The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

by   Andries E. Brouwer, et al.
University of Delaware

We prove a conjecture by Van Dam and Sotirov on the smallest eigenvalue of (distance-j) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-j) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical P- and Q-polynomial association schemes.



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1 Introduction

This note looks at the smallest eigenvalue (or the second largest one in absolute value) of the graphs defined by the relations in classical - and -polynomial association schemes. The most well-known of these schemes is the Hamming scheme.

1.1 Hamming graphs

Let be integers. Let be a set of size . The Hamming scheme is the association scheme with vertex set , and as relation the Hamming distance. The relation graphs , where , have vertex set

, and two vectors of length

are adjacent when they differ in places.

The eigenmatrix of has entries , where

The eigenvalues of the graph are the numbers in column of , so are the numbers , . The graph is regular of degree , and this is the largest eigenvalue. Motivated by problems in semidefinite programming related to the max-cut of a graph, Van Dam & Sotirov [6] conjectured

Conjecture 1.1

Let and where is even when . Then the smallest eigenvalue of is .

Alon & Sudakov [1] proved this for and large and fixed. Dumer & Kapralova [13, Cor. 10], proved this for and all . Here we settle the full conjecture.

In most cases is not only the smallest eigenvalue, but also the second largest eigenvalue in absolute value. The only exception is the case , : the -matrix of is

and the eigenvalues of are , and .

The binary case was already settled by Dumer & Kapralova. We give a short and self-contained proof.

Theorem 1.2

([13, Cor. 10]) Let .

(i) If , then for all , .

(ii) If , then and for all , .

Corollary 1.3

Let and .

(i) One has for all , .

(ii) One has if and only if is even or .

The nonbinary case is settled here.

Theorem 1.4

Let and .

(i) One has for all , .

(ii) One has for all , unless .

1.2 Johnson graphs

The Johnson graphs are the graphs with as vertices the -subsets of a fixed -set, adjacent when they meet in a -set. W.l.o.g. we assume (since is isomorphic to ), and then these graphs are distance-regular of diameter . The eigenmatrix has entries , where

For , the distance- graphs of the Johnson graph are the graphs with the same vertex set as , where two vertices are adjacent when they have distance in , that is, when they meet in a -set. For this graph is known as the Kneser graph . Motivated by problems in semidefinite programming related to the max-cut of a graph, Karloff [19] conjectured in 1999 the following:

Conjecture 1.5

Let and . Then the smallest eigenvalue of is .

Here we prove this conjecture (Corollary 3.11), and more generally determine precisely in which cases is the smallest eigenvalue of (Theorem 3.10).

1.3 Graphs with classical parameters

For general information on distance-regular graphs, see [2]. In [2, §6.1], graphs with classical parameters are defined as distance regular graphs of diameter with parameters given by certain expressions in (see Section 4 for details).

The concept of graphs with classical parameters unifies a number of families of distance-regular graphs, such as the Hamming graphs, Johnson graphs, Grassmann graphs, dual polar graphs, bilinear forms graphs, etc.

1 0 Hamming graphs
1 1 Johnson graphs ,
Grassmann graphs ,
0 dual polar graphs ,
bilinear forms graph
alternating forms graphs
Hermitian forms graphs

Below we give the asymptotic behavior of the eigenmatrix of these graphs when are fixed and tends to infinity (Theorem 4.5). We also give a simple explicit expression for the eigenvalues , that perhaps has not been noticed before (Proposition 4.1).

Subsequently, we investigate each of the individual families, and determine smallest and second largest eigenvalues and/or other properties of the eigenvalues. Main results are Theorem 5.8 for the Grassmann graphs, Corollary 6.5 for the dual polar graphs, Theorem 7.5 for the bilinear forms graphs, Theorem 8.3 for the alternating forms graphs, and Theorem 9.5 for the Hermitian forms graphs.

2 The Hamming case

We prove the stated results for the Hamming graphs.

2.1 Identities

We collect some (well-known) identities used in the sequel.

The defining equation gives as a polynomial in of degree with leading coefficient . We give three expressions.

(see Delsarte [7, p. 39], and [8, (15)]).

One has the symmetry

In particular, and have the same sign.

There is also the symmetry

Proposition 2.1

Let . Then


2.2 Proofs

The occurrence of in Conjecture 1.1 is explained by the following proposition. Where it refers to or , it is assumed that or .

