A Characterization of the Hamming Graphs and the Dual Polar Graphs by Completely Regular Subgraphs

Graphs and Combinatorics (2012) 28:449–467DOI 10.1007/s00373-011-1064-8

ORIGINAL PAPER

A Characterization of the Hamming Graphsand the Dual Polar Graphs by CompletelyRegular Subgraphs

Akira Hiraki

Received: 29 September 2008 / Revised: 31 May 2011 / Published online: 30 June 2011© Springer 2011

Abstract In this paper we study a distance-regular graph Γ of diameter d ≥ 3which satisfies the following two conditions: (a) For any integer i with 1 ≤ i ≤ d − 1and for any pair of vertices at distance i in Γ there exists a strongly closed subgraphof diameter i containing them; (b) There exists a strongly closed subgraph Δ whichis completely regular in Γ . It is known that if Δ has diameter 1, then Γ is a regu-lar near polygon. We prove that if a strongly closed subgraph Δ of diameter j with2 ≤ j ≤ d − 1 is completely regular of covering radius d − j in Γ, then Γ is either aHamming graph or a dual polar graph.

Keywords Distance-regular graph · Completely regular · Hamming graph ·Dual polar graph

1 Introduction

Our notation and terminologies are standard. The reader is referred to the next sectionor [1,2,5] for the definitions.

It is well known that for any pair of vertices in a Hamming graph (resp. a dualpolar graph) there exists a strongly closed subgraph containing them whose diameteris equal to the distance between them. Moreover, any strongly closed subgraph is aHamming graph (resp. a dual polar graph) and it is completely regular in the originalgraph (see Sect. 5). In this paper, we consider a distance-regular graph Γ of diameterd = d(Γ ) ≥ 3 which satisfies the following hypothesis:

A. Hiraki (B)Division of Mathematical Sciences, Osaka Kyoiku University,Osaka 582-8582, Japane-mail: [email protected]

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450 Graphs and Combinatorics (2012) 28:449–467

Hypothesis 1 For any integer i with 1 ≤ i ≤ d − 1 and for any pair of vertices atdistance i in Γ there exists a strongly closed subgraph of diameter i containing them.

Moreover we study the case that there exists a strongly closed subgraph Δ whichis completely regular in Γ . A strongly closed subgraphΔ in a Hamming graph (resp.a dual polar graph) Γ , is an example of such a pair (Γ,Δ). Another example is a sin-gular line Δ of a regular near 2d-gon Γ with quadrangles. More examples are givenin [10]. Our purpose is to classify such pairs (Γ,Δ).We first consider the case thatΔhas diameter 1. We remark that a strongly closed subgraph of diameter 1 is a cliqueof size a1 + 2. It is known that a clique of size a1 + 2 is completely regular in Γif and only if Γ is a regular near polygon. We reprove this fact in Sect. 4. Next weconsider the cases thatΔ has diameter j with 2 ≤ j ≤ d −1. The case j = 2 had beenstudied by Suzuki in [12]. He studied a parallelogram-free distance-regular graph Γof diameter d ≥ 4 such that b1 > b2 and a2 �= 0. Then Γ satisfies Hypothesis 1.Under this assumption he proved that if there exists a strongly closed subgraph ofdiameter 2 which is completely regular of covering radius d − 2 in Γ , then Γ is eithera Hamming graph or a dual polar graph. In this paper, we study arbitrary diameter jwith 2 ≤ j ≤ d − 1.

The intersection numbers of the Hamming graphs and the dual polar graphs satisfy

ci =[

i1

], ai =

[i1

]a1 (1)

and

bi =([

d1

]−

[i1

])(1 + a1) (2)

for i = 1, . . . , d, where

[i1

]=

[i1

]q

:= qi−1 + · · · + q + 1 (3)

denotes the Gaussian binomial coefficient with basis q := c2 − 1. This almost char-acterizes these graphs (see [2, Theorems 9.2.5, 9.4.4, 9.4.5] and Theorem 20).

Let Γ be a distance-regular graph of diameter d ≥ 3 which satisfies Hypothesis 1.In [9] we showed several inequalities for the intersection numbers of Γ . Actually thesame inequalities in (17), (18) had been obtained. In some specific cases the equalitiesimply that Γ is either a Hamming graph or a dual polar graph by using a characteriza-tion of these graphs. In this paper, we will do similar argument and prove the followingresult.

Theorem 1 Let Γ be a distance-regular graph of diameter d ≥ 4 satisfying Hypoth-esis 1. If there exists a strongly closed subgraph Δ of diameter j with 2 ≤ j ≤ d − 2which is completely regular in Γ , then c2 > 1 and

ci =[

i1

], ai =

[i1

]a1 (4)

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Graphs and Combinatorics (2012) 28:449–467 451

hold for i = 1, . . . , d − 1, where

[i1

]=

[i1

]q

:= qi−1 + · · · + q + 1

denotes the Gaussian binomial coefficient with basis q := c2 − 1. Moreover thefollowing hold.

(i) Suppose a1 = 0 and c2 = 2. Then any strongly closed subgraph� of diameter twith 2 ≤ t ≤ d − 1 is the t-cube.

(ii) Suppose d ≥ 4 and a1 > 0. Then any strongly closed subgraph � of diameter twith 3 ≤ t ≤ d − 1 is either a Hamming graph or a dual polar graph.

(iii) Suppose d ≥ 5, a1 = 0 and c2 > 2. Then any strongly closed subgraph � ofdiameter t with 4 ≤ t ≤ d − 1 is the dual polar graph on [Dt (q)].

This theorem almost characterizes a strongly closed subgraph� of diameter j with2 ≤ j ≤ d − 2 which is completely regular in Γ . Next we should characterize Γitself. Suppose Γ is either a Hamming graph or a dual polar graph. Then Γ satisfiesHypothesis 1 and each strongly closed subgraph Δ of diameter j in Γ has coveringradius d − j . This condition characterizes these graphs.

Theorem 2 Let Γ be a distance-regular graph of diameter d ≥ 3. We assume d ≥ 4if a1 = 0 and c2 > 2. Then the following conditions are equivalent.

(i) Γ is either a Hamming graph or a dual polar graph.(ii) Γ satisfies Hypothesis 1. Moreover there exists a strongly closed subgraph of

diameter j with 2 ≤ j ≤ d − 1 which is completely regular of covering radiusd − j in Γ .

This paper is organized as follows. In Sect. 2, we recall some definitions and basicterminologies for distance-regular graphs, strongly closed subgraphs and completelyregular subgraphs. We give several basic results. In Sect. 3 we study a subgraph Δwhich satisfies some specific condition. We investigate the relation between such sub-graphs and completely regular subgraphs. Also we show that some inequalities forthe intersection numbers. In Sect. 4, we study a regular near polygons to settle thecase that a strongly closed subgraph of diameter 1 is completely regular. In Sect. 5,we first reprove the fact that any strongly closed subgraph in a Hamming graph (resp.a dual polar graph) is completely regular and compute its parameters for a completelyregular subgraph. Then we prove Theorem 1 and Theorem 2.

