Dam, E.R. van

Koolen, J.H.

English

Research Papers in Economics

DOI: 10.1007/s00222-005-0442-3
Invent. math. 162, 189–193 (2005)
A new family of distance-regular graphs with
unbounded diameter
E.R. van Dam1, ,J.H. Koolen2,  
1 Tilburg University, Dept. Econometrics and O.R., PO Box 90153, 5000 LE Tilburg,
The Netherlands (e-mail: Edwin.vanDam@uvt.nl)
2 POSTECH, Dept. Mathematics, Pohang 790-784, South Korea
(e-mail: koolen@postech.ac.kr)
Oblatum 18-XI-2004 & 2-II-2005
Published online: 7 April 2005 – © Springer-Verlag 2005
Abstract. We construct distance-regular graphs with the same – classical
– parameters as the Grassmann graphs on the e-dimensional subspaces of
a (2e + 1)-dimensional space over an arbitrary ﬁnite ﬁeld. This provides
the ﬁrst known family of non-vertex-transitive distance-regular graphs with
unbounded diameter.
1. Introduction
A simple connected graph is called distance-regular with intersection array
{b0,b1,...,bd−1; c1,c2,...,cd} if it has diameter d,a n df o ra n yt w o
vertices x, y at distance i the number of neighbours of y at distance i + 1
from x equals bi, and the number of neighbours of y at distance i − 1 from
x equals ci,f o ra l li = 0,...,d (where for convenience c0 = bd = 0).
Distance-regular graphs play an important role in algebraic combinatorics
because of the relation to design theory, coding theory, ﬁnite and Euclidean
geometry, and group theory (cf. [2]).
Let q be a prime power, and let n and e be integers such that n
2 ≥ e ≥ 2.
The Grassmann graph Jq(n,e) is the graph on the e-dimensional subspaces
of an n-dimensional vector space over the ﬁnite ﬁeld GF(q),w h e r et w o
  The research of E.R. van Dam has been made possible by a fellowship of the Royal
Netherlands Academy of Arts and Sciences
   This paper was written while J.H. Koolen was working at KAIST, Division of Ap-
plied Mathematics, Daejon, South Korea. J.H. Koolen is partially supported by Com2MaC-
KOSEF
Mathematics Subject Classiﬁcation (2000): 05E30190 E.R. van Dam, J.H. Koolen
e-dimensional subspaces are adjacent whenever they intersect in an (e−1)-
dimensional subspace. The Grassmann graphs are distance-regular graphs
with diameter e and intersection array {b0,b1,...,be−1;c1,c2,...,ce}
with
bi = q2i+1

e − i
1

n − e − i
1

and ci =

i
1
2
,
where
 n
m

=
(qn−1)···(qn−m+1−1)
(qm−1)···(q−1) is the number of m-dimensional subspaces
of an n-dimensional space over GF(q) (cf. [2, pp.268–269]).
Metsch [5] showed that the Grassmann graphs are uniquely determined
as distance-regular graphs with above parameters if e  = 2, n
2,o rn−1
2 (for
all q)a n d(e,q)  = (n−2
2 ,2),(n−2
2 ,3),o r(n−3
2 ,2).F o re = 2,n ≥ 4t h e
Grassmann graphs are in general not determined by their parameters, as the
line graph of a 2-
 n
1

