Admissibility Cnditins and Asympttic Behavir f Strngly Regular Graphs VASCO MOÇO MANO Department f Mathematics University f Prt Oprt PORTUGAL vascmcman@gmailcm LUÍS ANTÓNIO DE ALMEIDA VIEIRA Department f Civil Engineering University f Prt Oprt PORTUGAL lvieira@feuppt Abstract: Let G be a strngly regular graph with three distinct eigenvalues and A its adjacency matrix We assciate a three dimensinal Euclidean Jrdan algebra t A and establish new admissibility inequalities n the parameters and spectrum f G Finally, we extract sme asympttic cnclusins abut the spectrum f G Key-Wrds: Graph thery, Strngly regular graph, Euclidean Jrdan algebra, Matrix analysis, Cmbinatrics Intrductin Strngly regular graphs were intrduced in a 96 paper entitled, Strngly regular graphs, partial gemetries and partially balanced designs, [], frm R C Bse The study f these graphs ften fcus n btaining suitable admissibility cnditins ver their parameters The mst used and nt trivial admissibility cnditins are the Krein cnditins and the abslute bunds see [] In ur wrk we explre the clse and interesting relatinship f a three dimensinal Euclidean Jrdan algebra t the adjacency matrix f a strngly regular graph, in rder t btain sme new inequalities fr the existence f strngly regular graphs Euclidean Jrdan algebras were brn back in 94, when Pascual Jrdan, Jhn vn Neumann and Eugene Wigner published their entitled paper On an algebraic generalizatin f the quantum mechanical frmalism, [] In this paper, the authrs tried t deduce the Hermitian matrices prperties, in a quantum mechanics cntext It is remarkable that since then, Euclidean Jrdan algebras have had such a wide range f applicatins Fr instance there are applicatins t the thery f statistics see [4], t interir pint methds see [5] r [6] and t cmbinatrics see [7] Detailed literature n Euclidean Jrdan algebras can be fund in Kechers lecture ntes, [8], and in the mngraph by Faraut and Krányi, [9] This paper is rganized as fllws In Sectin we present sme basic cncepts cncerning strngly regular graphs and Euclidean Jrdan algebras and we frmulate ur prblem Then, in Sectin, we present the slutin fr ur prblem by assciating a three dimensinal Euclidean Jrdan algebra t the the adjacency matrix f a strngly regular graph Next, we deduce new admissibility cnditins ver the parameters f strngly regular graphs We finish this paper with sme experimental results and sme asympttic cnclusins, in Sectin 4 Prblem Frmulatin A Brief Intrductin n Euclidean Jrdan Algebras In this sectin we present relevant cncepts fr ur wrk which can be seen, fr instance in [9] Let W be a real vectr space with finite dimensin and a bilinear mapping u,v u v If fr all u and v in W we have I u v v u and II u u v u u v, with u u u, ISBN: 978--6804-00- 44
then W is called a Jrdan algebra If W cntains an element, e, such that fr all u in W we have e u u e u, then e is called the unit element f W Given a Jrdan algebra W with unit element, if there is an inner prduct <, > that verifies the equality < u v,w > < v,u w >, fr any u, v, w in W, then W is called an Euclidean Jrdan algebra An element c in an Euclidean Jrdan algebra W, with unit element e, is an idemptent if c c c Tw idemptents c and d are rthgnal if c d 0 We call the set { c,c,,c k } a cmplete system f rthgnal idemptents if I c i c i, i {,, k}, II c i c j 0, i j and III c c c k e Let W be an Euclidean Jrdan algebra with unit element e and u in W Then, there exist unique distinct real numbers,,, k, and a unique cmplete system f rthgnal idemptents { c,c,,c k } such that with c j, j,, k, real numbers see [9], Therem III The j s are the eigenvalues f u and the decmpsitin is the spectral decmpsitin f u u c c k c k, General Cncepts n Strngly Regular Graphs Alng this paper we nly cnsider nn-empty, simple graphs graphs with n lps nr parallel edges and nt cmplete graphs graphs that have sme nn-adjacent pair f vertices, herein called graphs Cnsidering a graph G, we dente its vertex set by VG and its edge set by EG An edge whse endpints are the vertices x and y is dented by xy and, in such case, the vertices x and y are adjacent r neighbrs The number f vertices f G, VG, is called the rder f G If all vertices f G have k neighbrs, then G is a k- regular graph Let G be a graph f rder n Then G is a n, k,a,c-strngly regular graph if it is k -regular and any pair f adjacent vertices have a cmmn neighbrs and any pair f nn-adjacent vertices have c cmmn neighbrs The parameters f a n, k,a,c-strngly regular graph are nt independent and are related by the equality kk a n k c We assciate t G an n by n matrix A # $, where each a ij, if v i v j E G, therwise a ij 0, a ij called the adjacency matrix f G The eigenvalues f A are simply called the eigenvalues f G It is well knwn see, fr instance, [] that