2 Preliminaries

First we recall our notation and terminologies. Let Γ = (VΓ, EΓ ) be a connectedgraph with usual distance ∂Γ and diameter d = d(Γ ). For a vertex u inΓ we denote byΓ j (u) the set of vertices which are at distance j from u, whereΓ−1(u) = Γd+1(u) = ∅.For two vertices x and y in Γ with ∂Γ (x, y) = i , let

C(x, y) := Γi−1(x) ∩ Γ1(y), A(x, y) := Γi (x) ∩ Γ1(y),

B(x, y) := Γi+1(x) ∩ Γ1(y).

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Definition 3 Let i be an integer with 0 ≤ i ≤ d.

(i) We say ci (Γ )-exists if ci (Γ ) = |C(x, y)| is a constant whenever ∂Γ (x, y) = i .(ii) We say ai (Γ )-exists if ai (Γ ) = |A(x, y)| is a constant whenever ∂Γ (x, y) = i .

(iii) We say bi (Γ )-exists if bi (Γ ) = |B(x, y)| is a constant whenever ∂Γ (x, y) = i .

A connected graph Γ of diameter d is said to be distance-regular if ci (Γ )-exists andbi (Γ )-exists for all i = 0, 1, . . . , d. Then Γ is a regular graph of valency k = k(Γ ) =b0(Γ ) and ai (Γ )-exists with ai (Γ ) = k(Γ )− ci (Γ )− bi (Γ ) for all i = 0, 1, . . . , d.We remark that c0(Γ ) = a0(Γ ) = bd(Γ ) = 0 and c1(Γ ) = 1. The constantsci (Γ ), ai (Γ ) and bi (Γ ) (i = 0, 1, . . . , d) are called the intersection numbers of Γ .

The rest of this paper let Γ be a distance-regular graph of diameter d = d(Γ )≥ 2and valency k = k(Γ ) ≥ 3. We write ci , ai and bi the intersection numbers ci (Γ ),

ai (Γ ) and bi (Γ ) of Γ .Next we recall the definitions of a completely regular subgraph and a strongly closed

subgraph. We refer the reader to [2, Chapter 11], [4], [5, Section 11.7] and [6,10,11] formore information about completely regular subgraphs and strongly closed subgraphs.

Definition 4 Let Y be a non-empty subset of vertices in Γ . We identify Y with theinduced subgraph on it. Let z ∈ VΓ . Define

∂Γ (z,Y ) = ∂Γ (Y, z) = min{∂Γ (z, v) | v ∈ Y }. (5)

Let

w(Y ) = wΓ (Y ) = max{∂Γ (y, y′) | y, y′ ∈ Y } (6)

and

t (Y ) = tΓ (Y ) = max{∂Γ (Y, x) | x ∈ VΓ } (7)

which are called the width and the covering radius of Y in Γ , respectively. Set

Γi (Y ) = {x ∈ VΓ | ∂Γ (Y, x) = i} (8)

for i = 0, 1, . . . , t (Y ). Set Γi (Y ) = ∅ if i < 0 or t (Y ) < i .

(i) A subgraph Y is called completely regular if for any integer i with 0 ≤ i ≤ t (Y )

γi = |Γi−1(Y ) ∩ Γ1(x)|, αi = |Γi (Y ) ∩ Γ1(x)|, βi = |Γi+1(Y ) ∩ Γ1(x)| (9)

are constants whenever x ∈ Γi (Y ).(ii) A subgraph Y is called strongly closed if C(u, v)∪ A(u, v) ⊆ Y for any u, v ∈ Y .

(iii) Let m be an integer with 1 ≤ m ≤ d − 1. We say the condition (SC)m holds iffor any pair of vertices at distance m there exists a strongly closed subgraph ofdiameter m containing them.

The following are well known basic facts.

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Lemma 5 Let Γ be a distance-regular graph of diameter d ≥ 2 and let Δ be asubgraph in Γ . Then the following hold.

(i) w(Δ)+ t (Δ) ≥ d.(ii) Let Ξ be a subgraph in Δ with the covering radius tΔ(Ξ) = max{∂Δ(z, Ξ) |

z ∈ Δ}. Then t (Ξ) ≤ t (Δ)+ tΔ(Ξ).(iii) A subgraph Δ is completely regular if and only if

ψi, j = |Δ ∩ Γ j (x)| (10)

is a constant whenever x ∈ Γi (Δ) for all i, j with 0 ≤ i ≤ t (Δ) and 0 ≤ j ≤ d.

Proof (i) Let z ∈ Δ and x ∈ Γd(z). Let y ∈ Δ such that ∂Γ (x, y) = ∂Γ (x,Δ).Then

d = ∂Γ (x, z) ≤ ∂Γ (x, y) + ∂Γ (y, z) ≤ t (Δ) + w(Δ).

(ii) Let x be a vertex of Γ such that ∂Γ (x, Ξ) = t (Ξ) and let z be a vertex of Δsuch that ∂Γ (x, z) = ∂Γ (x,Δ). Then

t (Ξ) = ∂Γ (x, Ξ) ≤ ∂Γ (x, z) + ∂Γ (z, Ξ)

≤ ∂Γ (x, z) + ∂Δ(z, Ξ) ≤ t (Δ)+ tΔ(Ξ).

(iii) This is proved in [5, Theorem 11.7.1]. The lemma is proved. ��We remark that the width w(Δ) = max{∂Γ (x, y) | x, y ∈ Δ} may not coincide

with the diameter d(Δ) = max{∂Δ(x, y) | x, y ∈ Δ} of Δ in general.Let Δ be a strongly closed subgraph of Γ of diameter m = d(Δ). For any pair

(x, y) of vertices in Δ any shortest path between x and y in Γ is contained in Δ.Thus the distance ∂Δ inΔ coincides with the distance ∂Γ in Γ . In particular, the widthw(Δ) coincide with the diameter d(Δ). Since C(x, y) ∪ A(x, y) ⊆ Δ, ci (Δ)-existsand ai (Δ)-exists with ci (Δ) = ci and ai (Δ) = ai for all i with 1 ≤ i ≤ m. Moreover,ifΔ is a regular graph of valency k(Δ), then bi (Δ)-exists with bi (Δ) = k(Δ)−ci −ai

for all i with 1 ≤ i ≤ m and thusΔ is distance-regular with valency k(Δ) = cm +am .Hence we have the following fact.

