd, r), or equivaletly, the highest power of 2 dividig is greater tha or equal to the highest power of 2 dividig r. Remark 2.9. Observe that usig Propositio 2.8, the graph C,r) c is osigular, wheever ad r are both odd ad gcd,r +1) = 1. Such umbers ca be easily computed. For example, a class of such graphs ca be obtaied by choosig two positive itegers s ad t with s > t ad defiig = 2 s 2 t +1 ad r = 2 t. 3. Geeralizatios. I this sectio, we look at a few classes of graphs/digraphs, which are geeralizatios of the graphs that appear i Sectio 2. We first start with a class of circulat digraphs. Let A = circq), where q = 1,...,1, 0,...,0, 1,...,1, 0,...,0 ), for certai }{{}}{{}}{{}}{{} r s t r+t+s) o-egative itegers r,s,t ad. Let the correspodig digraphs be deoted C r,s,t. The a few coditios uder which the C r,s,t digraphs are sigular appears ext. Propositio 3.1. The C r,s,t digraph is sigular if 1. gcd,t,r) > 1, or 2. gcd,t) = 1 ad oe of the followig coditio holds: a) there exists d 2 such that d s ad t = lr, for some positive iteger l mod d). b) is eve, there exists a eve iteger d such that r + s) is a odd multiple of d 2 ad t = lr, for some positive iteger l 1 mod d). Proof. Part 1: Observe that if A = AC r,s,t ) the 3.1) x)γ A x) = x r )+x r+s x t ). If gcd,r,t) = k > 1, the ζ /k is a root of equatio 3.1), ad hece, the C r,s,t digraph is sigular. This completes the proof of the first part. Part 2a: Let gcd,t) = 1 ad suppose that there exists a positive iteger d 2 such that d s ad t = lr, for some positive iteger l mod d). So, there exists β Z such that l = βd. I this case, the RHS of equatio 3.1) evaluated at ζ /d reduces to 0. Thus, the required result holds.

10 256 A.K. Lal ad A Satyaarayaa Reddy Part 2b: Let gcd,t) = 1 ad = 2m. Also, suppose that there exists a eve positive iteger d such that r+s is a odd multiple of d 2 ad t = lr, for some positive iteger l 1 mod d). The there exists β Z such that l = βd +1. I this ) case, the RHS of equatio 3.1) evaluated at ζ /d reduces to ζ r/d ) 1+ζ /2). Thus, uder the give coditios, the correspodig digraph C r,s,t is sigular. Hece, the proof of the lemma is complete Fig Sigular circulat 2,1,2)-graph o 8 vertices ad 2,3,2)-graph o 10 vertices. Thus, the above result gives coditios uder which the C r,s,t -digraphs, for oegative values of r,s ad t, are sigular. For certai values of r,s ad t, these digraphs correspod to graphs. For example, for a fixed positive iteger r, the 1. C r,1,r -digraph is sigular, wheever is chose so that gcd,r) 2 ad 2r+1. Usig Remark 1.5.2, oe obtais a circulat graph with adjacecy matrix circ0, 1,...,1, 0,...,0, 1,...,1). I particular, for r = 2 ad }{{}}{{}}{{} r 2r+1) r = 8, the C 2,1,2 -graph show i Figure 3.1 is sigular. 2. C 2r,2r+1,2r -digraphissigular,wheeverisevead 6r+1. Ithiscase, circ0,...,0, 1,...,1, 0,...,0, 1,...,1, 0,...,0) is the adjacecy matrix of }{{}}{{}}{{}}{{}}{{} r+1 2r 6r+1) 2r r a circulat graph. I particular, for r = 1 ad = 10, the C 2,3,2 -graph show i Figure 3.1 is sigular. We ow defie aother class of circulat digraphs ad obtai coditios uder which the circulat digraphs are sigular. Let i, j, k ad l be o-egative itegers such that j > l ad kj+i+l< ad let C i,j,k,l be a circulat digraph o vertices with γ ACi,j,k,l )x) = k i+l x s+tj = x i s l+1,s 1 Φ sx) t k+1)j,t j Φ tx) as its t=0s=i represeter polyomial. Thus, we have the followig theorem which we state without proof.

11 No-sigular Circulat Digraphs 257 Theorem 3.2. The circulat digraph C i,j,k,l, defied ) above, is sigular if, ad oly if, either gcdl+1,) 2 or gcd k +1, gcd,j) 2. Remark 3.3. Note that we ca vary the o-egative itegers i,j,k ad l to defie quite a few classes of circulat digraphs. For example, the graphs Gr,t) i [3] are a particular case of C i,j,k,l ad Theorem 3.2 is a geeralizatio of Remark Ackowledgmet. We gratefully ackowledge the very helpful commets ad costructive criticism of the aoymous referee. REFERENCES [1] J.A. Body ad U.S.R. Murty. Graph Theory with Applicatios. The Macmilla Press Ltd., Lodo, [2] Ph.J. Davis. Circulat Matrices. Wiley, New York, [3] M. Doob. Circulat graphs with det AG)) = degg): Codetermiats with K. Liear Algebra ad its Applicatios, 340:87 96, [4] D.S. Dummit ad R.M. Foote. Abstract Algebra, secod editio. Joh Wiley ad Sos, [5] D. Geller, I. Kra, S. Popescu, ad S. Simaca. O circulat matrices. Availiable at sori/eprits/circulat.pdf. [6] I. Kra ad S. Simaca. O circulat matrices. Notices of the America Mathematical Society, 593): , [7] R.P. Kursha ad A.M. Odlyzko. Values of cyclotomic polyomials at roots of uity. Mathematica Scadiavica, 49:15 35, [8] V.V. Prasolov. Polyomials. Spriger, [9] L. Aquio-Ruivivar. Sigular ad osigular circulat graphs. Joural of Research i Sciece, Computig ad Egieerig, 33), 2006.