Integer sequences from walks in graphs

Transcription

1 otes on umbe Theoy and Discete Mathematics Vol. 9, 3, o. 3, Intege seuences fom walks in gahs Enesto Estada, and José A. de la Peña Deatment of Mathematics and Statistics, Univesity of Stathclyde Glasgow G XH, U.K. Cento de Investigación en Matemáticas (CIMAT) A. C., Guanauato 364, México Abstact: We define numbes of the tye O () and E () (,,, ) and the coesonding intege seuences. We ove that these intege seuences, e.g., S () O (), O (),, O (), and S E () E (), E (),, E (), coesond to the numbe of odd and even walks in comlete gahs K. We then ove that thee is a uniue family of gahs which have exactly the same seuence of odd walks between connected nodes and of even walks between ais of nodes at distance two, esectively. These gahs ae the cown gahs: G n K K n. Keywods: Intege seuences, Gah walks, Cown gahs. AMS classification: 5C5; B99; 5C76; 5B5; 5C8 Intoduction Intege seuences aise fom a few diffeent souces, such as enumeation oblems, numbe theoy, game theoy, hysics, and so foth []. Enumeation oblems in gah theoy ae an immense souce of intege seuences as can be seen fom an insection of the Online Encycloedia of Intege Seuences [, 3]. In that aticula case an investigation in the context of gah theoy o combinatoics gives ise to a seuence of integes, which is then oely investigated. Howeve, hee we ose a diffeent oblem. Fo instance, let be a natual numbe ( > ) and let us define the following numbes (,,, ): O (), (.) E (). (.) Using O () and E () let us now intoduce the following seuences: S O () O (), O (),, O (), (.3) S E () E (), E (),, E (),. (.4) Thee ae two S O () and thee S E () seuences eoted in the Online Encycloedia of Intege Seuences (OEIS) [3]. They ae: 78

2 Seuence OEIS code Cuent wok, 3,, 43, 7,... A7583 S O (), 7, 6, 547, 49,... A66443 S O (3),, 5,, 85, 34,... A45, S E () *,,, 8, 64,... A5857, S E (3) *, 3, 5, 89, 37,... A85, S E (4) * * The even seuences hee do not include zeo as in the OIES ones. They aea, howeve, elated to diffeent mathematical obects. Fo instance, A7583, A45 and A85 aea associated to the fomulae ( n ) / 3, (4 n ) / 3 and (6 n ) / 5, esectively. Howeve, A66443 aeas as the numbe of distinct walks of length n along edges of a unit cube between two fixed adacent vetices and A5857 eesents the numbes whose base 9 eesentation is... Fo > 3 no seuence of the tye S O () is eoted in the OEIS and the same is tue fo S E () when > 4. So, the uestions ae: Is thee a geneal famewok in which all of these seuences can be goued togethe? Ae these seuences elated to any oety of a cetain kind of gah? Of couse, these uestions ae not always ossible to be answeed. Howeve, in the aticula case that we can associate such seuences with a oety of a tye of gah we can yet ask anothe fundamental uestion. Given an intege seuence that eesents a oety of a given kind of gah, can we constuct anothe family of gahs having the same seuence fo this oety? These uestions ae investigated hee fo the aticula seuences (.3) and (.4). We find hee that they coesond to the numbe of odd and even-length walks between ais of nodes in comlete gahs. We then ove that fo each comlete gah K n thee is a uniue gah, not isomohic to K n, which has exactly the same seuence of walks. These gahs ae the biatite double cove of the comlete gahs, which ae known as cown gahs. Peliminay esults Fo the sake of self-containment of this wok we ove hee a few basic esults which ae needed fo oving the main esult stated in the next section. We stat by finding the geneal exessions fo the numbes O () and E (). Lemma.: The numbes O () and E () ae ositive integes given by the following fomulae: O ( ), (.) E ( ),,,.... (.) Poof: The ocedue is uite simila fo both numbes, thus we ae showing only it fo O (). This numbe can be exessed as: 79