Proposition 2.2

Let and .

(i) if and only if .

(ii) if and only if or .

(ii) if and only if .

(iii) if and only if or .

(iv) Let . Then .

Proof. (i) Since has the same sign as , this follows from .

(ii) Since , the claim says that precisely for the two specified values of . But this condition is quadratic in , and is up to a constant factor .

(ii) Clear from (ii), since has positive leading coefficient.

(iii) The condition is equivalent to . Again it is quadratic in . Up to a constant factor it is .

(iv) We want to show that . Since this is the pair of conditions and .

The former is up to a positive constant factor equivalent to .

For the latter it suffices to see that . Up to a positive constant factor this is equivalent to . ∎

If , then , and .

In order to prove Theorems 1.2 and 1.4, we need three lemmas.

Lemma 2.3


Proof. Since unless , we have

. ∎

Lemma 2.4

Let and . If , then .

Proof. Apply Proposition 2.1. Put . One has . If and , then , and the conclusion follows if . Now , so we need , and that was one of the hypotheses. ∎

For the scheme is imprimitive, and the graphs

are bipartite for odd

, and disconnected for even . One has the additional symmetry .

Lemma 2.5

Let and . Then .

We prove Lemma 2.5 in the proof of Theorem 1.2.


1.2 ([13, Cor. 10]) Let .

(i) If , then for all , .

(ii) If , then and for all , .

Proof. (i) By the symmetry we may suppose .

We prove Lemma 2.5 and part (i) of the theorem simultaneously. Since , and , both statements are equivalent for all .

Prove the statement of the lemma by induction of . The conclusion follows by adding the inequalities for and , using that , except possibly when or . If , the claim is that , which is true. Instead of treating we use symmetry and take and prove the statement in (i) by induction on , using Proposition 2.2 (iv) and Lemma 2.4. Here we may suppose by the symmetry .

(ii) By symmetry, when and is odd. The 3-term recurrence reduces to for odd , so that and decreases with increasing . ∎


1.3 Let and .

(i) One has for all , .

(ii) One has if and only if is even or .

Proof. Since , part (i) follows from part (i) of the theorem, and part (ii) from . ∎

Next, consider the nonbinary case.


1.4 Let and .

(i) One has for all , .

(ii) One has for all , unless .

If then , for , and .

Proof. Since (and is the largest eigenvalue), part (i) follows from part (ii). The case was handled in Proposition 2.2, so we may assume .

For one has , and the statement is true.

For one has and . To show the claim it suffices to show that , and this follows from , unless , in which case still holds, unless .

So, we may assume . This implies that .

If then we can apply Lemma 2.4 (and induction on ) to conclude that , and we are done. So, assume .

One has , where the last factor is negative. From Lemma 2.3 we see that when .

Using and and and we see that it suffices to have , so suffices. The finitely many with can be checked separately. ∎

2.3 Large

Proposition 2.6

For fixed , let be sufficiently large. Then is positive for , and has sign for . For each , the smallest eigenvalue of is .

Proof. We have . When tends to infinity, and are fixed, this sum is dominated by its first nonzero term. So if , and if . ∎

How large is ‘sufficiently large’? The value is the unique smallest eigenvalue of for all when .

2 3 4 5 6 7 8 9 10 12 14 16 18 20 30 40 50 60 100
2 3 4 5 7 9 12 15 18 26 35 45 57 70 156 277 433 623 1730
Lemma 2.7

Suppose . Then

(i) for ,

(ii) ,

(iii) for .

Proof. If , then the terms decrease monotonically when increases, so that the sign of is that of the first nonzero term and the difference between and the first nonzero term is smaller than the next term.

For we have
, so that

and . So, it suffices to see
. This holds for , and for , , and for we can drop the factor , and the conclusion holds. ∎

2.4 Coincidences

A general matrix in the Bose-Mesner algebra of a -class association scheme (see [2, Chapter 2] for a definition) will have distinct eigenvalues, and generate , in the sense that each element of is a polynomial of degree at most in . Cases where some relation matrix has fewer eigenvalues (and hence generates a proper subalgebra) are of interest.

Look at the Hamming scheme. For , the main expected coincidences between the for fixed and are given in the following lemma.

Lemma 2.8

Let .

(i) If is even, then .

