aa r X i v : . [ m a t h . C O ] J u l Two distance-regular graphs
Andries E. Brouwer & Dmitrii V. Pasechnik2011/06/11
Abstract
We construct two families of distance-regular graphs, namely the sub-graph of the dual polar graph of type B ( q ) induced on the vertices farfrom a fixed point, and the subgraph of the dual polar graph of type D ( q )induced on the vertices far from a fixed edge. The latter is the extendedbipartite double of the former. We shall use ∼ to indicate adjacency in a graph. For notation and definitions ofconcepts related to distance-regular graphs, see [BCN]. We repeat the definitionof extended bipartite double.The bipartite double of a graph Γ with vertex set X is the graph with vertexset { x + , x − | x ∈ X } and adjacencies x δ ∼ y ǫ iff δǫ = − x ∼ y . Thebipartite double of a graph Γ is bipartite, and it is connected iff Γ is connectedand not bipartite. If Γ has spectrum Φ, then its bipartite double has spectrum( − Φ) ∪ Φ. See also [BCN], Theorem 1.11.1.The extended bipartite double of a graph Γ with vertex set X is the graphwith vertex set { x + , x − | x ∈ X } , and the same adjacencies as the bipartitedouble, except that also x − ∼ x + for all x ∈ X . The extended bipartite doubleof a graph Γ is bipartite, and it is connected iff Γ is connected. If Γ has spectrumΦ, then its extended bipartite double has spectrum ( − Φ − ∪ (Φ + 1). See also[BCN], Theorem 1.11.2. D ( q ) Let V be a vector space of dimension 8 over a field F , provided with a nonde-generate quadratic form of maximal Witt index. The maximal totally isotropicsubspaces of V (of dimension 4) fall into two families F and F , where thedimension of the intersection of two elements of the same family is even (4 or 2or 0) and the dimension of the intersection of two elements of different familiesis odd (3 or 1).The geometry of the totally isotropic subspaces of V , where A ∈ F and B ∈ F are incident when dim A ∩ B = 3 and otherwise incidence is symmetrizedinclusion, is known as the geometry D ( F ). The bipartite incidence graph on1he maximal totally isotropic subspaces is known as the dual polar graph oftype D ( F ).Below we take F = F q , the finite field with q elements, so that graph andgeometry are finite. We shall use projective terminology, so that 1-spaces, 2-spaces and 3-spaces are called points, lines and planes. Two subspaces arecalled disjoint when they have no point in common, i.e., when the intersectionhas dimension 0. Proposition 2.1
Let Γ be the dual polar graph of type D ( F q ) . Fix elements A ∈ F and B ∈ F with A ∼ B . Let ∆ be the subgraph of Γ inducedon the set of vertices disjoint from A or B . Then ∆ is distance-regular withintersection array { q , q − , q − q, q − q + 1; 1 , q, q − , q } . The distance distribution diagram is ✒✑✓✏ q ✒✑✓✏ q - q − q ✓✒ ✏✑ q ( q − q − q q − ✓✒ ✏✑ q ( q − q − q +1 q ✓✒ ✏✑ ( q − q + 1)( q − Proof:
There are q elements A ∈ F disjoint from A and the same numberof B ∈ F disjoint from B , so that ∆ has 2 q vertices.Given A ∈ F , there are q + q + q + 1 elements B ∈ F incident to it. Ofthese, q + q + 1 contain the point A ∩ B and hence are not vertices of ∆. So,∆ has valency q .Two vertices A, A ′ ∈ F have distance 2 in ∆ if and only if they meet in aline, and the line L = A ∩ A ′ is disjoint from B . If this is the case, then L isin q + 1 elements B ∈ F , one of which meets B , so that A and A ′ have c = q common neighbours in ∆.Given vertices A ∈ F and B ∈ F that are nonadjacent, i.e., that meet ina single point P , the neighbours A ′ of B at distance 2 to A in ∆ correspond tothe lines L on P in A disjoint from B and nonorthogonal to the point A ∩ B .There are q + q + 1 lines L on P in A , q + 1 of which are orthogonal to thepoint A ∩ B , and one further of which meets B . (Note that the points A ∩ B and A ∩ B are nonorthogonal since neither point is in the plane A ∩ B and V does not contain totally isotropic 5-spaces.) It follows that c = q −
