AUTOMORPHISMS OF THE CYCLE PREFIX DIGRAPH

Transcription

1 AUTOMORPHISMS OF THE CYCLE PREFIX IGRAPH Wllam Y C Chen Center for Combnatorcs The e Laborator of Pure Mathematcs and Combnatorcs of Mnstr of Educaton Nana Unverst, Tanjn 30007, P R Chna Vance Faber Bg Pne e, FL Bngqng L epartment of Rs Management and Insurance Nana Unverst, Tanjn 30007, P R Chna AMS Classfcaton: B83, 05C05 eords: Ccle pref dgraph, automorphsm groups Note Ths s a verson of an unpublshed report from 00 We place t here to gve t der access

2 Abstract Ccle pref dgraphs have been proposed as an effcent model of smmetrc nterconnecton netors for parallel archtecture It has been dscovered that the ccle pref netors have man attractve communcaton propertes In ths paper, e determne the automorphsm group of the ccle pref dgraphs We sho that the automorphsm group of a ccle pref dgraph s somorphc to the smmetrc group on ts underlng alphabet Our method can be appled to other classes of graphs bult on alphabets ncludng the hpercube, the autz graph,and the de Brujn graph eords: Ccle pref dgraph, automorphsm group AMS Subject Classfcaton: 05C5, 68M0 Introducton The ccle pref dgraphs,or the ccle pref netors, have been recentl proposed as model of smmetrc nterconnecton netors th a large number of vertces and small dameter [5] It has been dscovered that such netors possess a number of attractve propertes for effcent communcaton n parallel computng The man advantage of usng a smmetrc netor s that each processor sees the netor communcaton the same as an other processor n the netor [] Ths mples that all the processors can essentall eecute the same communcaton program For ths reason, a smmetrc netor s sometmes called a homogeneous netor Beond the verte smmetr propert of a dgraph, the automorphsm group actuall provdes a complete pcture of the smmetres of the dgraph An mportant characterzaton of smmetrc dgraphs s due to Sabduss hch states that a dgraph s verte smmetrc f and onl f t can be represented b a Cale coset dgraph [9] Ths characterzaton leads to the dscover of the ccle pref dgraph, and the reader s referred to [5] for the Cale coset dgraph

3 defnton of Hoever, n the present paper, e shall use the representaton of Γ n terms of sequences, analogous to the representatons of hpercubes, the de Brujn graphs and the autz graphs Throughout ths paper, e assume that The sequence defnton of goes as follos A verte of s represented b a partal permutaton on,,, +, hch s a sequence of dstnct elements A partal permutaton of elements s also called a -permutaton Snce the adjacenc n a dgraph could be ambguous f one does not specf the drecton, f u, v s an arc n a dgraph, e shall sa that u s adjacent to v, hereas v s net to u The adjacenc relaton of can be descrbed as follos:, m, f f m ;,,, We sa that s net to va a rotaton operaton for, and m s sad to be net va a shft operaton for m,,, In partcular, f =, the verte s sad to be net to notaton: va a full rotaton Moreover, e shall use the follong R = for ; R m = m for m,,, In ths paper, e ll dscuss the general case,, r 0, n hch the 3

4 adjacenc relaton s defned as:, m, f f r + ; m,,, Note that s a specal case of, for r = 0 From the above sequence representaton, one sees that, s verte smmetrc [5]: Gven an to vertces X = and Y =,one can alas fnd a permutaton π on,,, + such that π = Clearl, such a permutaton π nduces an automorphsm of, because t smpl relabels the underlng alphabet of, It s eas to see that ever verte n, has both ndegree and outdegree r, and the dgraph has + = + L + vertces Moreover,, has dameter + r [3] and man other nce propertes ncludng Hamltonct, hgh connectvt, small de dameter, and regular reachablt [,3,4,5,7,6,8] The man concern of ths paper s th the automorphsm group of, We shall sho that the automorphsm group of, s somorphc to the smmetrc group on ts underlng set,,, + Our approach to determne the automorphsm group of the ccle pref dgraphs ma be emploed to characterze the automorphsm group of other classes of netors,such as the hpercube, the autz graph and the de Brujn graph The Automorphsm Group of, Gven a dgraph G= V, E, an automorphsm of G s a permutaton π on V such that u, v E f and onl f π u, π v E All the automorphsms of G form the automorphsm group of G We sa that G s smmetrc or verte 4