(ii) If , then for all odd .

(iii) If , then for .

(iv) If , then for all .

Proof. We only have to show (iii), and this follows from Proposition 2.2 (ii), and the 3-term recurrence given in Proposition (2.1). ∎

If , then also and we have a further coincidence (when is odd and ). Integral zeros of Krawtchouk polynomials play a role e.g. in the study of the existence of perfect codes or the invertibility of Radon transforms, and have been studied by many authors, cf. [4, 12, 15, 16, 20, 27, 28]. For there are infinite families. For fixed there are zeros only for finitely many . Recall that if and only if .

Lemma 2.9

([4, Th. 4.6] and [12, Ex. 10]) Let , , .

(i) if and only if .

(ii) if and only if , for some integral .

(iii) if and only if , for some integral .

(iv) if .

The family given last has . There are also infinite families with for ([15]).

For arbitrary there are fewer obvious coincidences.

Lemma 2.10

Let .

(i) If , then for all .

(ii) If , then if and only if .

(iii) If , then .

Proof. (i) The matrix only has the single eigenvalue 1.

(ii) Note that is quadratic in .

(iii) This is what Proposition 2.2 (ii) says. ∎

We look for cases where some has fewer distinct eigenvalues than expected (given the above lemmas), or just has few distinct eigenvalues. Below we list cases where has precisely distinct eigenvalues, while , for .

Conjecture 2.11

If is connected, it has more than distinct eigenvalues.

2.4.1 Three distinct eigenvalues

If has three distinct eigenvalues, it is strongly regular, or (in case and even) it is the disjoint union of two isomorphic connected components, both strongly regular.

For example, the -matrix of was given above,

and is strongly regular with parameters and spectrum .

For one gets

and the graph has two connected components, both isomorphic to the graph on the 64 binary vectors of length 7 and even weight, adjacent when they differ in 4 places. The graph is strongly regular with parameters and spectrum .

Cases with three eigenvalues (the connected graphs among these are strongly regular—we give the standard parameters ):

4 2 2 2 copies of
5 2 2 2 copies of the Clebsch graph
5 2 4 2 copies of the complement of the Clebsch graph
7 2 4 2 copies of
4 3 2
4 3 3 :
3 4 2 :

More generally, if we take the Hamming scheme with , and call two distinct vertices adjacent if their distance is even, we obtain a strongly regular graph (as was observed in [18, Case III]), namely the graph , where the sign is . Indeed, the weight of a quaternary digit is given by the (elliptic) binary quadratic form . For this graph is .

2.4.2 Four/five/six distinct eigenvalues

In Table 1 below we list further cases in which has fewer than distinct eigenvalues.

comment 5 2 3 L2.8 (iii) 6 2 2,4 L2.8 (i) 7 2 2,6 L2.8 (i) 8 2 4 L2.8 (i),(ii) 11 2 6 L2.8 (i),(iii) 5 3 3 5 4 2 L2.10 (ii) 4 6 2 L2.10 (ii)
Four eigenvalues
comment 6 2 3 L2.8 (ii) 8 2 2,6 L2.8 (i) 9 2 2,4,6,8 L2.8 (i) 10 2 4 L2.8 (i) 10 2 8 L2.8 (i) 11 2 4 L2.8 (i) 11 2 8 L2.8 (i) 12 2 6 L2.8 (i),(ii) 15 2 8 L2.8 (i),(iii) 7 3 2 L2.10 (ii) 7 3 5 L2.10 (iii) 5 4 3 5 4 4 L2.10 (iii) 6 5 2 L2.10 (ii) 5 6 3 5 8 2 L2.10 (ii)
Five eigenvalues
comment 7 2 3 9 2 5 L2.8 (iii) 10 2 2,6 L2.8 (i) 11 2 2,10 L2.8 (i) 12 2 4 L2.8 (i) 12 2 8 L2.8 (i) 16 2 8 L2.8 (i),(ii) 19 2 10 L2.8 (i),(iii) 7 3 3 7 3 6 7 4 2 L2.10 (ii) 7 4 4 6 5 3 6 5 5 L2.10 (iii) 7 6 2 L2.10 (ii) 6 10 2 L2.10 (ii) 6 10 3
Six eigenvalues
Table 1: Cases where has fewer than distinct eigenvalues

For example, the eigenmatrix of is