On the (non-)existence of tight distance-regular graphs: a local approach

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 3$. Juri\v{s}i\'c and Vidali conjectured that if $\Gamma$ is tight with classical parameters $(D,b,\alpha,\beta)$, $b\geq 2$, then $\Gamma$ is not locally the block graph of an orthogonal array nor the block graph of a Steiner system. In the present paper, we prove this conjecture and, furthermore, extend it from the following aspect. Assume that for every triple of vertices $x, y, z$ of $\Gamma$, where $x$ and $y$ are adjacent, and $z$ is at distance $2$ from both $x$ and $y$, the number of common neighbors of $x$, $y$, $z$ is constant. We then show that if $\Gamma$ is locally the block graph of an orthogonal array (resp. a Steiner system) with smallest eigenvalue $-m$, $m\geq 3$, then the intersection number $c_2$ is not equal to $m^2$ (resp. $m(m+1)$). Using this result, we prove that if a tight distance-regular graph $\Gamma$ is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of $\Gamma$ is bounded by a function in the parameter $b=b_1/(1+\theta_1)$, where $b_1$ is the intersection number of $\Gamma$ and $\theta_1$ is the second largest eigenvalue of $\Gamma$.


Introduction
Let Γ denote a distance-regular graph with diameter D ≥ 3, intersection numbers a i , b i , c i (0 ≤ i ≤ D), and Koolen, and Terwilliger [8] showed that Γ satisfies the following inequality: (1) We say Γ is tight whenever Γ is nonbipartite and equality holds in (1).Tight distance-regular graphs have been studied with considerable attention and characterized in various ways; see [6,7,16,17].A notable characterization is that, for each vertex x in a tight distance-regular graph, its local graph at x is a connected strongly regular graph with eigenvalues see [8,Theorem 12.6].Suppose that Γ is tight with D ≥ 3, and let ∆ denote a local graph of Γ.We observe that ∆ is a connected strongly regular graph with eigenvalues a 1 , r, s.Throughout this paper, we assume that r and s are integers.Because if they are not, ∆ is a conference graph, which implies that Γ is a Taylor graph; see [12,13].Therefore, further discussion of Γ in this paper is unnecessary when r and s are not integers.
Suppose that s ≤ −2, that is, the smallest eigenvalue of ∆ is less than or equal to −2.In the present paper, we prove this conjecture and extend it to the case where a tight distance-regular graph Γ has no classical parameters; see Theorem 6.3 and Corollary 5.6.Furthermore, we extend the conjecture from the following viewpoint.Let Γ be a distance-regular graph with diameter D ≥ 3. Note that a tight distance-regular graph is 1-homogeneous in the sense of Nomura [8,Theorem 11.7].We consider a regular property for Γ that is a more general concept than the 1-homogeneous property: we say the (triple) intersection number γ(Γ) exists if, for every triple of vertices (x, y, z) of Γ such that x and y are adjacent and z is at distance 2 from both x and y, the number of common neighbors of x, y, and z is constant and equal to γ(Γ).To avoid the degenerate case, we assume that there exists at least one such triple (x, y, z) in Γ (i.e., a 2 = 0) when we say γ(Γ) exists.The result of our extension is the main result of this paper and is as follows: Theorem 1.2.Let Γ be a distance-regular graph with diameter D ≥ 3, valency k, and intersection number c 2 .Assume that Γ is locally strongly regular with smallest eigenvalue −m, where m ≥ 3, and the intersection number γ(Γ) exists.Then the following (i) and (ii) hold.
(i) If Γ is locally the block graph of an orthogonal array and k > m 2 , then c 2 = m 2 .
(ii) If Γ is locally the block graph of a Steiner system and k > m(m + 1), then c 2 = m(m + 1).
Theorem 1.2 is relevant to the problem of determining an upper bound on the diameter of a tight distanceregular graph.In the theory of distance-regular graphs, establishing an upper bound for the diameter of distance-regular graphs in terms of some intersection numbers is an important problem.In particular, with respect to the valency k = b 0 , various bounds for the diameter have been known and have contributed to the theory of distance-regular graphs; see [14].One of the significant results of these contributions is the proof of the Bannai-Ito conjecture [1, p. 237] by Bang, Dubickas, Koolen, and Moulton [2].