the eigenvalues f the n, k,a,c-strngly regular graph G are k, and, where and are given by a c a c 4 k c and a c a c 4 k c Admissibility Cnditins Over Strngly Regular Graphs In the last sectin we presented equatin that is an example f an admissibility cnditin that must be satisfied by the parameters f any strngly regular graph There are many ther admissibility cnditins ver the parameters f a strngly regular graph Sme f the mst imprtant admissibility cnditins are knwn as the Krein cnditins, btained in 97 by Sctt, Jr, [0] Hwever, there are still many parameter sets fr which we dnt knw if they crrespnd t a strngly regular graph r nt In this wrk we deduce sme new cnditins t claim the inadmissibility f certain parameter sets f strngly regular graphs We are able t deduce them by assciating an Euclidean Jrdan algebra t the adjacency matrix f a strngly regular graph ISBN: 978--6804-00- 45
Prblem Slutin An Euclidean Jrdan Algebra Assciated t a Strngly Regular Graph Let G be a n,k,a,c-strngly regular graph such that 0 < c < k < n and let A be the adjacency matrix f G with three distinct eigenvalues, namely k, and, given by the frmulae frm Subsectin Let k and be the psitive eigenvalues and be the negative eigenvalue f A Nw we cnsider the real Euclidean Jrdan subalgebra f W spanned by I n and the natural pwers f A Since A has three distinct eigenvalues, then W is a three dimensinal Euclidean Jrdan algebra Let { E, E, E } be the unique cmplete system f rthgnal idemptents f W assciated t A, with # E k k k n, k # A k # A # k # I J n n E # k E # k k # A # k A k# # k I and n k # A # k A k# # k I, n where J n is the matrix whse entries are all equal t We rewrite the idemptents under the basis { } f W, btaining I n, A, J n A I n E n I n n A n J A I n n, E n k n # E n k n # A k n # J A I n n and I n n k n # I n k n n # A k n # J A I n n the set f square matrices f rder n Nw let p be a natural number such that p and dente by M n with real entries Fr B in M n, we dente by B p and B p the Hadamard pwer f rder p f B and the Krnecker pwer f rder p f B, respectively, with B B and B B Here we intrduce a cmpact ntatin fr the Hadamard and the Krnecker pwers f the elements f Let x and be natural numbers such that and x Then, we cnsider E x E x and E x E x New Admissibility Cnditins fr Strngly Regular Graphs Let G be a strngly regular graph, W the Euclidean Jrdan algebra spanned by the natural pwers f I n and { } the unique cmplete system f rthgnal idemptents assciated t A Then, we A, and E, E, E cnsider the fllwing spectral decmpsitin f A, A ke E E Nw let S l, with l a natural number, be the fllwing sum S l J n l E J n l E J n l 4 E 5 J n l 6 E l, ISBN: 978--6804-00- 46
where each summand is a Krnecker prduct with l factrs The sum S l given by: Nw we bserve that Let q l, q l Since the set S l and q l J n l E J n l E J n l 4 E 5 J n l 6 E l, S l be the real numbers such that l i J n E S l i i q l E i i F { E i E i E i l : i,i,,i l #{,, } } has a principal submatrix l is a cmplete system f rthgnal idemptents that is a basis f the real Euclidean Jrdan algebra V l spanned by I l and the natural pwers f A, then the minimal plynmial f S l is Since S l is a principal submatrix f S l f 0 n # l l n l i i and since f is the minimal plynmial f S l then we cnclude, by the interlacing Therem see, fr instance, [, Therem 45], that the eigenvalues f S l nnnegative are all Nw since # > and >, then n k n # n k n # <, < and k n # < i Therefre, the series J n # E is cnvergent and we call its sum S Cnsider the real numbers q, q and q i such that S q E q E q E Because q lim l q l #, q lim l q l # q lim l q l #, and the eigenvalues f S l and are nnnegative, it fllws that q 0, q 0 and q 0 Let S E S and q, q and q such that S q E q E q E Then we deduce that: n # q n # q n # q n # Nw, since min the parameters q, q n k n k n # n k n # n k k n # k n k ; n # n k n # n k n k n # n k n # n k n # k ; n # n k n # n k n k n # n k n # n k # n # k # n # n k A B min A min B, fr any matrices A and B in M n see [, p ], and since and q are nnnegative, then we cnclude that the parameters q, q and q are ISBN: 978--6804-00- 47