Lemma 6 Let Γ be a distance-regular graph of diameter d ≥ 3. Let Δ be a stronglyclosed subgraph of diameter m := d(Δ) in Γ . Then the distance ∂Δ in Δ coincideswith the distance ∂Γ inΓ . In particular, ci (Δ)-exists and ai (Δ)-exists with ci (Δ) = ci

and ai (Δ) = ai for all i with 0 ≤ i ≤ m. Moreover if Δ is regular, then it is dis-tance-regular with ci (Δ) = ci , ai (Δ) = ai and bi (Δ) = bi − bm for all i with0 ≤ i ≤ m. ��

We remark that there exist several examples of non-regular strongly closed sub-graphs in a distance-regular graph (see [11, Theorems 1.4–1.5], [7, §2] and [8]).Suppose that a subgraph Δ is completely regular. Then

α0 = |Γ0(Δ) ∩ Γ1(v)| = |Δ ∩ Γ1(v)| = ψ0,1

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is a constant whenever v ∈ Γ0(Δ) = Δ. Hence any completely regular subgraph isa regular graph (it may be disconnected). It follows, by Lemma 6, that if a stronglyclosed subgraph Δ is completely regular, then it is distance-regular.

3 Completely Regular Subgraphs

Throughout this section Γ denotes a distance-regular graph of diameter d ≥ 2. Firstwe prove the following lemma.

Lemma 7 Let Γ be a distance-regular graph of diameter d ≥ 2 and let Δ be a sub-graph in Γ . Then Δ is strongly closed if and only if each vertex x in Γ1(Δ) has theunique neighbor x∗ in Δ and ∂Γ (v, x) = ∂Γ (v, x∗)+ 1 holds for any v ∈ Δ.

Proof Suppose that Δ is strongly closed. Since x ∈ Γ1(Δ), there exists a vertexx∗ ∈ Δ such that x∗ ∈ Γ1(x). Let v ∈ Δ\{x∗}. Since C(v, x∗) ∪ A(v, x∗) ⊆ Δ andx �∈ Δ, we have x ∈ B(v, x∗). Hence ∂Γ (v, x) = ∂Γ (v, x∗)+ 1 holds. In particular,x∗ is the unique neighbors of x in Δ. Conversely if Δ is not strongly closed, thenthere exists vertices v, v′ ∈ Δ and w ∈ (C(v, v′) ∪ A(v, v′))\Δ. Then w ∈ Γ1(Δ)

and v ∈ Δ such that ∂Γ (v,w) �= ∂Γ (v, v′)+ 1. The lemma is proved. ��

In this section we study a subgraph of Γ which satisfies the following condition.

Definition 8 Let Y be a subgraph of Γ. Let x be a vertex of Γ and Z ⊆ VΓ .A subgraph Y is called conical with respect to x if there exists a vertex x∗ in Y suchthat ∂Γ (x, y) = ∂Γ (x, x∗) + ∂Γ (x∗, y) holds for any y ∈ Y . A subgraph Y is calledconical with respect to Z if it is conical with respect to z for any z ∈ Z .

We remark that a vertex x∗ is the unique vertex in Y such that ∂Γ (x, x∗) = ∂Γ (x,Y )since

∂Γ (x,Y ) = ∂Γ (x, u) = ∂Γ (x, x∗)+ ∂Γ (x∗, u) ≥ ∂Γ (x,Y )+ ∂Γ (x

∗, u)

holds for any vertex u ∈ Y such that ∂Γ (x, u) = ∂Γ (x,Y ).It is straightforward to see that any subgraph Y is conical with respect to Y itself. We

regard Y is conical with respect to Z when Z = ∅. Lemma 7 shows that a subgraphΔis strongly closed if and only if it is conical with respect to Γ1(Δ). A familiar exampleof a subgraph which is conical with respect to VΓ is a singular line of a regular near2d-gon Γ . Also any strongly closed subgraph Δ in a Hamming graph (resp. a dualpolar graph) Γ is conical with respect to VΓ . We will reprove these facts in Sects. 4and 5, respectively. In this section we observe a subgraph Δ which is conical withrespect to VΓ . Our purpose is to prove the following result.

Proposition 9 Let Γ be a distance-regular graph of diameter d ≥ 3. LetΔ be a sub-graph of widthw(Δ) and covering radius t (Δ) in Γ . Suppose that 2 ≤ w(Δ) ≤ d −1and Δ is conical with respect to VΓ . Then c2 > 1 and Δ is a completely regularstrongly closed subgraph with w(Δ)+ t (Δ) = d. In particular,

ci =[

i1

], ai =

[i1

]a1, bi =

([d1

]−

[i1

])(1 + a1) (11)

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hold for i = 1, . . . , d, where

[i1

]=

[i1

]q

:= qi−1 + · · · + q + 1

denotes the Gaussian binomial coefficient with basis q := c2 − 1.

To show this we prove several lemmas.

Lemma 10 Let Γ be a distance-regular graph of diameter d ≥ 2. Let Δ be a sub-graph of Γ and x ∈ VΓ \Δ. Let i = ∂Γ (x,Δ). SupposeΔ is conical with respect to x.Let x∗ be the unique vertex ofΔ such that ∂Γ (x, x∗) = ∂Γ (x,Δ). Then the followinghold.

(i) Let z be a vertex with ∂Γ (x, z)+ ∂Γ (z, x∗) = ∂Γ (x, x∗). ThenΔ is conical withrespect to z. In particular, x∗ is the unique vertex of Δ such that ∂Γ (z, x∗) =∂Γ (z,Δ).

(ii) Let y ∈ VΓ . SupposeΔ is conical with respect to y. Let y∗ be the unique vertexof Δ such that ∂Γ (y, y∗) = ∂Γ (y,Δ). Then

∂Γ (x∗, y∗) ≤ ∂Γ (x, y)− |∂Γ (x,Δ)− ∂Γ (y,Δ)| (12)

holds.(iii)

Γi−1(Δ) ∩ Γ1(x) = C(x∗, x). (13)

(iv) Suppose that Δ is conical with respect to Γi (Δ) ∩ Γ1(x). Then

Γi (Δ) ∩ Γ1(x) = A(x∗, x) ∪⎛⎝ ⋃

z∈Δ1(x∗)C(z, x)\C(x∗, x)

⎞⎠ , (14)

where the right-hand side of (14) is disjoint.

Proof (i) Let v ∈ Δ. Then

∂Γ (z, v) ≤ ∂Γ (z, x∗)+ ∂Γ (x∗, v)

= ∂Γ (x, x∗)− ∂Γ (x, z)+ ∂Γ (x∗, v) = ∂Γ (x, v)− ∂Γ (x, z) ≤ ∂Γ (z, v).