Bannai-Ito Conjecture.There are finitely many distance-regular graphs with fixed valency at least three.
To prove this conjecture, they demonstrated that the diameter of the distance-regular graph is bounded by a univariate function with the variable valency k.Returning our attention to the present paper, we will discuss an upper bound on the diameter in a tight distance-regular graph using a specific parameter, distinct from valency k.Specifically, by utilizing the result of Theorem 1.2, we will show that when a tight distance-regular graph is not locally the block graph of an orthogonal array or a Steiner system, its diameter is bounded by a function of the parameter b = b 1 /(1 + θ 1 ).We present this finding in the following theorem.
Theorem 1.3.Let Γ be a tight distance-regular graph with diameter D ≥ 3, intersection number b 1 , and We This paper is organized as follows.In Section 2, we present basic definitions and some known results about distance-regular graphs.Section 3 discusses the block graph of an orthogonal array and its properties.
We then analyze the structure of the µ-graph of an amply regular graph that is locally the block graph of an orthogonal array.Following that, Section 4 covers the block graph of a Steiner system and its properties.
We also analyze the structure of the µ-graph of an amply regular graph that is locally the block graph of a Steiner system.In Section 5, we revisit results related to the triple intersection number of a distance-regular graph and dedicate this section to proving our main result, Theorem 1.2.We conclude this section with a discussion of the case of tight distance-regular graphs with diameter three.Section 6 provides the proof of Conjecture 1.1 using Theorem 1.2.Finally, the paper concludes in Section 7 with the proof of Theorem 1.3 and a discussion of further direction.

Preliminaries
In this section, we review the basic definitions and some known results concerning distance-regular graphs that we will use later.For more background information, refer to [3].
Throughout this section, let Γ denote a finite, undirected, connected, and simple graph.We denote V (Γ) by the vertex set of Γ.For vertices x, y ∈ V (Γ), the distance between x and y, denoted as ∂(x, y), is the length of a shortest path from x to y in Γ.The diameter D of Γ is the maximum value of ∂(x, y) for all pairs of vertices x and y of Γ. Suppose that Γ has diameter D. For x ∈ V (Γ) and an integer 0 ≤ i ≤ D, . For an integer k ≥ 0 we say Γ is regular with valency k (or k-regular ) if |Γ(x)| = k for every x ∈ V (Γ).
We now recall some special regular graphs.We say the graph Γ is distance-regular whenever for all integers 0 ≤ h, i, j ≤ D and for all vertices x, y ∈ V (Γ) with ∂(x, y) = h, the number p h i,j = |Γ i (x) ∩ Γ j (y)| is independent of x and y.The numbers p h i,j are called the intersection numbers of Γ.By construction, we observe that p h i,j = p h j,i for 0 ≤ i, j, h ≤ D. We abbreviate Observe that Γ is regular with valency k = b 0 .Moreover, we note that a 0 = b D = c 0 = 0, c 1 = 1, and (i) Every pair of adjacent vertices has precisely λ common neighbors.
(ii) Every pair of vertices at distance 2 has precisely µ common neighbors.
(iii) Every pair of nonadjacent vertices has precisely µ common neighbors.
Let Γ be κ-regular with ν vertices.We say Γ is amply regular with parameters (ν, κ, λ, µ) if (i) and (ii) hold.We also say Γ is strongly regular with parameters (ν, κ, λ, µ) if (i) and (iii) hold.Observe that every distance-regular graph is amply regular with λ = a 1 and µ = c 2 .Moreover, every distance-regular graph with D ≤ 2 is strongly regular.If Γ is a connected strongly regular graph with parameters (ν, κ, λ, µ) and diameter two, then it has precisely three distinct eigenvalues κ > r > s, satisfying The following is an example of a strongly regular graph for later use in this paper.
Example 2.1.A generalized quadrangle is an incidence structure such that: (i) every pair of points is on at most one line, and hence every pair of lines meets in at most one point, (ii) if p is a point not on a line L, then there is a unique point p ′ on L such that p and p ′ are collinear.If every line contains s + 1 points, and every point lies on t + 1 lines, we say that the generalized quadrangle has order (s, t), denoted by GQ(s, t).