x nnnegative and, therefre the eigenvalues f S are als nnnegative, where x is a natural number Nw let i q xs, fr i,, { }, be the real numbers such that S x i q xs E i i Analysing the parameters q xs, q xs, q xs and after sme algebraic manipulatin we are able t establish the fllwing therem that is a new admissibility cnditin fr the existence f strngly regular graphs Therem : Let G be a n, k,a,c-strngly regular graph, such that 0 < c < k < n, whse adjacency matrix has the eigenvalues k, and Then % n # $ 0 n # $ n k # # n k # k # # n # k % n # $ n # $ % n # $ 0 n # $ fr any natural number x n k # # n k # k # # n # k % n # $ n # $ x # x # x x #n k # # #n k # % n # $ n # $ n # k # ; #n k # # #n k # % n # $ n # $ #$ # ; Nte that prceeding in a similar way with the ther idemptents f we wuld btain ther necessary cnditins fr the existence f a n, k,a,c-strngly regular graph Regarding the inequalities frm Therem we can prvide a simple explanatin abut the difference f the expnents in inequality and The reasn is that, since the middle parcel is the nly ne that is negative, if we multiply by a psitive number, k, and we wish t fail, then we must have an dd expnent in rder fr that parcel t remain negative As fr inequality, we have an ppsite situatin Multiplying a negative parcel by a negative number,, wuld give us a trivial inequality Therefre we make the middle parcel psitive by applying an even expnent The fllwing results are btained frm the inequalities f Therem Therem : Let G be a n, k,a,c-strngly regular graph, such that 0 < c < k < n, whse adjacency matrix has the eigenvalues k, and Then 4 n k # 0 This result is nly relevant if <, therwise it represents a trivial inequality Nte that if 0, then k has t be larger than n Next we present anther result ver the parameters f a strngly regular graph that satisfy k < n Therem : Let G be a n, k,a,c-strngly regular graph, such that 0 < c < k < n, whse adjacency matrix has the eigenvalues k, and If k < n, then 5 4# < 8n n k # # # # # x x # $ k ISBN: 978--6804-00- 48
Analysing inequality 5 we bserve that, cnsidering n, k and fixed, the left hand side is a plynmial in f degree, with psitive cefficients, and the right hand side is a plynmial in f degree with psitive cefficients Therefre we may cnclude that if is smaller than, then cannt be t large 4 Cnclusin In this sectin we present a few examples f parameter sets n, k,a,c that dnt verify inequalities, and 5 In Table we cnsider the parameter sets P 0,00,,97, P 585, 784,, 75, P 989,988,987,0989 and P 4 999900,9999000,8999,998900 Fr each set we present the respective eigenvalues and, and the parameters q ys cnsider 8n q n k # # # # # btained frm inequality 5 f Therem and q ys 4# fr sme values f y We als, Parameters P P P P 4 00 007 0 000 95 70 000 0 0 7 q S 9 0 7 0 8 0 5 0 9 q S q 5 S q 6 S 5 0 5 5 0 5 54 0 8 50 0 6 0 0 4 6 0 0 0 0 0 7 0 0 4 95 0 4 q 0 4 9 0 6 8 0 8 40 0 4 Table : Numerical results fr P, P, P and P 4 Frm the data presented in Table we cnfirm the results expressed in Therem, namely we cnfirm that if is much smaller than the abslute value f, then we cnclude that the parameter set n, k,a,c des nt crrespnd t a strngly regular graph Regarding ur results we can shw that ur parameters q xs asympttic behavir when k < n and 0 In fact, with x, we have: % k n # k lim q x S #n$ 0 n $ # k #n kk n $ # #n k Then, in rder fr inequality t fail, we must have k n k n k < n kk n n k # n n k < n k $ n nk < 0 < 0 $ n k n But since n > 0, we must have k n < 0, and s k < n and q xs, display an interesting ISBN: 978--6804-00- 49
As fr the parameter q xs, we can prceed in a similar manner t btain an analgus cnclusin This prves an interesting asympttic behavir, because we can cnclude that fr a n, k,a,c parameter set such that k < n and sufficiently small, inequalities and will fail and therefre the crrespnding strngly regular graph des nt exist In ther wrds, if is sufficiently small, then k n, which cnfirms the infrmatin given by inequality 4 f Therem The parameter sets that we used in Table match these requirements and the crrespnding results are precisely as expected References: R C Bse, Strngly regular graphs, partial gemetries and partially balanced designs, Pacific J Math, Vl, 94, pp 9-64 C Gdsil and G Ryle, Algebraic Graph Thery, Springer, 00 P Jrdan, J v Neuman and E Wigner, On an algebraic generalizatin f the quantum mechanical frmalism, Annals f Mathematics, Annals f Mathematics, Vl5, 94, pp 9-64 4 H Massan and E Neher, Estimatin and testing fr lattice cnditinal independence mdels n Euclidean Jrdan algebras, Ann Statist, Vl6, 998, pp05-08 5 L Faybusvich, Euclidean Jrdan algebras and Interir-pint algrithms, J Psitivity, Vl, 997, pp -57 6 L Faybusvich, Linear systems in Jrdan algebras and primal-dual interir-pint algrithms, Jurnal f Cmputatinal and Applied Mathematics, Vl86, 997, pp48-75 7 D M Cards and L A Vieira, Euclidean Jrdan algebras with strngly regular graphs, Jurnal f Mathematica Sciences, Vl0, 004, pp 88-894 8 M Kecher, The Minnesta Ntes n Jrdan Algebras and Their Applicatins, Springer, 999 9 J Faraut and A Krányi, Analysis n Symmetric Cnes, Oxfrd Science Publicatins, 994 0 L L Sctt Jr, A Cnditin n Higman s parameters, Ntices f Amer Math Sc, Vl0 A-97, 97, pp 7-0-45 R Hrn and C R Jhnsn, Matrix Analysis, Cambridge University Press, 985 R Hrn and C R Jhnsn, Tpics in Matrix Analysis, Cambridge University Press, 99 ISBN: 978--6804-00- 50