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Bipartite Q-Polynomial Distance-Regular Graphs
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  See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/226014229 Bipartite Q-Polynomial Distance-RegularGraphs  Article   in  Graphs and Combinatorics · March 2004 DOI: 10.1007/s00373-003-0538-8 CITATIONS 8 READS 18 1 author: John Simon CaughmanPortland State University 33   PUBLICATIONS   723   CITATIONS   SEE PROFILE All content following this page was uploaded by John Simon Caughman on 30 May 2014. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the srcinal documentand are linked to publications on ResearchGate, letting you access and read them immediately.  Bipartite  Q -polynomial Distance-regular Graphs John S. Caughman, IV  1 Abstract.  Let Γ denote a bipartite distance-regular graph with diameter  D  ≥  12. We show Γ is Q -polynomial if and only if one of the following (i)-(iv) holds:(i) Γ is the ordinary 2 D -cycle.(ii) Γ is the Hamming cube  H  ( D, 2).(iii) Γ is the antipodal quotient of   H  (2 D, 2).(iv) The intersection numbers of Γ satisfy c i  =  q  i − 1 q  − 1  , b i  =  q  D − q  i q  − 1 (0 ≤ i ≤ D ) , where  q   is an integer at least 2.We obtain the above result using the Terwilliger algebra of Γ. 1 Introduction This article continues our investigation of bipartite  Q -polynomial distance-regular graphs [5], [6], [7], [8]. Let us briefly summarize our results to date. Let Γ denote a bipartite  Q -polynomial distance-regular graphwith diameter  D  ≥  3. In [7] we determined the irreducible modules for the Terwilliger algebra of Γ. In [5] we found the possible  Q -polynomial orderings of the eigenvalues of Γ. In [6] we assumed  D  ≥  4 and that Γis the quotient of an antipodal distance-regular graph. Under these assumptions we showed that Γ is eitherthe 2 D -cycle or the antipodal quotient of the Hamming cube  H  (2 D, 2). In [8] we considered the formulasfor the intersection numbers of Γ given in Leonard’s theorem (the type I case). In the present bipartite case,the intersection numbers of Γ are determined by  D  and two complex scalars  q   and  s ∗ . We showed that  q   and s ∗ are real provided  D  ≥  4. In [3] we showed that with respect to any vertex, the distance-2 graph inducedon the  D th subconstituent of Γ is distance-regular and  Q -polynomial.In the present paper we prove the following theorem. 1.1 Theorem.  Let  Γ  denote a bipartite distance-regular graph with diameter   D  ≥  12 . Then  Γ  is   Q - polynomial if and only if one of the following (i)-(iv) holds:(i)  Γ  is the ordinary   2 D -cycle.(ii)  Γ  is the Hamming cube   H  ( D, 2) .(iii)  Γ  is the antipodal quotient of   H  (2 D, 2) .