The desired result is proved.(ii) We may assume that ∂Γ (y,Δ) ≤ ∂Γ (x,Δ). Then

∂Γ (x∗, y∗) = ∂Γ (x, y∗)− ∂Γ (x, x∗) ≤ ∂Γ (x, y)+ ∂Γ (y, y∗)− ∂Γ (x, x∗).

Since ∂Γ (x, x∗) = ∂Γ (x,Δ) and ∂Γ (y, y∗) = ∂Γ (y,Δ), the desired result is proved.

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(iii) We have C(x∗, x) ⊆ Γi−1(Δ) ∩ Γ1(x). Let u ∈ Γi−1(Δ) ∩ Γ1(x). Then thereexists v ∈ Δ such that ∂Γ (u, v) = i − 1. The uniqueness of x∗ implies that v = x∗and u ∈ C(x∗, x) since

∂Γ (x,Δ) ≤ ∂Γ (x, v) ≤ ∂Γ (x, u)+ ∂Γ (u, v) = 1 + (i − 1) = ∂Γ (x,Δ).

(iv) We remark that Γi−1(Δ) ∩ Γ1(x) = C(x∗, x) by (iii). The right-hand side of(14) is contained in the left-hand side of (14). Let u ∈ Γi (Δ) ∩ Γ1(x). Then Δ isconical with respect to u by our hypothesis. Let u∗ be the unique vertex in Δ suchthat ∂Γ (u, u∗) = ∂Γ (u,Δ). It follows, by (ii), that ∂Γ (x∗, u∗) ≤ ∂Γ (x, u) = 1. Sou is contained in the right-hand side of (14). The uniqueness of u∗ implies that theright-hand side of (14) is disjoint. The lemma is proved. ��Lemma 11 Let Γ be a distance-regular graph of diameter d ≥ 2. Let Δ be a sub-graph of Γ and x ∈ VΓ \Δ. Let i = ∂Γ (x,Δ). Suppose Δ is conical with respect tox. Let x∗ be the unique vertex of Δ such that ∂Γ (x, x∗) = ∂Γ (x,Δ). Let v ∈ Δ\{x∗}and s := ∂Γ (v, x∗). Suppose that Δ is strongly closed and it is conical with respectto Γi (Δ) ∩ Γ1(x). Then

⎛⎝ ⋃w∈C(v,x∗)

(C(w, x)\C(x∗, x))

⎞⎠ ⊆ (

C(v, x)\C(x∗, x))

(15)

and⎛⎝ ⋃w∈A(v,x∗)

(C(w, x)\C(x∗, x))

⎞⎠ ⊆ (

A(v, x)\A(x∗, x))

(16)

hold, where the left-hand sides of (15) and (16) are disjoint. In particular,

cs(ci+1 − ci ) ≤ cs+i − ci (17)

and

as(ci+1 − ci ) ≤ as+i − ai (18)

hold. Then the following three conditions are equivalent.

(a) The equality holds in (15) (resp. in (16)),(b) The equality holds in (17) (resp. in (18)),(c) C(v, x) ∩ Γi+1(Δ) = ∅ (resp. A(v, x) ∩ Γi+1(Δ) = ∅).

Moreover ifΔ is conical with respect to Γi+1(Δ)∩Γ1(x), then these three conditionshold.

Proof Here we only prove (16). We can prove (15) by just changing the sets A( , ) intoC( , ). Let w ∈ A(v, x∗). Then ∂Γ (w, x) = i + 1. Take any u′ ∈ C(w, x)\C(x∗, x).

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It follows, by (14), that u′ ∈ Γi (Δ) ∩ Γ1(x). So Δ is conical with respect to u′.Then w is the unique vertex of Δ with ∂Γ (u′, w) = ∂Γ (u′,Δ). Thus ∂Γ (u′, x∗) =∂Γ (u′, w) + 1 = i + 1, ∂Γ (u′, v) = ∂Γ (u′, w) + ∂Γ (w, v) = i + s and u′ ∈A(v, x)\A(x∗, x). Hence (16) (resp. (15)) is proved. It is clear that C(x∗, x) ⊆C(v, x). Let x ′ ∈ A(x∗, x). Then Δ is conical with respect to x ′. Thus ∂Γ (x ′, v) =∂Γ (x ′, x∗) + ∂Γ (x∗, v) = i + s and A(x∗, x) ⊆ A(v, x). Hence we obtain (17) and(18). Take any vertex z in the right-hand of (15) (resp. (16)). If z ∈ Γi (Δ), then z iscontained in the left-hand side of (15) (resp. (16)) by (14). The equivalence of threeconditions (a),(b),(c) is straightforward. Let p ∈ Γi+1(Δ)∩Γ1(x). IfΔ is conical withrespect to p, then ∂Γ (p, v) = ∂Γ (p, x∗) + ∂Γ (x∗, v) = i + 1 + s and p ∈ B(v, x).The lemma is proved. ��Corollary 12 Let Γ be a distance-regular graph of diameter d ≥ 2. LetΔ be a regu-lar subgraph of valency k(Δ) with width w(Δ) and covering radius t (Δ) in Γ . Thenthe following hold.

(i) Let i be an integer with 1 ≤ i ≤ t (Δ). Suppose that Δ is conical with respect toΓi (Δ). Then

|Γi−1(Δ) ∩ Γ1(x)| = ci , |Γi (Δ) ∩ Γ1(x)| = ai + k(Δ)(ci+1 − ci ) (19)

and

|Γi+1(Δ) ∩ Γ1(x)| = bi − k(Δ)(ci+1 − ci ) (20)

are constants whenever x ∈ Γi (Δ).(ii) Suppose that Δ is conical with respect to VΓ . Then Δ is a completely regular

strongly closed subgraph with t (Δ)+ w(Δ) = d.

Proof (i) This follows by Lemma 10(iii), (iv).(ii) Lemma 7 implies that Δ is strongly closed. Let z ∈ VΓ such that ∂Γ (z,Δ) =

t (Δ). Let z∗ be the unique vertex of Δ such that ∂Γ (z, z∗) = ∂Γ (z,Δ) = t (Δ).Lemma 6 implies that Δ is distance-regular and thus there exists v ∈ Δ such that∂Γ (z∗, v) = d(Δ) = w(Δ). Since

d ≤ t (Δ)+ w(Δ) = ∂Γ (z, z∗)+ ∂Γ (z∗, v) = ∂Γ (z, v) ≤ d

by Lemma 5(i), the desired result follows by (i). ��Lemma 13 Let Γ be a distance-regular graph of diameter d ≥ 3. Let Δ be a sub-graph of width j := w(Δ) and covering radius t (Δ) in Γ such that 2 ≤ j ≤ d − 1.Let m be an integer with 2 ≤ m ≤ d − j + 1. Suppose that Δ is conical with respectto Γi (Δ) for i = 1, . . . ,m. Then the following hold.