1, andalso that ∆ has diameter 4, and is distance-regular. ✷ The geometry induced by the incidence relation of D ( F ) on the verticesof ∆, together with the points and lines contained in the planes disjoint from A ∪ B , has Buekenhout-Tits diagram (cf. [P]) ssss ❅❅(cid:0)(cid:0) F F ptsAfAf that is, the residue of an object A ∈ F is an affine 3-space, where the objectsincident to A in F play the rˆole of points. Similar things hold more generally for D n ( F ) with arbitrary n , and even more generally for all diagrams of sphericaltype. See also [BB], Theorem 6.1.Let P be a nonsingular point, and let φ be the reflection in the hyperplane H = P ⊥ . Then φ is an element of order two of the orthogonal group that fixes H pointwise, and consequently interchanges F and F . For each A ∈ F we2ave φ ( A ) ∼ A . The quotient Γ /φ is the dual polar graph of type B ( q ), and wesee that more generally the dual polar graph of type D m +1 ( q ) is the extendedbipartite double of the dual polar graph of type B m ( q ). The quotient ∆ /φ isa new distance-regular graph discussed in the next section. It is the subgraphconsisting of the vertices at maximal distance from a given point in the dualpolar graph of type B ( q ). For even q we have B ( q ) = C ( q ), and it followsthat the symmetric bilinear forms graph on F q is distance-regular, see [BCN]Proposition 9.5.10 and the diagram there on p. 286. B ( q ) First a very explicit version of the graph of this section.
Proposition 3.1 (i) Let W be a vector space of dimension over the field F q ,provided with an outer product × . Let Z be the graph with vertex set W × W where ( u, u ′ ) ∼ ( v, v ′ ) if and only if ( u, u ′ ) = ( v, v ′ ) and u × v + u ′ − v ′ = 0 .Then Z is distance-regular of diameter on q vertices. It has intersection array { q − , q − q, q − q + 1; 1 , q, q − } and eigenvalues q − , q − , − , − q − with multiplicities , q ( q + 1)( q − , ( q − q + 1)( q − , q ( q − q − ,respectively.(ii) The extended bipartite double ˆ Z of Z is distance-regular with intersectionarray { q , q − , q − q, q − q + 1; 1 , q, q − , q } and eigenvalues ± q , ± q , with multiplicities , q ( q − , q − q + 1)( q − , respectively.(iii) The distance-1-or-2 graph Z ∪ Z of Z , which is the halved graph of ˆ Z , is strongly regular with parameters ( v, k, λ, µ ) = ( q , q ( q − , q ( q + q − , q ( q − . The distance distribution diagram of Z is ✒✑✓✏ q − ✓✒ ✏✑ q − q − q − q q ✓✒ ✏✑ ( q − q − q − q − q − q +1 q − ✓✒ ✏✑ ( q − q +1)( q − q − q Proof:
Note that the adjacency relation is symmetric, so that Z is an undi-rected graph. The computation of the parameters is completely straightforward.Clearly, Z has q vertices. For a, b ∈ W the maps ( u, u ′ ) ( u + a, u ′ +( a × u )+ b )are automorphisms of Z , so Aut( Z ) is vertex-transitive.The q − ,
0) are the vertices ( v,
0) with v = 0. The commonneighbours of (0 ,
0) and ( v,
0) are the vertices ( cv,
0) for c ∈ F q , c = 0 ,
1. Hence a = q − q − q −
1) vertices at distance 2 from (0 ,
0) are the vertices ( u, u ′ )with u, u ′ = 0 and u ′ ⊥ u . The common neighbours of (0 ,
0) and ( u, u ′ ) arethe ( v,
0) with v × u = u ′ , and together with ( v,
0) also ( v + cu,
0) is a commonneighbour, so c = q . Vertices ( u, u ′ ) and ( v, v ′ ), both at distance 2 from (0 , = v ⊥ u ′ and v = u and v × u = u ′ and v ′ = u × v + u ′ , sothat a = q − q − q − q − q + 1) vertices have distance 3 to (0 , w, w ′ ) with w w ′ or w = 0 = w ′ . The neighbours ( u, u ′ ) of ( w, w ′ )that lie at distance 2 to (0 ,
0) satisfy 0 = u ⊥ w ′ and (0 =) u ′ = w × u + w ′ ,3o that c = q −