3 8 ( ) ( ). O (.3) In ode to show that the numbe is a ositive intege it is enough to ealize that ( ) > O, (.4) which oves the esult. In a simila way the oof fo E () is conducted. ow we ove the following esult that elates the seuences S O () and S E () with the numbe of walks in comlete gahs. Let us conside a simle gah without multile links o self-loos G (V, E) with nodes (vetices) v i V, i,, n and edges {v i, v } E. A walk of length k is a seuence of (not necessaily distinct) nodes v, v,, v k, v k, such that fo each i,,, k thee is an edge fom v i to v i. If v v k the walk is named a closed walk. A comlete gah K n is the gah having n nodes and evey ai of nodes is connected by an edge. Theoem.: The seuence S O () and S E () give, esectively, the numbe of walks of odd and even lengths between ais of nodes in a comlete gah K n. Poof: Let M (, ) be the numbe of odd walks of length between the nodes and in K. Then, it is known that ( ) ( ) ( ), M λ, (.5) whee () is the th enty of the othonomalized eigenvecto associated with the λ eigenvalue. The incial eigenvalue of the adacency matix of K is and its coesonding othonomalized eigenvecto is ( ), whee is an all-ones vecto. The est of the eigenvalues ae eual to. Then, ( ) ( ) ( )( ) ( ) ( )., M (.6) In a simila way the esult fo the numbe of walks of even length M (, ) between two nodes in K is oved.

4 We now tun ou attention to the gahs G H K, which ae known as the biatite double cove of the gah H. Hee denotes the tenso (Konecke oduct) of the adacency matices of the two gahs. In the next section we ove the main esult of this wok, which is elated to the numbe of walks in the biatite double coves of comlete gahs: G K K. These gahs ae known as cown gahs. We state now a few known o easy to ove oeties of cown gahs [4 9]: ) G has nodes, whee is the numbe of nodes in the oiginal K ; ) G is biatite and egula with degee k ; 3) The adacency matix of G is ( ) AG A( K ) A( K ) ; 4) The diamete of G is 3 and its distance matix is A ( ) ( K ) 3I A( K ) DG 3I A( K ) A ( K ) ; 5) The numbe of walks of length between the nodes i and in the gah G is given by: M i k, if i, d and i is odd k k, if, i i d and is even k, if i, and is odd k, if i, and is even k whee k is the degee of a node in G. 6) The sectum of G is s(g ) {[k], [], [ ], [ k] }; 7) The gahs G ae distance-egula gahs. 3 Main esult ow, we state the main esult of this wok. Let S O () O (), O 3 (),, O (), be the seuence of walks of odd length in a comlete gah K. Then, by Lemma. and Theoem. we obtain that O ( ). (3.) Theoem 3.: Let G be a gah with adacency matix A (a i ). Then, fo evey edge of G the vitual owe ( a ) O ( ), fo evey, if and only if G is the biatite double cove gah of a comlete gah, G K K. 8

5 Poof: We divide the oof in seveal stes. Poosition 3.: Let G be a connected, biatite almost comlete gah with nodes. Then the following hold: (a) The sectal adius ρ ; (b) G is a egula gah with constant degee k fo any node. Poosition 3.3: Let G be a connected, biatite gah satisfying: (c) G is a egula gah with constant degee k fo any node ; ( ) ( ) (d) Fo evey edge we have a fo,. Then G K K. We oceed with the oof of the Poositions. Poof of Poosition 3.: Let us wite B A(K ) (b s ) fo the adacency matix of K. Then fo any edge in G we have ( ) ( ) ρ lim a lim b. (3.) Let be any vetex of G and choose an edge. The matix s A% A is double stochastic, ρ its (, ) enty a% measues the obability to go fom node to node in the gah G. Theefoe, k. We need the following Lemma fo the oof of Poosition 3.3. Lemma 3.4: Let G be a biatite gah satisfying (c) and (d) above. Then fo any ai of nodes (, ) in G the following holds: a ( ). Poof: Assume othewise that a ( ) >. Obseve that a ( ) a a a ( ) and we get the following full subgah H of G: s s s. Hence 8