(i)

cs(ch+1 − ch) = cs+h − ch, as(ch+1 − ch) = as+h − ah (21)

hold for any integers s and h with 1 ≤ s ≤ j and 1 ≤ h ≤ m − 1.

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(ii) If c2 = 1, then (c1, a1) = · · · = (cm+ j−1, am+ j−1).

(iii) Suppose c2 > 1. Then Δ is regular. In particular,

ci =[

i1

], ai =

[i1

]a1

hold for i = 1, . . . ,m + j − 1.

Proof We remark that Δ is strongly closed by Lemma 7.(i) Let h be an integer with 1 ≤ h ≤ m − 1. Take y, z ∈ Δ with ∂Γ (y, z) = j and

x ∈ Γh(y) ∩ Γh+ j (z). Since ∂Γ (x,Δ) ≤ ∂Γ (x, y) = h ≤ m − 1, Δ is conical withrespect to x . Let x∗ be the unique vertex ofΔ such that ∂Γ (x, x∗) = ∂Γ (x,Δ). Since

h + j = ∂Γ (x, z) = ∂Γ (x, x∗)+ ∂Γ (x∗, z) ≤ h + j,

we have ∂Γ (x,Δ) = h and x∗ = y. For any integer s with 1 ≤ s ≤ j take v ∈Γs(y) ∩ Γ j−s(z). We remark that Δ is conical with respect to Γh(Δ) ∪ Γh+1(Δ) andv ∈ Δ such that ∂Γ (v, x∗) = s. The equations in (21) follow by applying Lemma 11.

(ii) Put h = 1 in (i). Then (c1, a1) = · · · = (c j+1, a j+1) holds. Put s = j in (i).Then we obtain (c j+h, a j+h) = (ch, ah) for h = 1, . . . ,m − 1 by induction on h. Thedesired result is proved.

(iii) Let (u, w) be an edge inΔ. We count the number of edges between (Γ1(u)\Δ)and (Γ1(w)\Δ). Then we have

|Γ1(u)\Δ|(c2 − 1) = |Γ1(w)\Δ|(c2 − 1)

and thus |Γ1(u) ∩Δ| = |Γ1(w) ∩Δ|. Since Δ is connected, it is regular. The rest ofthe assertion follows by (i) and the induction on i . The lemma is proved. ��

A connected graph E is called an expanded tree of order s if each edge lies on aclique of size s + 1 and there are no induced cycles except triangles.

The case in Lemma 13(ii) is exceptional trivial case. Let

r = max{i | (ci , ai , bi ) = (c1, a1, b1)}.

Suppose r ≥ 2. Let j be an integer with 2≤ j ≤r . Then any two vertices at distance jthere exists a strongly closed subgraph � which is an expanded tree of order a1 + 1.Conversely any subgraph Δ of diameter j which is an expanded tree of order a1 + 1is strongly closed. For a description of these fact we refer the reader to [8, §2]. It isstraightforward to see from the definition of r that Δ is conical with respect to Γi (Δ)

for any i with i + j ≤ r + 1.

Proof of Proposition 9 Put m = d − j + 1 in Lemma 13. If c2 = 1, then (c1, a1) =(cd , ad) by Lemma 13(ii). This is a contradiction. Thus we have c2 > 1 and Δ isregular and

ci =[

i1

], ai =

[i1

]a1

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hold for i = 1, . . . , d by Lemma 13(iii). Hence

bi = (cd + ad)− (ci + ai ) =[

d1

](1 + a1)−

[i1

](1 + a1)

holds for i = 1, . . . , d. The desired result is proved by Corollary 12(ii). ��The rest of this section we study a subgraph Δ which is conical with respect to �

for some subgraph �.

Lemma 14 Let Γ be a distance-regular graph of diameter d ≥ 2 and x be a vertexof Γ . Let Ξ and � be subgraphs of Γ such that Ξ ⊆ �. Then the following hold.

(i) Suppose � is conical with respect to x. Let x∗ be the unique vertex of � suchthat ∂Γ (x, x∗) = ∂Γ (x,�). Then Ξ is conical with respect to x if and only if itis conical with respect to x∗

(ii) Let � be a subgraph of Γ. Suppose that Ξ is conical with respect to � and that� is conical with respect to �. Then Ξ is conical with respect to �.

Proof (i) Suppose thatΞ is conical with respect to x . Let y be the unique vertex ofΞsuch that ∂Γ (x, y) = ∂Γ (x, Ξ). Then ∂Γ (x, y) = ∂Γ (x, x∗) + ∂Γ (x∗, y). Hence Ξis conical with respect to x∗ by Lemma 10(i). Conversely suppose that Ξ is conicalwith respect to x∗. Let u be the unique vertex ofΞ such that ∂Γ (x∗, u) = ∂Γ (x∗, Ξ).Then ∂Γ (x, u) = ∂Γ (x, x∗)+ ∂Γ (x∗, u) and

∂Γ (x, v) = ∂Γ (x, x∗)+ ∂Γ (x∗, v) = ∂Γ (x, x∗)+ ∂Γ (x

∗, u)+ ∂Γ (u, v)

= ∂Γ (x, u)+ ∂Γ (u, v)

for any v ∈ Ξ . Hence Ξ is conical with respect to x .(ii) Take any vertex x in �. Since � is conical with respect to x , there exists the

unique vertex x∗ of� such that ∂Γ (x, x∗) = ∂Γ (x,�). ThenΞ is conical with respectto x∗. It follows, by (i), that Ξ is conical with respect to x . The lemma is proved. ��

In [11, Theorem 1.1] Suzuki proved that if there exists a non-regular stronglyclosed subgraph of diameter m then bm−1 = bm holds. Conversely if there existsa regular strongly closed subgraph of diameter m, then it is distance-regular and0 < bm−1(Δ) = bm−1 − bm . Hence a strongly closed subgraph of diameter m isregular if and only if bm−1 > bm . It follows that if there exists a regular stronglyclosed subgraph of diameter m, then any strongly closed subgraph of diameter m isregular. In particular, the following result holds.

Lemma 15 Let Γ be a distance-regular graph of diameter d ≥ 3 and let j be aninteger with 1 ≤ j ≤ d − 2. Suppose the condition (SC) j+1 holds and there existsa regular strongly closed subgraph Δ of diameter j in Γ . Then for any pair (y, z)of vertices at distance j in Δ and for any v ∈ B(z, y) there exists a strongly closedsubgraph � of diameter j + 1 containing v and Δ.