5 transtve f ever verte of G loos the same - strctl speang, for an to vertces u and v there ests an automorphsm π of G such that π u = v The man result of ths paper s to determne the automorphsm group of the ccle pref dgraph, Namel, e ll prove the follong: Theorem For r +, the automorphsm group of, s somorphc to the smmetrc group S + on the underlng set {,,, + } We establsh a fe lemmas Lemma Let α be an automorphsm of, and u, v to vertces n, If v s net to u va a shft operaton then α v s net to α v va a shft operaton as ell Proof: Snce, s verte smmetrc, thout loss of generalt e ma assume that u= and v =, here s an nteger satsfng < + Let α = and let α = Snce s net to n and α s an automorphsm, s net n Note that the dstance from to s at least because does not appear n and ether a rotaton or shft operaton can move an element n a sequence at most one step forard from left to rght Assume that s net to va a rotaton operaton, sa, R for some <, that s, = R Then t s clear that R = It follos that R = Therefore, the dstance from to s at most Snce 5

6 an automorphsm preserves dstance, α be an automorphsm must be net to va a shft operaton Snce the adjacenc of to vertces n, s realzed ether b a rotaton or shft operaton, Lemma s equvalent to the statement that f v s net to u va a rotaton operaton, then α v s net to α v va a rotaton operaton as ell Lemma 3 Let α be an automorphsm and be to, vertces of, such that = α Let S m be a shft operaton th m {,, L, + } \ {,, L, } If,, appears n Sm, then must appear n α Sm at the same poston as n Sm Proof Snce = α, b Lemma e have α Sm = t for some t Clearl, and are at the same poston for Lemma 4 Let α be an automorphsm and be to, vertces of, such that = α Let R, r +, be a rotaton operaton If,, appears n R, then must appear n α R at the same poston as n R Proof Let = R = + 6

7 From Lemma, t follos that α s net to α va a rotaton operaton,sa, α = R α = R, j j here r + j Snce R = +, t taes at least steps to move from the frst poston to the th poston Hence the dstance from R to s at least On the other hand, e have R R = Thus, the dstance from R to s eactl Smlarl,the dstance from α = R s j Snce an automorphsm preserves j dstance,e have = j Therefore, e have α = R = + We no reach the clam b comparng th We ma llustrate Lemmas 3 and 4 b the follong dagram: S or R α α 3 S or R 7

8 Lemma 5 Let α be an automorphsm of, Then there ests a permutaton = + on the underlng set {,, L, + } of, such that for = +, +,, + α =, 4 α 5 = Proof We choose as the mage of under the acton of α For +, snce s net to va a shft operaton, then α s net to α = va a shft operaton as sgnfed n 5 Snce α s a one-to-one map on the vertces of,, the elements +,, + are dstnct to each other as ell as to,,, It follos that the above chosen sequence s a permutaton on {,, L, + } We call the permutaton constructed from the automorphsm α n the above lemma the derved permutaton of α Our goal s to sho that the automorphsm α s unquel determned b ts derved permutaton For convenence, e also rte as Let us llustrate the above lemma b the follong dagram S α α 6 S 8

9 efnton 6 Let α be an automorphsm of, and = + a permutaton on {,, L, + } Gven a verte n,, e sa that α and are compatble for mage of, namel, f α and have the same α = 7 Lemma 7 Let α be an automorphsm of, and =,,, + be ts derved permutaton Suppose α and are compatble for a verte If s net to va a rotaton operaton, then α and are compatble for as ell Proof Suppose α = and = R, r + j B Lemma 4, e have the follong dagram: R R α α 8 R R That s, α = as desred Lemma 8 Let α be an automorphsm of,, r +, and = + be ts derved permutaton Suppose α and are compatble 9