(iv) The intersection numbers of   Γ  satisfy  c i  =  q  i − 1 q   − 1  , b i  =  q  D − q  i q   − 1 (0  ≤  i  ≤  D ) , where   q   is an integer at least 2. We remark that the intersection numbers given in case (iv) above are realized by both the bipartite dualpolar graphs [2, p.205] and the Hemmeter graphs [14]. 2 Preliminaries 1 Dept. of Mathematical Sciences, Portland State University, Portland, OR 97207-0751.Email: caughman@mth.pdx.edu. AMS 1991 Subject Classification: Primary 05E30. 1  In this section, we recall some basic definitions and results concerning distance-regular graphs. For moreinformation we refer the reader to [1] or [2].Let  X   denote a nonempty finite set. Let Mat X ( C ) denote the  C -algebra consisting of the matrices over C  which have rows and columns indexed by  X  . Let  V   denote the  C -vector space consisting of the columnvectors over C which have coordinates indexed by  X  . Observe that Mat X ( C ) acts on  V   by left multiplication.Endow  V   with the Hermitean inner product satisfying  u,v   =  u t v  ( u,v  ∈  V   ) , where  u t denotes the transpose of   u  and  v  denotes the complex conjugate of   v .Let Γ = ( X,R ) denote a finite, connected, undirected graph, without loops or multiple edges, with vertexset  X  , edge set  R , path-length distance function  ∂  , and diameter  D  :=  max { ∂  ( x,y ) |  x,y  ∈  X  } . For each x  ∈  X   and each integer  i , set Γ i ( x ) :=  { y  ∈  X   |  ∂  ( x,y ) =  i } . We abbreviate Γ( x ) = Γ 1 ( x ). We say Γ is bipartite  whenever there exists a partition  X   =  X  + ∪  X  − such that the subgraphs induced on  X  + and X  − contain no edges. From now on assume Γ is bipartite with diameter  D  ≥  3. The graph Γ is said to be distance-regular , with  intersection numbers  c i ,  b i  (0  ≤  i  ≤  D ), whenever for all integers  i  (0  ≤  i  ≤  D )and for all  x,y  ∈  X   with  ∂  ( x,y ) =  i , we have  | Γ i − 1 ( x ) ∩ Γ( y ) |  =  c i  and  | Γ i +1 ( x ) ∩ Γ( y ) |  =  b i . From now onassume Γ is distance-regular. Following convention, we set  k := b 0 . Note that  c i  +  b i  =  k  (0  ≤  i  ≤  D ), andthat  c 0  = 0,  b D  = 0.We now recall the Bose-Mesner algebra of Γ. For  0  ≤  i  ≤  D  let  A i  denote the matrix in Mat X ( C ) with x,y  entry( A i ) xy  =   1 if   ∂  ( x,y ) =  i, 0 if   ∂  ( x,y )   =  i  ( x,y  ∈  X  ) . We call  A i  the  i th  distance matrix  of Γ. We abbreviate  A  =  A 1  and call  A  the  adjacency matrix  of Γ. By [2, p127],  A 0 ,...,A D  form a basis for a subalgebra  M   of Mat X ( C ). We call  M   the  Bose-Mesneralgebra  of Γ. By [1, pp.59,64],  M   has a second basis  E  0 ,..., E  D  such that: (i)  E  0  =  | X  | − 1 J  ; (ii)  E  i E  j  = δ  ij E  i  (0  ≤  i,j  ≤  D ); (iii)  E  i  =  E  i  (0  ≤  i  ≤  D ); (iv)  E  0  +  E  1  +  ···  +  E  D  =  I  , where  I   denotes identitymatrix and  J   denotes the all 1’s matrix. We refer to  E  0 ,..., E  D  as the  primitive idempotents  of Γ. For0  ≤  i  ≤  D  let  θ i  denote the scalar in  R  satisfying  AE  i  =  θ i E  i . We call  θ i  the  eigenvalue  of   A  associatedwith  E  i . We remark  − k  ≤  θ i  ≤  k  and  θ 0  =  k  [2, p.45]. Let  E   denote a primitive idempotent of Γ, and write E   =  | X  | − 1  Di =0 θ ∗ i A i .  