123


Proof We remark that each strongly closed subgraph of diameter j is regular. Since∂Γ (z, v) = j + 1 and the condition(SC) j+1 holds, there exists a strongly closed sub-graph of diameter j + 1 containing z and v. SetΞ := Δ∩�. Then y, z ∈ Ξ and thusΞ is a strongly closed subgraph of diameter j . Lemma 6 implies that the both of Δand Ξ are distance-regular with ci (Δ) = ci (Ξ) = ci and bi (Δ) = bi (Ξ) = bi − b j

for all i = 0, . . . , j . Hence Ξ = Δ and Δ ⊆ �. ��Lemma 16 Let Γ be a distance-regular graph of diameter d ≥ 2. Let i and j bepositive integers with i + j ≤ d. Suppose the condition (SC)h holds and any stronglyclosed subgraph of diameter h is regular for h = 1, . . . , i + j − 1. LetΔ be a regularstrongly closed subgraph of diameter j . Then the following hold.

(i) Let x ∈ Γi (Δ). If Δ ∩ Γi+ j (x) �= ∅, then Δ is conical with respect to x.(ii) If Δ is completely regular, then Δ is conical with respect to Γi (Δ).

Proof (i) We prove the assertion by induction on i . The case i = 1 is true by Lemma 7.We assume that i ≥ 2. Let y ∈ Δ with ∂Γ (x, y) = i and take z ∈ Δ ∩ Γi+ j (x). Then

∂Γ (x, z) ≤ ∂Γ (x, y)+ ∂Γ (y, z) ≤ i + j = ∂Γ (x, z).

Hence ∂Γ (x, z) = ∂Γ (x, y)+∂Γ (y, z) and ∂Γ (y, z) = j . Let v ∈ C(x, y) ⊆ B(z, y).Since the condition (SC) j+1 holds, there exists a strongly closed subgraph � ofdiameter j + 1 containing v and Δ by Lemma 15. Let u be a vertex of � such that∂Γ (x, u) = ∂Γ (x,�). Then

i + j = ∂Γ (x, z) ≤ ∂Γ (x, u)+ ∂Γ (u, z) ≤ ∂Γ (x,�)+ d(�)

≤ (i − 1)+ ( j + 1).

Thus ∂Γ (x,�) = i − 1 and z ∈ � ∩ Γi+ j (x). Then � is conical with respectto x by the inductive hypothesis. We remark that v is the unique vertex of� such that∂Γ (x, v) = ∂Γ (x,�). Since v ∈ Γ1(Δ), Δ is conical with respect to v. Hence Δ isconical with respect to x by Lemma 14(i).

(ii) Let u, v ∈ Δ such that ∂Γ (u, v) = j and take w ∈ Γi (u) ∩ Γi+ j (v). Thenw ∈ Γi (Δ) and

ψi,i+ j = |Δ ∩ Γi+ j (w)| ≥ |{v}| > 0.

Hence we have Δ ∩ Γi+ j (x) �= ∅ for any x ∈ Γi (Δ) by Lemma 5(iii). The desiredresult follows by (i). ��

4 Regular Near Polygons

A distance-regular graph Γ of diameter d ≥ 2 is said to be (the collinearity graph of)a regular near polygon if each edge lies on a clique of size a1 + 2 and that each cliqueC of size a1 +2 and for any vertex x in Γ with ∂Γ (x,C) ≤ d −1 there exists a uniquevertex x∗ in C such that ∂Γ (x, x∗) = ∂Γ (x,C). Such a graph is also called a regular

123


near n-gon, where n = 2d + 1 if there exists a vertex y and a clique L of size a1 + 2such that ∂Γ (y, L) = d, and n = 2d otherwise. For more information of regular nearpolygons we refer the reader to [2, Chapter 6.4] and [3].

In this section we prove the following result.

Proposition 17 Let Γ be a distance-regular graph of diameter d ≥ 2. Suppose thateach edge lies on a clique of size a1 +2. Then the following conditions are equivalent.

(i) Γ is a regular near n-gon with n = 2d (resp. n = 2d + 1).(ii) Any strongly closed subgraph Δ of diameter 1 is conical with respect to Γi (Δ)

for i = 1, . . . , d − 1. In particular, Δ is completely regular of covering radiust (Δ) = d − 1 (resp. t (Δ) = d).

(iii) There exists a strongly closed subgraph Δ of diameter 1 and covering radiust (Δ) = d − 1 (resp. t (Δ) = d) which is conical with respect to Γi (Δ) fori = 1, . . . , d − 1.

Then we can reprove the following fact as a direct consequence.

Proposition 18 Let Γ be a distance-regular graph of diameter d ≥ 3 which satisfiesHypothesis 1 and each strongly closed subgraph is regular. Suppose there exists astrongly closed subgraph Δ of diameter 1 which is completely regular in Γ . Then Γis a regular near n-gon, where n = 2d if t (Δ) = d − 1, n = 2d + 1 if t (Δ) = d.

We start from the following lemma.

Lemma 19 Let Γ be a distance-regular graph of diameter d ≥ 2. Let Δ be a cliquein Γ . Suppose thatΔ is conical with respect to Γi (Δ) for all i = 1, . . . , d −1. ThenΔis a completely regular strongly closed subgraph of covering radius t (Δ) ∈ {d −1, d}with

γi = ci , αi = ai + (a1 + 1)(ci+1 − ci ), βi = bi − (a1 + 1)(ci+1 − ci ) (22)

for i = 1, . . . , d − 1. If t (Δ) = d, then

γd = cd(a1 + 2), αd = b0 − cd(a1 + 2). (23)

In particular, t (Δ) = d − 1 if and only if bd−1 = (a1 + 1)(cd − cd−1).

Proof Since Δ is conical with respect to Γ1(Δ), it is strongly closed by Lemma 7and hence |Δ| = a1 + 2 and k(Δ) = a1 + 1. Then we have t (Δ) ∈ {d − 1, d} byLemma 5(i). It follows, by Corollary 12(i), that

|Γi−1(Δ) ∩ Γ1(x)| = ci , |Γi (Δ) ∩ Γ1(x)| = ai + (a1 + 1)(ci+1 − ci )

and

|Γi+1(Δ) ∩ Γ1(x)| = bi − (a1 + 1)(ci+1 − ci )

123


are constants whenever x ∈ Γi (Δ) for all i = 1, . . . , d − 1. Therefore Δ is com-pletely regular if t (Δ) = d − 1. In particular, t (Δ) = d − 1 if and only if bd−1 =(a1 + 1)(cd − cd−1). Suppose t (Δ) = d. Let y ∈ Γd(Δ). ThenΔ ⊆ Γd(y). We showthat

Γd−1(Δ) ∩ Γ1(y) =⋃z∈Δ

C(z, y), (24)

where the right-hand side is disjoint. It is clear that the right-hand side of (24) is con-tained in the left-hand side. Take any vertexw in the left-hand side of (24). Then thereexists the unique vertex w∗ inΔ such that ∂Γ (w,w∗) = d − 1. Hence w ∈ C(w∗, y)and that w is contained in the right-hand side of (24). The uniqueness of w∗ showsthat the right-hand side of (24) is disjoint. The desired result is proved. ��

It follows, by [2, Theorem 6.4.1], that a distance-regular graph of diameter d ≥ 2such that each edge lies on a clique of size a1 + 2 is a regular near polygon if andonly if

ai = ci a1, bi = b0 − ci (1 + a1) (25)

hold for i = 1, . . . , d − 1. In particular, Γ is a regular near 2d-gon if and only ifad = cda1 holds.