1. This shows that Z is distance-regular with the claimedparameters. The spectrum follows.The fact that the extended bipartite double is distance-regular, and has thestated intersection array, follows from [BCN], Theorem 1.11.2(vi).The fact that Z is strongly regular follows from [BCN], Proposition 4.2.17(ii)(which says that this happens when Z has eigenvalue − ✷ For q = 2, the graphs here are (i) the folded 7-cube, (ii) the folded 8-cube,(iii) the halved folded 8-cube. All are distance-transitive. For q > q is a power of two, the graphs ˆ Z have the same parameters as certainKasami graphs, but for q > H be a vector space of dimension 7 over the field F q , provided with anondegenerate quadratic form. Let Γ be the graph of which the vertices are themaximal totally isotropic subspaces of H (of dimension 3), where two verticesare adjacent when their intersection has dimension 2. This graph is known asthe dual polar graph of type B ( q ). It is distance-regular with intersection array { q ( q + q + 1) , q ( q + 1) , q ; 1 , q + 1 , q + q + 1 } . (See [BCN], § Proposition 3.2
Let Γ be the dual polar graph of type B ( q ) . Fix a vertex π of Γ , and let ∆ be the subgraph of Γ induced on the collection of vertices disjointfrom π . Then ∆ is isomorphic to the graph Z of Proposition 3.1. Its extendedbipartite double ˆ∆ (or ˆ Z ) is isomorphic to the graph of Proposition 2.1. Proof:
Let V be a vector space of dimension 8 over F q (with basis { e , . . . , e } ),provided with the nondegenerate quadratic form Q ( x ) = x x + x x + x x + x x . The point P = (0 , , , , , , , −
1) is nonisotropic, and P ⊥ is the hy-perplane H defined by x = x . Restricted to H the quadratic form becomes Q ( x ) = x x + x x + x x + x .The D -geometry on V has disjoint maximal totally isotropic subspaces E = h e , e , e , e i and F = h e , e , e , e i . Fix E and consider the collection ofall maximal totally isotropic subspaces disjoint from E . This is precisely thecollection of images F A of F under matrices (cid:18) I A I (cid:19) , where A is alternatingwith zero diagonal (cf. [BCN], Proposition 9.5.1(i)). Hence, we can label the q vertices F A ∩ H of ∆ with the q matrices A .Two vertices are adjacent when they have a line in common, that is, whenthey are the intersections with H of maximal totally isotropic subspaces in V ,disjoint from E , that meet in a line contained in H . Let A = a b c − a d e − b − d f − c − e − f . Then det A = ( af − be + cd ) , and if det A = 0 but A = 0, then ker A hasdimension 2, and is spanned by the four vectors (0 , f, − e, d ) ⊤ , ( − f, , c, − b ) ⊤ ,( e, − c, , a ) ⊤ , ( − d, b, − a, ⊤ . Writing the condition that matrices A and A ′ belong to adjacent vertices we find the description of Proposition 3.1 if we take u = ( c, e, f ) and u ′ = ( − d, b, − a ). ✷ History
In 1991 the second author constructed the graphs from Section 2 and the firstauthor those from Section 3. Both were mentioned on the web page [ac], butnot published thus far. These graphs have been called the Pasechnik graphsand the Brouwer-Pasechnik graphs, respectively, by on-line servers.
References [BB] R. J. Blok & A. E. Brouwer,
The geometry far from a residue , pp. 29–38in: Groups and Geometries, L. di Martino, W. M. Kantor, G. Lunardon,A. Pasini, M. C. Tamburini (eds.), Birkha¨user Verlag, Basel, 1998.[ac] A. E. Brouwer,
Additions and corrections to [BCN], [BCN] A. E. Brouwer, A. M. Cohen & A. Neumaier,
Distance-regular graphs ,Springer, Heidelberg, 1989.[P] A. Pasini,
Diagram geometries , Oxford University Press, 1994.Addresses of authors:Andries E. BrouwerDept. of Math.Techn. Univ. EindhovenP. O. Box 5135600MB EindhovenNetherlands [email protected]
Dmitrii V. PasechnikSchool of Physical and Mathematical SciencesNanyang Technological University21 Nanyang LinkSingapore 637371 [email protected]@ntu.edu.sg