We refer to  θ ∗ 0 , θ ∗ 1 ,..., θ ∗ D  as the  dual eigenvalue sequence  for  E  . We note  θ ∗ i  ∈  R (0  ≤  i  ≤  D ). We also note  θ ∗ 0  equals the rank of   E   and is therefore nonzero [1, p.62].Observe  M   is closed under entry-wise multiplication. We say Γ is  Q-polynomial  (with respect to thegiven ordering  E  0 ,...,E  D  of the primitive idempotents) whenever for 0  ≤  i  ≤  D , the matrix  E  i  is an entry-wise polynomial in  E  1  with degree exactly  i . Suppose Γ is  Q -polynomial with respect to  E  0 ,...,E  D . Then thedual eigenvalues associated with  E  1  are mutually distinct [1, p.263]. We say Γ is  Q-polynomial  wheneverthere exists an ordering of the primitive idempotents with respect to which Γ is  Q -polynomial.Fix any vertex  x  ∈  X  . For 0  ≤  i  ≤  D  let  E  ∗ i  =  E  ∗ i  ( x ) denote the diagonal matrix in Mat X ( C ) with  y,y entry ( A i ) xy  ( y  ∈  X  ). Let  T   =  T  ( x ) denote the subalgebra of Mat X ( C ) generated by  A  and  E  ∗ 0 ,...,E  ∗ D .We call  T   the  Terwilliger algebra  of Γ with respect to  x . We remark that  T   is finite dimensional andsemisimple.By a  T  -module we mean a subspace  U   ⊆  V   such that  BU   ⊆  U   for all  B  ∈  T  . Let  U   denote a  T  -module and let  U  ′ denote a  T  -module contained in  U  . Then the orthogonal complement of   U  ′ in  U   is a T  -module. From this we find  U   is an orthogonal direct sum of irreducible  T  -modules. Taking  U   =  V   wefind  V   is an orthogonal direct sum of irreducible  T  -modules. Let  W   denote an irreducible  T  -module. Bythe  multiplicity  with which  W   appears in  V   , we mean the number of irreducible  T  -modules in this sumwhich are isomorphic to  W  .For the rest of this article, we will refer to the following definition. 2.1 Definition.  Let Γ = ( X,R ) denote a bipartite distance-regular graph with valency  k  ≥  3 and diameter D  ≥  3. Fix any  x  ∈  X   and let  T   denote the Terwilliger algebra of Γ with respect to  x . Let  E  0 ,...,E  D  denotean ordering of the primitive idempotents of Γ. We assume Γ is  Q -polynomial with respect to  E  0 ,...,E  D .For 0  ≤  i  ≤  D  let  θ i  denote the eigenvalue associated with  E  i . Let  θ ∗ 0 , θ ∗ 1 ,..., θ ∗ D  denote the dual eigenvaluesequence associated with  E  1 .2  3 The Irreducible Modules for the Terwilliger Algebra With reference to Definition 2.1, let  W   denote an irreducible  T  -module. By the  endpoint  of   W   we mean min { i | 0  ≤  i  ≤  D, E  ∗ i  W    = 0 } . By the  dual endpoint  of   W   we mean  min { i | 0  ≤  i  ≤  D, E  i W    = 0 } . By the diameter  of   W   we mean  |{ i | 0  ≤  i  ≤  D, E  ∗ i  W    = 0 }|− 1.Now let  W   denote an irreducible  T  -module with endpoint  r , dual endpoint  t , and diameter  d . In [7], weobtained the following results. The sequence E  ∗ r v, E  ∗ r +1 v, ...,E  ∗ r + d v  (1)is a basis for  W  , where  v  is any nonzero vector in  E  t W   [7, Theorem 9.3]. With respect to this basis, thematrix representing  A  is of the form  0  b 0 ( W  )  0 c 1 ( W  ) 0  b 1 ( W  ) c 2 ( W  )  · ·· · ··  0  b d − 1 ( W  ) 0  c d ( W  ) 0  ,  (2) where c i ( W  ) =  θ t ( θ ∗ r + i +1 − θ ∗ r +1 ) − θ t +1 ( θ ∗ r + i − θ ∗ r ) θ ∗ r + i +1 − θ ∗ r + i − 1 (1 ≤ i ≤ d − 1) ,  (3) b i ( W  ) =  θ t ( θ ∗ r +1 − θ ∗ r + i − 1 ) +  θ t +1 ( θ ∗ r + i − θ ∗ r ) θ ∗ r + i +1 − θ ∗ r + i − 1 (1 ≤ i ≤ d − 1) ,  (4) and  b 0 ( W  ) =  c d ( W  ) =  θ t  [7, Theorem 10.3]. We remark the  c i ( W  ),  b i ( W  ) are rational if   D  ≥  5 by [8,Lemma 3.3] and [5, Lemma 3.3]. By [7, Corollary 12.2], b i − 1 ( W  ) c i ( W  )  >  0 (1  ≤  i  ≤  d ) . We comment on  r ,  t , and  d . We have 2 t  +  d  =  D,  so  D  − d  is even [7, Theorem 9.4]. Furthermore, wehave ( D − d ) / 2  ≤  r  ≤  D − d  [7, p.86]. The isomorphism class of   W   as a  T  -module is determined by  r  and  d [7, Theorem 13.1].We use the following notation. 3.1 Definition.  With reference to Definition 2.1, for 0  ≤  r,d  ≤  D  let  mult ( r,d ) denote the multiplicity withwhich  W   appears in  V   , where  W   denotes an irreducible  T  -module of endpoint  r  and diameter  d . If no such W   exists we interpret  mult ( r,d ) = 0.We now give formulae for the eigenvalues and dual eigenvalues. Let ˜ H  (2 D, 2) denote the quotient of  H  (2 D, 2). 3.2 Lemma.  With reference to Definition 2.1, assume   D  ≥  4  and   Γ  is not one of   H  ( D, 2) ,  ˜ H  (2 D, 2) . Thenthere exist scalars   q,s ∗ ∈ R  such that (i),(ii) hold below.(i) | q  |  >  1 , s ∗ q  i  = 1 (2  ≤  i  ≤  2 D  + 1) .  (5) (ii) θ i  =  h ( q  D − i − q  i ) ,  (6) θ ∗ i  =  θ ∗ 0  +  h ∗ (1 − q  i )(1 − s ∗ q  i +1 ) q  − i (7) for   0  ≤  i  ≤  D , where  h  = 1 − s ∗ q  3 ( q   − 1)(1 − s ∗ q  D +2 ) , θ ∗ 0  =  h ∗ ( q  D − 1)(1 − s ∗ q  2 ) q  ( q  D − 1 + 1)  , h ∗ = ( q  D +  q  2 )( q  D +  q  ) q  ( q  2 − 1)(1 − s ∗ q  2 D ) .  (8)3  We abbreviate   β   =  q   +  q  − 1 . Proof.  By [7, Corollary 15.2] there exist  q,s ∗ ∈  C  such that  q    = 0,  q  i  = 1 (1  ≤  i  ≤  D ),  s ∗ q  i  = 1(2  ≤  i  ≤  2 D ) and such that (6)-(8) hold. Moreover  s ∗ q  2 D +1  = 1 by [6, Theorem 1.1]. Observe  q  2  = 1 so q    = 1,  q    =  − 1. By [8, Lemma 1.1, Theorem 1.2] we have  q,s ∗ ∈  R . Suppose  s ∗ = 0. Then  | q  |  >  1 since q   is an integer [2, Proposition 6.2.1]. Suppose  s ∗  = 0. Replacing ( q,s ∗ ) by ( q  − 1 ,s ∗− 1 ) if necessary we mayassume  | q  | ≥  1. We mentioned  q   ∈ R  and  q    = 1,  q    =  − 1 so  | q  |  >  1. 