If Γ is a regular near polygon, then ad−1 = cd−1a1 holds. Thus bd−1 = (a1 + 1)(cd − cd−1) holds if and only if ad = cda1, i.e., Γ is a regular near 2d-gon.

Proof of Proposition 17 (i) ⇒ (ii). Let Δ be a strongly closed subgraph of diame-ter 1 in Γ . Then it is a clique of size a1 + 2 and conical with respect to Γi (Δ) fori = 1, . . . , d−1 by the definition of regular near polygons. HenceΔ is completely reg-ular by Lemma 19. In particular, t (Δ) = d−1 if and only if bd−1 = (a1+1)(cd −cd−1)

holds, i.e., Γ is a regular near 2d-gon.(ii) ⇒ (iii). This is clear.(iii) ⇒ (i). Lemma 19 implies that Δ is completely regular of covering radius

t (Δ) ∈ {d − 1, d}. Let i be an integer with 1 ≤ i ≤ d − 2. Let x ∈ Γi (Δ) and x∗ bethe unique vertex inΔ such that ∂Γ (x, x∗) = i . Take v ∈ Δ\{x∗}. Then ∂Γ (x∗, v) = 1.Since Δ is conical with respect to Γi (Δ) ∪ Γi+1(Δ),

a1(ci+1 − ci ) = ai+1 − ai

holds by Lemma 11. Hence ah = cha1 and bh = b0 − ch(1 + a1) hold for allh = 1, . . . , d − 1 by induction on h. The desired result is proved. ��Proof of Proposition 18 Put j = 1 in Lemma 16. Then Δ is conical with respect toΓi (Δ) for i = 1, . . . , d − 1. Hence the desired result is proved by Proposition 17.

��We remark that a Hamming graph and a dual polar graph are regular near 2d-gons.

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5 Proof of the Theorems

First we recall the following result (see [2, Theorems 9.2.5, 9.4.4, 9.4.5]). Thedefinitions of the Hamming graphs and the dual polar graphs will be found in [2,Chapter 9.2, 9.4].

Theorem 20 Let Γ be a distance-regular graph of diameter d ≥ 3 and c2 > 1 suchthat

ci =[

i1

], ai =

[i1

]a1, bi =

([d1

]−

[i1

])(1 + a1) (26)

for i = 1, . . . , d, where

[i1

]=

[i1

]q

:= qi−1 + · · · + q + 1

denotes the Gaussian binomial coefficient with basis q := c2 − 1. Then the followinghold.

(i) Suppose a1 = 0 and c2 = 2. Then Γ is the d-cube.(ii) Suppose a1 > 0 and each edge lies on a clique of size a1 + 2. Then Γ is either a

Hamming graph or a dual polar graph.(iii) Suppose a1 = 0 and q := c2 − 1 > 1. If d ≥ 4 and for any three distinct vertices

the number of their common neighbors is 0, 1 or q + 1, then q is a prime powerand Γ is the dual polar graph on [Dd(q)]. ��

Here we consider the following hypothesis. The above theorem implies that underthis hypothesis the Hamming graphs and the dual polar graphs can be characterizedby their intersection numbers.

Hypothesis 2 Let Γ be a distance regular graph of diameter d ≥ 3. Assume that Γsatisfies one of the following conditions.

(i) a1 = 0 and c2 = 2 hold.(ii) a1 > 0 and each edge lies on a clique of size a1 + 2.

(iii) a1 = 0, c2 > 2, d ≥ 4 and for any three distinct vertices the number of theircommon neighbors is 0, 1 or c2.

Remark 21 Let Γ be a distance-regular graph of diameter d ≥ 3. Then the followinghold.

(i) A strongly closed subgraph of diameter 1 is a clique of size a1 +2. So each edgelies on a clique of size a1 + 2 if and only if the condition (SC)1 holds.

(ii) Suppose c2 = 2 and a1 = a2 = 0. Then a strongly closed subgraphs ofdiameter 2 is a quadrangle. Hence the condition (SC)2 always holds as anypair of vertices at distance 2 lies on a quadrangle.

123


(iii) Suppose q := c2 − 1 ≥ 2 and a1 = a2 = 0. Then a strongly closed subgraphsof diameter 2 is the complete bipartite graph Kq+1,q+1. Hence for any threedistinct vertices the number of their common neighbors is 0, 1 or q + 1 if andonly if the condition (SC)2 holds.

(iv) We assume d ≥ 4 if a1 = 0 and c2 > 2. Then (i)-(iii) imply that if Γ satisfiesHypothesis 1, then it satisfies Hypothesis 2.

(v) Let m be an integer with 2 ≤ m ≤ d − 1. If Γ satisfies the condition (SC) j

for all j = 1, . . . ,m, then any strongly closed subgraph � of diameter m alsosatisfies the condition (SC) j for all j = 1, . . . ,m − 1 (see [6, Theorem 1] or[8, Corollary 7]).

In this section we prove Theorem 1 and Theorem 2. It can be verified from thedefinition that any pair of vertices in a Hamming graph (resp. a dual polar graph)there exists a strongly closed subgraph containing them whose diameter is equal tothe distance between them. A brief description of this fact for a Hamming graph isgiven in [8, §2].

Proposition 22 Let Γ be either a Hamming graph or a dual polar graph of diameterd ≥ 2. Then the condition (SC)i holds for all i = 1, . . . , d − 1 and each stronglyclosed subgraph is distance-regular. ��

This fact is also proved in [3] if a1 > 1 since a Hamming graph and a dual polargraph are regular near polygons with quadrangles. Also it is known that these sub-graphs are completely regular in the original graph. Here we reprove this fact andcompute their parameters for completely regular subgraphs.

Proposition 23 Let Γ be either a Hamming graph or a dual polar graph of diameterd ≥ 3. Let j be an integer with 1 ≤ j ≤ d − 1 and Δ be a strongly closed sub-graph of diameter j in Γ . ThenΔ is conical with respect to VΓ with covering radiust (Δ) = d − j . In particular, it is completely regular with

γi =[

i1

], αi =

[i1

]a1 + qi

[j1

](1 + a1) (27)

and

βi =([

d1

]−

[i1

]− qi

[j1

])(1 + a1) (28)

hold for i = 1, . . . , t (Δ), where

[i1

]=

[i1

]q

:= qi−1 + · · · + q + 1

denotes the Gaussian binomial coefficient with basis q := c2 − 1.