The point graph of a generalized quadrangle is the graph with the points of the quadrangle as its vertices, where two points are adjacent if and only if they are collinear.The point graph of a GQ(s, t) is strongly regular with parameters We recall the notion of a complete multipartite graph.A clique in Γ is a subset of V (Γ) such that every pair of distinct vertices is adjacent.A clique of size n is referred to as a complete graph K n .A coclique of Γ is a subset of V (Γ) such that no two vertices are adjacent.A complete bipartite graph K m,n is a graph whose vertex set can be partitioned into two cocliques, say an m-set M and an n-set N , where each vertex in M is adjacent to all vertices in N .A complete multipartite graph K t×n is a graph whose vertex set can be partitioned into cocliques {M i } t i=1 of size n, where each vertex in M i is adjacent to all vertices in M j (1 ≤ j = i ≤ t).We note that K 2×m is the same as K m,m .
Next, we recall the concepts of a local graph and a µ-graph.For a vertex x ∈ V (Γ), let ∆(x) denote the subgraph of Γ induced on Γ(x).We call ∆(x) the local graph of Γ at x. Let P be a property of a graph or a family of graphs.We say Γ is locally P whenever every local graph of Γ has the property P or belongs to the family P.For example, we say Γ is locally complete multipartite or locally strongly regular.Suppose that Γ is amply regular with parameters (ν, κ, λ, µ).For two vertices x, y with ∂(x, y) = 2, the subgraph of Γ ).Let Γ be a regular graph with v vertices, valency k, and smallest eigenvalue −m.We recall the Q-polynomial property.Let Γ be distance-regular with diameter D ≥ 3. We abbreviate the vertex set as X = V (Γ).We denote Mat X (R) as the R-algebra consisting of real matrices, where both rows and columns are indexed by X.For each integer 0 ≤ i ≤ D, define the matrix A i ∈ Mat X (R) with (x, y)-entry 1 if ∂(x, y) = i and 0 otherwise.Observe that

with equality if and only if every vertex outside
It is known that the matrices {A i } D i=0 form a basis for a commutative subalgebra M of Mat X (R).We call M the Bose-Mesner algebra of Γ.The algebra M has a second basis where the matrices E i (0 ≤ i ≤ D) are called the primitive idempotents of Γ.We note that M is closed under the entrywise multiplication • since A i • A j = δ i,j A i .Thus, there exist real numbers q h i,j such that An ordering {E i } D i=0 is called Q-polynomial whenever for all distinct h, j (0 ≤ h, j ≤ D) we have q h 1,j = 0 if and only if |h − j| = 1.We say Γ is Q-polynomial whenever there is a Q-polynomial ordering of the primitive idempotents; cf.[3, p. 235].Suppose Γ is a tight distance-regular graph.In [16], several characterizations of Γ with the Q-polynomial property were introduced.In [8, Section 13(vi)], the authors provided many examples of Γ, both with and without the Q-polynomial property.Here, we recall one example of Γ that does not have the Q-polynomial property, which will be used later in this paper.We finish this section with one comment.Let Γ be a graph with valency k and diameter D. It is well-known that the number of vertices is bounded in terms of k and D: ( The right-hand side of ( 5) is called the Moore bound.We call Γ a Moore graph if the equality in (5) holds.
For more detailed information about Moore graphs, see [14].

The block graph of an orthogonal array
In this section, we discuss the block graph of an orthogonal array and its properties.We then analyze the structure of the µ-graphs of an amply regular graph that is locally the block graph of an orthogonal array.
An orthogonal array, denoted as OA(m, n), is a structured m × n 2 array with entries chosen from the set Moreover, the spectrum of the block graph of OA(m, n) is Using Lemma 2. For 1 ≤ i ≤ n, define C i := S 1,i ∩ Γ(z).Applying Lemma 2.3, we find that for each i, the size of C i is either is a partition of the vertex set of M into m canonical cliques of size m.Note that, without loss of generality, we may permute the entries of OA(m, n) so that C i = ∅ for all i > m, and thus O consists of the entries {1, 2, . . ., m} and each vertex in M is incident to m canonical cliques.Therefore, we conclude that M is the block graph of OA(m, m).

The block graph of a Steiner system
In this section, we discuss the block graph of a Steiner system and its properties.We then analyze the structure of µ-graph of an amply regular graph that is locally the block graph of a Steiner system.A Steiner system S(2, m, n) is a 2-(n, m, 1) design, that is, a collection of m-sets taken from a set of size n, satisfying the property that every pair of elements from the n-set is contained in exactly one m-set.In this context, the elements of the n-set are referred to as points, and the m-sets are referred to as blocks of the system.