6 ( 3 ) a. Indeed, using whee all neighbouing vetices to and ae,,. Let us calculate hyothesis (d) we get ( 3) ( ) ( )( ) a a, (3.3) which yields eithe one of the following two situations: ( ) ) a, fo some. Without loss of geneality, fo we have a a a (3.4) ( ) s s s Since thee ae no edges connecting any two of,,, because G is biatite, thee should exist vetices,, ( ) not in H, maybe not aiwise diffeent, such that a s, a s, fo each s,, ( ). We can assume that a, a, fo each,,. and thee ae no aths of length two between and. Reeating the calculation done in (3.3) we get ( ) ( ) ( )( ) 3 ( )( ) a a 3, (3.5) which is a contadiction. ) Fo i in the set,,, we have a ( ) ( ) ( ) ( ) ( ) 5 3 ( ) i. Then 5 3 a a a ( )( ), (3.6) which also yields a contadiction. Poof of Poosition 3.3: We shall now descibe the stuctue of the matix A. Fo that ( ) uose we shall calculate all vitual owes a. Fist, fo each vetex we have ( ) a k i. Let, be diffeent vetices in G. By the Lemma above, a ( ). Assume that a ( ) and take any vetex with edges. Then fo the neighbous, K, of we get, using Poosition 3., ( 3) ( ) ( )( ) i ( ) 83 a a, (3.7) ( ) which imlies euality a. We obseve that the numbe of vetices of the gah G is n. Indeed, since G is biatite, we get two non-emty classes V, V of totally disconnected vetices in the set,, n. Let,, s be thee diffeent vetices in V and i edges. Then ( ) ( ) a a. (3.8) s Since this numbe is not zeo, thee is a vetex k in V which is connected to and s, and ( ) a. Theefoe A ( ) has a o deending if, belong o not to the same class V s i. Hence each of V and V have the same cadinality. Moeove it is clea that the adacency matix A has enties a o deending if, ae diffeent and in distinct classes o belong to the same class V i, esectively. As desied, it follows that A A(K ) A(K ).

7 4 Closing emaks We have intoduced two new intege seuences which we ove coesond to the seuences of even and odd walks in comlete gahs K n. The fundamental uestion osted hee is whethe thee ae othe gahs with exactly the same seuence of walks as those of the comlete gahs. We have oved that the answe is ositive and the gahs having such seuences ae uniue. They coesond to the biatite double coves of the comlete gahs, known as cown gahs: G n K K n. Acknowledgement Enesto Estada thanks CIMAT fo wam hositality duing Januay-Febuay 3 and fo a visiting ofessoshi osition duing this eiod. Refeences [] Sloane,. J. A. My favoite intege seuences. Seuences and Thei Alications (Poceedings of SETA 98), edited by C. Ding, T. Helleseth, and H. iedeeite, Singe-Velag, London, 999, 3 3. [] Sloane,. J. A. The Online Encycloedia of Intege Seuences. otices of the ACM, Vol. 5, 3, [3] The Online Encycloedia of Intege Seuences. htt://oeis.og/. [4] Walle, D. A. Double coves of gahs, Bul. Aust. Math. Soc., Vol. 4, 976, [5] Haemes, W. H. Matices fo gahs, design and codes. Infomation Secuity, Coding Theoy and Related Combinatoics, D. Cnković and V. Tonchev (Eds.), IOS Pess,, [6] Bouwe, A. E., A. M. Cohen, A. eumaie, Distance Regula Gahs. Singe- Velag, Belin, ew Yok, 989. [7] Cvetković, D. M., M. Doob, H. Sachs, Secta of Gahs, V.E.B. Deutsche Velag de Wissenschaften, Belin, 979. [8] Fiol, M. A. Algebaic chaacteizations of distance-egula gahs. Disc. Math., Vol. 46,, 9. [9] Agawal, S., P. K. Jha, M. Vikam, Distance egulaity in diect-oduct gahs. Al. Math. Lett., Vol. 3,,