To prove this proposition we need the following lemmas.

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Lemma 24 Let Γ be a distance-regular graph of diameter d ≥ 2. Suppose thereexists a regular strongly closed subgraph Δ of diameter d − 1. Then

b1 ≥ (b0 − bd−1)(c2 − 1). (29)

The equality in (29) holds if and only if Δ is completely regular of covering radius 1.

Proof Δ is distance-regular with valency k(Δ) = b0(Δ) = b0 −bd−1 and it is conicalwith respect to Γ1(Δ) by Lemma 6 and Lemma 7. It follows, by Corollary 12(i), that

0 ≤ |Γ2(Δ) ∩ Γ1(x)| = b1 − (b0 − bd−1)(c2 − 1)

for any x ∈ Γ1(Δ). Hence (29) holds. In particular, the equality in (29) holds if andonly ifΔ has covering radius 1. IfΔ has covering radius 1, then it is completely regularby Corollary 12(i). ��Lemma 25 Let Γ be a distance-regular graph of diameter d ≥ 3. Let j be an integerwith 1 ≤ j ≤ d − 2. Suppose there exist regular strongly closed subgraphs Δ and �of diameter j and j + 1, respectively such that Δ ⊆ �. Then

b1 − b j+1 ≥ (b0 − b j )(c2 − 1). (30)

The equality in (30) holds if and only if Δ is completely regular of covering radius1 in �.

Proof Lemma 6 implies that� is a distance-regular graph of diameter j +1 such that

ci (�) = ci , bi (�) = bi − b j+1

for i = 0, 1, . . . , j + 1. The desired result is proved by applying Lemma 24 to �.��

Proof of Proposition 23 We remark that Δ is regular of valency

k(Δ) = b0 − b j =[

j1

](1 + a1).

by Lemma 6 and (2). We prove that Δ is conical with respect to VΓ by induction ond − j . If j = d − 1, then

b1 = (b0 − bd−1)(c2 − 1)

holds by (1) and (2). Hence the desired result is proved by Lemma 24 and Lemma 7.Suppose that j ≤ d − 2. Then there exists a strongly closed subgraph � of diame-ter j + 1 containing Δ by Lemma 15 and Proposition 22. It follows, by the inductivehypothesis, that� is conical with respect to VΓ with covering radius d(�) = d− j−1.Since

b1 − b j+1 = (b0 − b j )(c2 − 1)

123


holds by (1) and (2),Δ is completely regular of covering radius 1 in� by Lemma 25.Hence � ⊆ Δ ∪ Γ1(Δ) and that Δ is conical with respect to� by Lemma 7. Then Δis conical with respect to VΓ by Lemma 14(ii). Thus the desired result is proved by(1), (2), Proposition 9 and Corollary 12. ��Proof of Theorem 1 Lemma 16(ii) implies thatΔ is conical with respect to Γi (Δ) fori = 1, . . . , d − j . SinceΔ is regular, we have b j−1 > b j . Put m = d − j in Lemma 13.If c2 = 1, then

(c1, a1) = · · · = (cd−1, ad−1)

which contradicts b j−1 > b j . Hence c2 > 1 and we obtain (4) by Lemma 13. Let �be a strongly closed subgraph of diameter t with t ≤ d − 1. Then

ci (�) = ci =[

i1

], ai (�) = ai =

[i1

]a1 (31)

hold for i = 1, . . . , t by Lemma 6. Hence

bi (�) = b0(�)− ci (�)− ai (�) = (ct + at )− ci − ai

hold for i = 1, . . . , t . We remarked, in Remark 21, that� satisfies the condition (SC) j

for j = 1, . . . , t − 1 and thus it satisfies Hypothesis 2. The desired result is proved byTheorem 20. ��

Finally we prove the following result.

Proposition 26 Let Γ be a distance-regular graph of diameter d ≥ 3. We assumed ≥ 4 if a1 = 0 and c2 > 2. Then the following conditions are equivalent.

(i) Γ is either a Hamming graph or a dual polar graph.(ii) Γ satisfies Hypothesis 1. Any strongly closed subgraph Δ of diameter j with

1 ≤ j ≤ d − 1 is completely regular of covering radius d − j in Γ . Moreover itis conical with respect to VΓ .

(iii) Γ satisfies Hypothesis 1. Moreover there exists a strongly closed subgraph ofdiameter j with 2 ≤ j ≤ d − 1 which is completely regular of covering radiusd − j in Γ .

(iv) Γ satisfies Hypothesis 2. Moreover there exists a strongly closed subgraph ofdiameter d − 1 which is completely regular of covering radius 1.

(v) Γ satisfies Hypothesis 2. Moreover there exists a strongly closed subgraph Δ ofdiameter j with 2 ≤ j ≤ d − 1 which is conical with respect to VΓ .

Proof (i) ⇒ (ii). This is proved in Proposition 22 and 23.(ii) ⇒ (iii), (iv), (v). These are clear.(iii) ⇒ (v). We remark that Hypothesis 1 implies Hypothesis 2. Lemma 16(ii) shows

that Δ is conical with respect to Γi (Δ) for i = 1, . . . , d − j . Since Δ has coveringradius d − j ,

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VΓ =d− j⋃i=0

Γi (Δ).

Hence Δ is conical with respect to VΓ .(iv) ⇒ (v). Since Δ has covering radius 1, VΓ = Δ ∪ Γ1(Δ). It follows, by

Lemma 7, that Δ is conical with respect to VΓ .(v) ⇒ (i). This is a direct consequence of Proposition 9 and Theorem 20. ��

Proof of Theorem 2 This is a direct consequence of Proposition 26. ��

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145–176 (1983)4. Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Research

Reports Suppl. no. 10 (1973)5. Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall Inc., New York (1993)6. Hiraki, A.: A distance-regular graph with strongly closed subgraphs. J. Algebraic Combin. 14,

127–131 (2001)7. Hiraki, A.: A characterization of the doubled Grassmann graphs, the doubled Odd graphs, and the Odd

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Combin. 29, 1603–1616 (2008)9. Hiraki, A.: A characterization of some distance-regular graphs by strongly closed subgraphs. Eur.

J. Combin. 30, 893–907 (2009)10. Hiraki, A.: Completely regular subgraphs in distance-regular graphs. Preprint11. Suzuki, H.: On strongly closed subgraphs of highly regular graphs. Eur. J. Combin. 16, 197–220 (1995)12. Suzuki, H.: Parallelogram-free distance-regular graphs having completely regular strongly regular

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A Characterization of the Hamming Graphs and the Dual Polar Graphs by Completely Regular Subgraphs