,q + 1,1

design has the same parameters. Here we
shall present new distance-regular graphs with the parameters of the Grass-
mann graphs for n = 2e+1,e ≥ 2. This will provide the ﬁrst known family
of non-vertex-transitive distance-regular graphs with unbounded diameter.
Indeed, besides the families of distance-transitive graphs (among which
is the family of Grassmann graphs), the only known families of distance-
regular graphs with unbounded diameter are the quadratic forms graphs,
the Doob graphs (pseudo Hamming graphs), the Hemmeter graphs (pseudo
Dm(q) graphs), and the Ustimenko graphs (halved Hemmeter graphs)
(cf. [2, p. 195]), and these are all vertex-transitive. It is speciﬁcally the
non-vertex-transitivity which indicates that the solution of Bannai’s prob-
lem (cf. [1, p. 293], [2, p. viii]) of determining all Q-polynomial distance-
regular graphs with large diameter may be further away than expected.
Another property of the new graphs is that the corresponding Terwilliger
algebras (cf. [6] and the references therein) depend on the base points. No
other known distance-regular graph with large diameter has this property
(since these graphs are all vertex-transitive).
2. A partial linear space and its line graph
The origin of the discovery of the new distance-regular graphs lies in the
constructionofcospectralgraphsoftheDoubledGrassmanngraphs(cf.[3]).
For n = 2e + 1, the incidence structure of the e-dimensional subspaces
versus the (e+1)-dimensional subspaces, where incidence is symmetrized
containment, is a partial linear space which has as point graph Jq(2e+1,e),
as line graph Jq(2e + 1,e + 1) (which is isomorphic to Jq(2e + 1,e)
through orthogonal complementing), and as incidence graph the Doubled
Grassmanngraph. Byadjusting thispartiallinear space(intherightmanner)
in such a way that the point graph doesn’t change, we are able to obtain as
line graph a new distance-regular graph. The incidence graph of the partial
linear space is cospectral with the Doubled Grassmann graph, but it is not
distance-regular (cf. [3]).A new family of distance-regular graphs with unbounded diameter 191
The partial linear space is deﬁned as follows. Let V be a (2e + 1)-
dimensional vector space over GF(q), and ﬁx a hyperplane H in V.T h e
points are the e-dimensional subspaces of V, and there are two types of
lines. First, we have the (e + 1)-dimensional subspaces of V that are not
contained in H; and a line of this type is incident to its e-dimensional
subspaces. Second, we have the (e−1)-dimensional subspaces of H; a line
of this type is incident to the e-dimensional subspaces of H that contain it.
It is easy to check that this deﬁnes a partial linear space having as many

2e+1
e

points as lines, such that each line is incident to

e+1
1

points, and
the other way around. Moreover, two points are collinear if and only if they
intersect in an (e − 1)-dimensional subspace. Thus the point graph of the
partial linear space is the Grassmann graph Jq(2e +1,e).I fN is the point-
line incidence matrix of the partial linear space, then NNT −

e+1
1

I is the
adjacency matrix of the point graph. Since NT N and NNT have the same
spectrum, and NT N −

e+1
1

I is the adjacency matrix of the line graph, it
follows that the line graph has the same spectrum as the Grassmann graph.
This leads to the following.
Theorem. Let q be a prime power, and let e ≥ 2 be an integer. Let V be
a (2e + 1)-dimensional vector space over GF(q), and let H be a hyper-
plane in V. Let G be the graph whose vertices are the (e + 1)-dimensional
subspaces of V that are not contained in H and the (e − 1)-dimensional
subspaces of H; where two vertices of the ﬁrst kind are adjacent if they in-
tersect in an e-dimensional subspace; a vertex of the ﬁrst kind is adjacent to
a vertex of the second kind if the ﬁrst contains the second; and two vertices
of the second kind are adjacent if they intersect in an (e − 2)-dimensional
subspace. Then G is distance-regular, with the same parameters as the
Grassmann graph Jq(2e + 1,e). Moreover, G is not vertex-transitive, and
hence it is not isomorphic to the Grassmann graph.
Proof. It is clear that in G, two vertices of the ﬁrst kind are at distance i if
and only if they intersect in an (e+1−i)-dimensional subspace; a vertex of
the ﬁrstkind andavertex ofthe second kind areat distance i ifthey intersect
in an (e−i)-dimensional subspace; and two vertices of the second kind are
at distance i if they intersect in an (e − 1 − i)-dimensional subspace. By
using for example Lemma 9.3.2 in [2], it is straightforward now to check
that each vertex has ki = qi2
e+1
i
 e
i