10 0 for a verte If s net to va a shft operaton, then α and are compatble for as ell Proof From the gven condton, e ma rte as, },,, { L Suppose = α B Lemma 3, e have the follong dagram: S S j j α α It remans to sho that j = For convenence, e sa that a path contans f s contaned n ever verte n the path There are to cases: Case > B Lemma 3, 4 and 5, f e can fnd a path from to that contans, then e can get the follong dagram: j S S j α α α

11 From ths dagram, e obtan j = No e sho that there s a path from to that contans In fact, f {,, L, }, sa j =, then j j+ s net to va a rotaton operaton If {,, L, }, then s net to va a shft operaton Step b step,fnall e can get the follong path: L L L 9 hch s a path to that contans Case Analogous to the above case, f e can fnd a path from to that contans, then e have the follong agram S α α α S j j Smlar to Case, e ma fnd a path from to that contans Combnng the above to cases, Lemma 8 s proved

12 We are no read to prove Theorem Proof of Theorem Snce each permutaton on {,, L, + } nduces an automorphsm α of, th beng the derved permutaton of α, e onl need to sho that an automorphsm α of, s unquel determned b ts derved permutaton Let α be an automorphsm of, th derved permutaton =,,, + We am to sho that α and are compatble for an verte of,, that s, α = 0 Our strateg for provng 0 s as follos B defnton, t holds for the ntal verte Assumng that t holds for a a verte, f e can sho that t also holds for an verte net to, then 0 must be true for an verte because, s strongl connected Let us assume that satsfes 0 Suppose s a verte net to If s net to va a rotaton operaton, from Lemma 7 t follos that 0 holds for When s net to va a shft operaton, from Lemma 8 t follos that 0 also holds for Notng that dfferent derved permutatons lead to dfferent automorphsms, ths completes the proof Let us eamne some specal cases of the above theorem Let n = + When =, turns out to be the Cale graph on the smmetrc group S th generators, 3,, n In ths case, t s much easer to n see that the automorphsm group s S nbecause no shft operaton s nvolved

13 The smplest nontrval case s for = Note that s the autz graph, It s non that the automorphsm group of the autz graph, s somorphc to the smmetrc group on the alphabet {,, L, + } see[0] Acnoledgment Ths or as partall supported b the 973 project sponsored b the Mnstr of Scence of Technolog of Chna, and the Natonal Scence Foundaton of Chna The authors ould le to than r Q L L for valuable comments References [] S B Aers and B rshnamurth, A group-theoretc model for smmetrc nterconnecton netors, IEEE Trans on Computers,38 989, [] W Y C Chen, V Faber and E nll, Restrcted routng and de dameter of the ccle pref netor, F Hsu, et al, Eds, IMA CS Ser screte Math Theoret Comput Sc 995, 3-46 [3] F Comellas and M A Fol, Verte smmetrc dgraphs th small dameter, screte Appl Math , - [4] F Comellas and M Mtjana, Broadcastng n ccle pref dgraphs, screte ApplMath , 3-39 [5] V Faber, J Moore and W Y C Chen, Ccle pref dgraphs for smmetrc nterconnecton netors, Netors, 3 993, [6] F Hsu, On contaner dth and length n graphs, groups, and netors, IEICE Trans Fund Elect Commun Comput Sc E77-A 994, [7] M Jang and F Ruse, etermnng the Hamlton-connectedness of certan verte transtve graphs, screte Math , [8] S-C La, G J Chang, F Cao, F Hsu, Fault-tolerant routng n crculant netors and ccle pref netors, Ann Combn [9] G Sabduss, Verte transtve graphs, Monatsh Math ,

14 [0] J L Vllar, The underlng graph of a lne dgraph, screte Appled Math 37/38 99,