A straightforward counting argument reveals that the number of blocks in a Steiner system S(2, m, n) is given by n(n − 1)/m(m − 1), and each point occurs in exactly (n − 1)/(m − 1) blocks.
Moreover, the spectrum of this graph is The block graph of a Steiner system S(2, m, mn + m − n) with n ≥ m + 1 is called a Steiner graph S m (n).
By Lemma 4.1, the graph S m (n) is strongly regular with parameters Using ( 4) and ( 8), we can determine that the size of a maximum clique in the block graph of a Steiner system S(2, m, n) is (n − 1)/(m − 1).Constructing a Delsarte clique in the block graph of S(2, m, n) is straightforward: for each i ∈ {1, . . ., n}, we define S i as the set of all blocks in the design that contain the point i.These cliques S i are referred to as the canonical cliques of the block graph. .Note that none of the points of p 1 , p 2 , . . ., p m equals q.Consider the corresponding canonical clique S q of ∆.It follows that none of S p1 , S p2 , . . ., S pm equals S q .By Lemma 2.3, S q has m + 1 neighbors of y, denoted as B0 , B1 , . . ., Bm .
These blocks { Bi } m i=0 belong to B ′ , and each block Bi contains the point q, so we obtain m + 1 new vertices in M(x, y).This implies that the number of vertices of M(x, y) is at least (m 2 + 1) + (m + 1) = m 2 + m + 2.

However, this contradicts the fact that |B
Next, we consider the pair (P ′ , B ′ ).We will show that this pair forms a 2-(m 2 , m, 1) design, that is, each pair of points in P ′ is contained in exactly one block of B ′ .For each pair of distinct points p and q in P ′ , let B p,q denote the (unique) block in B that contains both p and q.We define B ′′ as the collection of blocks in B that contain pairs of points from P ′ , i.e., B ′′ = {B p,q ∈ B | p, q ∈ P ′ }.We assert that B ′ = B ′′ .First, it is clear that B ′ is a subset of B ′′ .Next, we determine the cardinality of B ′′ .To do this, consider the set 5 Proof of Theorem 1.2 In this section, we prove Theorem 1.2.To do this, we first recall and present some lemmas required for the proof without providing their proofs.
In particular, Γ has diameter 2 if and only if Γ is locally GQ(n − 1, n − 1).
Proof of Theorem 1.2.Let ∆ denote the local graph of Γ at a vertex x ∈ V (Γ).Since ∆ is strongly regular, we denote its parameters as (k, a 1 , λ, µ) and its eigenvalues as a 1 > r > −m, where a 1 is the intersection number of Γ.For notational convenience, we let n = r + m.Now, we consider each case: (i) ∆ is the block graph of an orthogonal array, and (ii) ∆ is the block graph of a Steiner system.
Case (i): Suppose ∆ is the block graph of an orthogonal array with k > m 2 .Assume that c 2 = m 2 ; we will derive a contradiction from this assumption.To this end, we consider the c 2 -graphs of Γ.By Lemma 3.2, every c 2 -graph of Γ is the block graph of OA(m, m), which is isomorphic to K m×m , where m ≥ 3.
We claim that m = 3.To show this, we consider the (triple) intersection number γ(Γ).We assert that γ(Γ) ≥ 2. Suppose that γ(Γ) = 1.Choose a vertex z at distance two from x, and then choose a vertex y that is adjacent to both x and z.Next, choose a Delsarte clique C of ∆ that contains y.Consider the subset We give a comment on the case when Γ has diameter D = 3 in Corollary 5. and In Corollary 5.6, the graph Γ with D = 3 corresponds to a Taylor graph.In this case, referring to the above discussion, it can yield the following stronger result.Proof.Throughout this proof, let ∆ denote a local graph of Γ with parameters (k, a 1 , λ, µ).Using (10), (11) along with µ = a 1 + rs from (3), the parameters (k, a 1 , λ, µ) are expressed in terms of m and n: First, we show that (i)-(iii) are equivalent.
Since ∆ is the local graph of Γ, it also has the parameter µ = m(n − m) from (12).From these two formulas for µ, it follows that n = 2m − 1.