vertices at distance i. One can also
check that the intersection parameters bi and ci are well-deﬁned and the
same as in the Grassmann graph, but this is tedious and elaborate.
Alternatively, we note that G is indeed the line graph of the above
described partial linear space, and thus has the same spectrum as Jq(2e +
1,e). We can then use a theorem by Fiol and Garriga [4, Thm. 4.4] which
implies that a graph which is cospectral with a distance-regular graph Γ
with diameter e is itself distance-regular if for every vertex the number of
vertices at distance e is the same as in Γ.S i n c eke in G is indeed the same192 E.R. van Dam, J.H. Koolen
as in the Grassmann graph, it thus follows that G is distance-regular. Since
the parameters of a distance-regular graph follow from its spectrum, G has
the same parameters as Jq(2e + 1,e).
To show that G is not vertex-transitive, we consider the maximal cliques
in G. It is straightforward to check that there are four types of such cliques,
and they can be described as follows. The ﬁrst type consists of all (e + 1)-
dimensional subspaces not contained in H containing a ﬁxed e-dimensional
subspace S,a n dt h e(e − 1)-dimensional subspaces contained in S ∩ H.I f
S ⊂ H, then the size of such a maximal clique is

e+1
1

; otherwise it is one
more. The second type consists of all (e + 1)-dimensional subspaces not
contained in H and contained in a ﬁxed (e+2)-dimensional subspace S not
contained in H. The size of such a maximal clique is

e+2
1

− 1. The third
type consists of all (e+1)-dimensional subspaces not contained in H which
are contained in a ﬁxed (e+2)-dimensional subspace S not contained in H,
which moreover are containing a ﬁxed (e − 1)-dimensional subspace S  of
S∩ H,p l u sS  itself. The size of this type of maximal clique is

3
1

.T h el a s t
type of clique consists of all (e−1)-dimensional subspaces of H containing
aﬁ x e d(e − 2)-dimensional subspace S of H. These largest cliques have
size

e+2
1

. Since they consist of (e − 1)-dimensional subspaces only, this
shows that G is not vertex-transitive, and hence that G is not isomorphic to
the Grassmann graph.    
It is clear that PΓL(2e + 1,q)2e, the subgroup of PΓL(2e + 1,q) that
ﬁxes H, is a group of automorphisms of the new graph. We expect that this
is the full group of automorphisms. In any case, the graph has two orbits
under the action of its automorphism group on the vertices, and four orbits
of edges.
3. Further remarks
WeclaimthattheTerwilligeralgebra(cf.[6]andthereferences therein)with
respect to an (e−1)-dimensional subspace is different from the Terwilliger
algebra with respect to an (e + 1)-dimensional subspace. We were able to
show that this is true asymptotically; besides this we checked it for the case
q = 2,e = 3. By counting the number of cliques of size four containing
aﬁxedvertex one can show that the number oftriangles in the local graph of
an (e − 1)-dimensional subspace is (in general) different from the number
of triangles in the local graph of an (e + 1)-dimensional subspace. This
implies that the spectra of these two local graphs are different, and hence
that the Terwilliger algebras with respect to these two vertices are different.
The Grassmann graphs Jq(n,e) can be seen as q-analogues of the John-
son graphs J(n,e). The latter are known to be determined as distance-
regular graphs by the parameters, except for the case n = 8, e = 2
(cf. [2, p. 258]). For the Johnson graph J(2e+1,e) one can deﬁne a partial
linear space, and adjust it, in an analogous way as we did for the Grass-A new family of distance-regular graphs with unbounded diameter 193
manngraph Jq(2e+1,e). However,theobtained distance-regular linegraph
is isomorphic to J(2e + 1,e). Indeed, the argument that the graph is not
vertex-transitive by considering the maximal cliques fails in this case.
Acknowledgements. The authors thank Willem Haemers for several fruitful conversations.
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A.: Distance-Regular Graphs.

Current research on algebraic combinatorics – Supplements to our book, Algebraic Combinatorics I.

Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra.

E.: Characterizing distanceregularity of graphs by the spectrum.

From local adjacency polynomials to locally pseudo-distanceregular graphs.

K.:Acharacterization of Grassmanngraphs.

A new family of distance-regular graphs with unbounded diameter.

A new family of distance-regular graphs with unbounded diameter.

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