3.3 Lemma.  With the assumptions of Lemma 3.2, c i  =  h ( q  i − 1)(1 − s ∗ q  D + i +1 )1 − s ∗ q  2 i +1  (1  ≤  i  ≤  D − 1) ,b i  =  h ( q  D − q  i )(1 − s ∗ q  i +1 )1 − s ∗ q  2 i +1  (1  ≤  i  ≤  D − 1) , and   c D  =  k  =  h ( q  D − 1) , where   h ,  q  , and   s ∗ are from Lemma 3.2. Proof.  Set  r  =  t  = 0 and  d  =  D  in (3), (4) and use (6), (7).The formulas in Lemmas 3.2 and 3.3 represent a special case of Leonard’s theorem [15]. 3.4 Theorem.  [7, Theorem 15.6] With reference to Definition 2.1 and Lemma 3.2, the following (i)-(vii)hold. (i)  mult (0 ,D ) = 1 . (ii)  mult (1 ,D − 2) = ( q  D − q  )(1 − s ∗ q  2 )( q  − 1)(1 − s ∗ q  D +2 ) . (iii)  mult (2 ,D − 2) = ( q  D − 1)( q  D − q  )(1 − s ∗ q  4 )(1 +  s ∗ q  D +1 ) q  ( q  2 − 1)(1 − s ∗ q  D +2 )(1 − s ∗ q  2 D )  . (iv)  mult (2 ,D − 4) = ( q  D − 1)( q  D − q  3 )(1 − s ∗ q  2 )(1 − s ∗ q  4 )(1 − s ∗ q  2 D +1 ) q  ( q  − 1)( q  2 − 1)(1 − s ∗ q  D +2 )(1 − s ∗ q  D +3 )(1 − s ∗ q  2 D ) . (v)  mult (3 ,D − 4) = ( q  D − 1)( q  D − q  )( q  D − q  3 )(1 − s ∗ q  2 )(1 − s ∗ q  6 )(1 +  s ∗ q  D +1 ) q  3 ( q  − 1)( q  2 − 1)(1 − s ∗ q  D +2 )(1 − s ∗ q  D +4 )(1 − s ∗ q  2 D − 2 )  . (vi) Suppose  D ≥ 6. Then mult (3 ,D − 6) = ( q  D − 1)( q  D − q  )( q  D − q  5 )(1 − s ∗ q  2 )(1 − s ∗ q  4 )(1 − s ∗ q  6 )(1 − s ∗ q  2 D +1 ) q  3 ( q  − 1)( q  2 − 1)( q  3 − 1)(1 − s ∗ q  D +2 )(1 − s ∗ q  D +3 )(1 − s ∗ q  D +4 )(1 − s ∗ q  2 D − 2 ) . (vii)  mult (4 ,D − 4) = ( q  D − 1)( q  D − q  )( q  D − q  2 )( q  D − q  3 )(1 − s ∗ q  2 )(1 − s ∗ q  8 )(1 +  s ∗ q  D +1 )(1 +  s ∗ q  D +2 ) q  4 ( q  2 − 1)( q  4 − 1)(1 − s ∗ q  D +3 )(1 − s ∗ q  D +4 )(1 − s ∗ q  2 D )(1 − s ∗ q  2 D − 2 )  . 4 The parameter  q >  1  if   D  ≥  6 4.1 Theorem.  With the assumptions of Lemma 3.2, further assume   D  ≥  6 . Then  q >  1 . Proof.  Recall  | q  |  >  1 by (5) so  q <  − 1 or  q >  1. We suppose  q <  − 1 and obtain a contradiction. We firstclaim that  θ ∗ 0   =  θ 0 . Recall  D  ≥  6 so if   θ ∗ 0  =  θ 0  then Γ is  H  ( D, 2) by [5, Lemma 10.1]. We are assumingΓ   =  H  ( D, 2) so  θ ∗ 0   =  θ 0 .Next we claim that  D  is odd and that θ ∗ 0  > θ ∗ D  > θ ∗ 2  > θ ∗ D − 2  >  ···  > θ ∗ D − 3  > θ ∗ 3  > θ ∗ D − 1  > θ ∗ 1 .  (9)By [5, Theorem 1.1] and since  θ ∗ 0   =  θ 0 , this claim will follow provided  θ D − 1  > θ 1 . We verify  θ D − 1  > θ 1 .Observe  β <  − 2 since  q <  − 1. By (6) we find  θ 0  +  θ 2  =  βθ 1 . Recall  θ 0  =  k  and  θ 2  >  − k  so  θ 0  +  θ 2  >  0.Therefore  θ 1  <  0. By (6)  θ D − 1  =  − θ 1 , so  θ D − 1  > θ 1 . Our claim follows.4
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