(iii) ⇒ (i): Using c 2 = 2m(m − 1) and the parameters in (12), express the equation Simplify ( 14) to get the equation (m − 1)(n − 2m + 1) = 0. We note that m = 1 since −m is the smallest eigenvalue of ∆.Therefore, we have n = 2m − 1.Using this equation, we find that the parameters in ( 6) and ( 13) are equal.Therefore, ∆ has the same parameters as the block graph of OA(m, n).
Next, we show that (iv)-(vi) are equivalent.(ii) The halved 6-cube has intersection array {15, 6 Proof of Conjecture 1.1 In this section, we consider tight distance-regular graphs with classical parameters and prove Conjecture 1.1.
We begin by recalling the notion of classical parameters.For a non-zero integer b, we define Let Γ be a distance-regular graph with diameter D ≥ 3. We say Γ has classical parameters (D, b, α, β) We note that if Γ has classical parameters (D, b, α, β), then Γ is tight if and only if β = 1 + α D−1 where g(m) = 1 2 m 3 (2m − 3) + 1 m 2 (m − 1) + 2 − m − 1.We note that a 1 is the valency of ∆ and the diameter of ∆ is two.Thus, by (5) we have Applying the inequality (24) to the right-hand side of (25), we find For notational convenience, we set m := −s and n := r − s.By Sims' result (cf.[15, Theorem 5.1]), ∆ belongs to one of the following families: (i) complete multipartite graphs with classes of size m, (ii) block graphs of orthogonal arrays OA(m, n), (iii) block graphs of Steiner systems S(2, m, mn + m − n), (iv) finitely many further graphs.If Γ has classical parameters (D, b, α, β), then in case (i), Γ is the complete multipartite graph K (n+1),m with D = 2 [3, Proposition 1.1.5].For cases (ii) and (iii), when Γ has diameter D = 3, we must have b = 1.This restriction implies that Γ is one of the following three graphs: the Johnson graph J(6, 3), the halved 6-cube, or the Gosset graph E 7 (1); see [11, Section 7].Hence, our focus lies on cases where D ≥ 4 and b ≥ 2. Jurišić and Vidali posed the following conjecture: Conjecture 1.1 ([11, Conjecture 2]).Let Γ be a tight distance-regular graph with classical parameters (D, b, α, β), b ≥ 2, and diameter D ≥ 4. For a vertex u of Γ, the local graph of Γ at u is not the block graph of an orthogonal array or a Steiner system.

assume b ≥ 2 .
If a local graph of Γ is neither the block graph of an orthogonal array nor the block graph of a Steiner system, then the valency k (and hence diameter D) of Γ is bounded by a function of b.In Remark 7.3, we give an explicit bound in terms of b for the valency of Γ. From Theorem 1.3, it follows that the diameter of a tight distance-regular graph with classical parameters (D, b, α, β), D ≥ 3, and b ≥ 2, is bounded by a function of b; see Corollary 7.4.
We refer to the sequence {b 0 , b 1 , . . ., b D−1 ; c 1 , c 2 , . . ., c D } as the intersection array of Γ. Next, consider the following regularity properties of the graphs below: with equality if and only if every vertex outside C has exactly µ/m neighbors in C, where µ is the number of common neighbors of any two nonadjacent vertices.The upper bound for the size of a clique in (4) is called the Hoffman bound (or Delsarte bound ).If a clique C in a distance-regular graph attains the Hoffman bound, we call C a Delsarte clique.Lemma 2.3.Let Γ be an amply regular graph with parameters (ν, k, a 1 , c 2 ).Assume that Γ is locally strongly regular with parameters (k, a 1 , λ, µ).For a vertex x of Γ, let ∆(x) be the local graph of Γ at x with smallest eigenvalue −m.If C is a Delsarte clique of ∆(x), then a vertex at distance two from x either has 1 + µ/m neighbors in C or no neighbors in C. Proof.Let z be a vertex of Γ at distance two from x. Suppose that the Delsarte clique C has a neighbor of z.We will show that the number of neighbors of z in C is 1 + µ/m.Select a vertex y ∈ C that is adjacent to z.Consider the local graph ∆(y) in Γ, and note that ∆(y) is strongly regular with smallest eigenvalue −m.Now, consider the vertex subset C ′ = C ∪ {x} \ {y} in Γ. Obviously, C ′ forms a clique in ∆(y) of the same size as C. Hence, C ′ is a Delsarte clique of ∆(y).Since ∆(y) is strongly regular and z ∈ ∆(y) is not an element of C ′ , Lemma 2.2(ii) implies that z has µ/m neighbors in C ′ .Therefore, z has precisely 1 + µ/m neighbors in C.
{1, . . ., n}.It possesses the property that the columns of every 2 × n 2 subarray contain all possible n 2 pairs exactly once.In other words, for each pair of rows, every pair of elements from the set {1, . . ., n} appears precisely once in a column.The block graph of an orthogonal array is a graph whose vertices are the columns of OA(m, n), where two columns are adjacent if and only if there exists a row where they share the same entry.We note that the block graph of OA(m, n) is the same concept as the Latin square graph L m (n); see[4, Section 8.4].Lemma 3.1 (cf.[5, Theorem 5.5.1]).If OA(m, n) is an orthogonal array with n ≥ m, then its block graph is a strongly regular graph with parameters n 2 , m(n − 1), (m − 1)(m − 2) + n − 2, m(m − 1) .

Lemma 3 . 2 .
2(ii) and Lemma 3.1, we find that the maximum clique size in the block graph of OA(m, n) is n.Constructing a Delsarte clique in the block graph of OA(m, n) is straightforward: for each i ∈ {1, . . ., n}, consider the set S r,i , which consists of the columns of OA(m, n) containing the entry i in row r.Note that these sets naturally form cliques. Furthermore, as each element in {1, . . ., n} appears exactly n times in each row, the size of each clique S r,i is n for all i and r.These cliques are referred to as the canonical cliques of the block graph of OA(m, n).Let Γ be an amply regular graph with parameters (v, k, a 1 , c 2 ) and locally the block graph of an orthogonal array OA(m, n).If c 2 = m 2 , then every c 2 -graph of Γ is the block graph of an orthogonal array OA(m, m), and therefore, is complete m-partite.Proof.Observe that for each row r (1 ≤ r ≤ m) in OA(m, n), the set S r,i (1 ≤ i ≤ n) forms a canonical clique of size n.Fix a vertex x of Γ, and let ∆ denote the local graph of Γ at x.By construction of OA(m, n), ∆ consists of n (disjoint) canonical cliques S r,1 , S r,2 , . . ., S r,n(1 ≤ r ≤ m).Note that every vertex of ∆ belongs to exactly m canonical cliques.Fix a row r = 1 and observe that each S 1,i is a canonical clique in ∆.Select a vertex z of Γ at distance two from the vertex x.Let M = M(x, z) denote the c 2 -graph of Γ induced by the vertices x and z.Since c 2 = m 2 , M consists of m 2 columns obtained from the orthogonal array OA(m, n).Let O be the m × m 2 array consisting of the vertices of M. We claim that O has the structure of an orthogonal array OA(m, m), which implies that M is a block graph of OA(m, m).To prove this claim, we will show that in each row of O, precisely m distinct symbols occur, each exactly m times.In other words, it is equivalent to proving that M consists of m disjoint canonical cliques, with each vertex of M being incident to precisely m canonical cliques.

Lemma 4 . 2 .
Let Γ be an amply regular graph with parameters (v, k, a 1 , c 2 ) and locally the block grpah of a Steiner system S(2, m, n).If c 2 = m(m + 1), then every c 2 -graph of Γ is the block graph of a Steiner system S(2, m, m 2 ), and therefore, is complete (m + 1)-partite.Proof.For a vertex x of Γ, let ∆ denote the local graph of Γ at x, that is, the block graph of a Steiner system S(2, m, n).We denote its corresponding Steiner system by (P, B), where P denotes the set of points and B denotes the set of blocks.Observe that B is the vertex set of the local graph ∆, and furthermore, |P| = n and |B| = n(n − 1)/(m(m − 1)).Select a vertex y of Γ at distance two from the vertex x.Let M(x, y) denote the c 2 -graph of Γ induced by the vertices x and y.Let B ′ denote the vertex set of M(x, y).Observe that B ′ is a subset of B with cardinality m(m + 1) since c 2 = m(m + 1).We define the subset P ′ of P byP ′ = p ∈ P p ∈ B∈B ′ B .We claim that |P ′ | = m 2 .To prove this claim, let us consider a vertex B in M(x, y).Since B is a block in B ′ , we can write it as B = {p 1 , p 2 , . . ., p m }, where p i ∈ P ′ (1 ≤ i ≤ m).Now, for the point p 1 we consider the canonical clique S p1 of ∆.By Lemma 4.1 and (7), ∆ is strongly regular with µ = m 2 .Applying Lemma 2.3, we find that there are exactly m + 1 neighbors of y in S p1 , denoted as B = B 0 , B 1 , . . ., B m .Observe that each B i contains m − 1 points, excluding the common point p 1 .It implies that the total number of points in m i=0 B i is m 2 .Since each B i belongs to B ′ , all m 2 points are elements of P ′ .Therefore, we have |P ′ | ≥ m 2 .Suppose that |P ′ | > m 2 .Recall the vertices B = {p 1 , p 2 , . . ., p m }, B 1 , . . ., B m .For 1 ≤ i ≤ m, let S pi denote the canonical clique of ∆ corresponding to the point p i .By construction, the canonical cliques containing the vertex B are precisely S p1 , S p2 , . . ., S pm , and each S pi has precisely m neighbors of y besides B. Therefore, we obtain m 2 + 1 vertices of M(x, y).Now, choose a point q ∈ P ′ such that q / ∈ B i for all 0 ≤ i ≤ m.Such a point can be chosen because | m i=0 B i | = m 2 and by our assumption |P ′ | > m 2

k ≤ 1 + g(m) 2 .Remark 7 . 3 . 4 ( 1 + 4 2 + 1 .Corollary 7 . 4 .
Since m = 1 + b, the valency k of Γ is bounded by a function in b.Since the diameter of a distance-regular graph is bounded in terms of its valency (cf.[2, Section 4]), we conclude that the diameter of Γ is bounded by a function in b.The result follows.Referring to the proof of Theorem 1.3, the valency k is bounded by a function ϕ in the variable b, whereϕ(b) = 1 b) 3 (2b − 1) + 1 b(1 + b) 2 + 2 − 2b − Since b = m − 1,we also find that the diameter of Γ is bounded by a function in the variable m, where −m is the smallest eigenvalue of a local graph of Γ.Let Γ be a tight distance-regular graph with classical parameters (D, b, α, β), D ≥ 3, b ≥ 2.Then, the diameter of Γ is bounded by a function in b.Proof.Let k > θ 1 > . . .> θ D be eigenvalues of Γ. From Remark 6.2, Γ satisfies that b = b 1 /(1 + θ 1 ).By Theorem 6.3, a local graph of Γ is neither the block graph of an orthogonal array nor the block graph of a Steiner system.Therefore, by Theorem 1.3, the diameter of Γ is bounded by a function in b.The result follows.
Steiner systemS(2, m, n) is said to be symmetric if the number of points is equal to the number of blocks; otherwise, it is regarded as non-symmetric.The block graph of a Steiner system S(2, m, n) is defined as the graph whose vertices correspond to the blocks of the system.Two blocks are adjacent in this graph if and only if they intersect at exactly one point.
Lemma 4.1 (cf.[5, Theorem 5.3.1]).The block graph of a non-symmetric Steiner system S(2, m, n) is a strongly regular graph with parameters 6. Recall a Taylor graph, that is, a distance-regular graph with intersection array {k, c 2 , 1; 1, c 2 ; k} with c 2 < k − 1.We note that a nonbipartite distance-regular graph with diameter 3 is tight if and only if it is a Taylor graph [9, Theorem 3.2].Let Γ be a Taylor graph.Then Γ is locally strongly regular with parameters (k, a 1 , λ, µ) and eigenvalues a 1 > r > s.Since Γ is a Taylor graph, its local graphs satisfy is locally the Steiner graph S 2 (4).(iii)The Taylor graph from the Kneser graph K(6, 2) has intersection array {15, 8, 1; 1, 8, 15}.Its local graph is strongly regular with parameters (15, 6, 1, 3) with eigenvalues 6, 1, −3.Note that m = 3 and n = 4. Neither n = 2m − 1 nor n = 2m is satisfied.Therefore, the Taylor graph from K(6, 2) is not locally the block graph of an orthogonal array or a Steiner graph.