A Cheeger inequality of a distance regular graph
using the Green’s function
Gil Chun Kim
Department of Mathematics
Dong-A University, Busan, Korea
1. Some definitions and facts
2. Green’s function
3. A Cheeger inequality of a distance regular graph
4. Explicit expression of the krawtchouk polynomial
𝑆 be a subset of set of vertices in a graph Γ = (𝑉, 𝐸), where 𝑉 is a set
of vertices of Γ and 𝐸 is a set of edges of Γ.
. Let
(𝑎) The edge boundary of 𝑆, denoted by 𝜕𝑆 is defined as follows:
𝜕𝑆 = 𝑥, 𝑦 ∈ 𝐸 Γ
𝑥 ∈ 𝑆 𝑎𝑛𝑑 𝑦 ∈ 𝑉 − 𝑆},
where 𝐸(Γ) is a edge set of Γ.
(𝑏) If 𝑆 ≠ ∅, then the volume of 𝑆, denoted by 𝑣𝑜𝑙(𝑆) is defined as follows:
𝑣𝑜𝑙 𝑆 = 𝑢∈𝑆 𝑘𝑢
where 𝑘𝑢 is a valency of 𝑢 in Γ. The volume of Γ is denoted by
𝑣𝑜𝑙 Γ = 𝑢 𝑘𝑢
(𝑐) The 𝐶ℎ𝑒𝑒𝑔𝑒𝑟 𝑟𝑎𝑡𝑖𝑜 of 𝑆, denoted by ℎ𝑆 is defined as
|𝜕𝑆|
ℎ𝑆 =
min{𝑣𝑜𝑙 𝑆 , 𝑣𝑜𝑙 Γ − 𝑣𝑜𝑙 𝑆 }
(𝑑) The 𝐶ℎ𝑒𝑒𝑔𝑒𝑟 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 of Γ, denoted by ℎΓ is defined as
ℎΓ = min ℎ𝑆
𝑆⊂𝑉
In graph theory, the Cheeger constant is an important geometric meaning. The
Cheeger constant give to us answer of the problem of separating the graph into two
large components by making a small edge-cut. The Cheeger constant of a connected
graph is strictly positive.
If the Cheeger constant is “small” but postive, then there are two large sets of
vertices with “few” edges between them.
On the other hand if the Cheeger constant is “large”, then there are two sets of
vertices with “many” edges between those two subsets.
We are interested in finding bounds on the Cheeger constants of graphs.
The Hamming graph is the graph that describes the distance – 1 relation
in the Hamming scheme 𝐻(𝑑, 𝑞)
The Hamming graph is a special class of graphs used in several branches
of Mathematics and Computer Science. The Hamming graphs are
interesting in connection with error-correcting codes and association
schemes.
The Cheeger constant of the Hamming graphs Γ(𝑑, 𝑞)
𝑞+1
2
ℎΓ =
𝑑(𝑞 − 1)
Hamming graph Γ(3,2)
𝑆 = 000, 100, 010, 110
𝜕𝑆 = 4, 𝑣𝑜𝑙 𝑆 = 12
4
1
ℎΓ =
=
12
3
The Johnson graph is also an interesting class of graphs.
A vertex set 𝑉 of 𝛤 𝑛, 𝑑 consists of all binary vectors of length 𝑛 with
Hamming weight 𝑑, such that 𝑉 = 𝑛𝑑 . The Johnson graph is the graph
that describes the distance – 1 relation in the Johnson scheme 𝐽(𝑛, 𝑑).
Two vertices are adjacent if they differ in two coordinates.
The Cheeger constant of the Johnson graphs Γ(𝑛, 𝑑)
ℎΓ ≤
ℎΓ ≤
(𝑛−𝑑+1)2 𝐴2
2𝑑(𝑛−𝑑) 𝑛
𝑑
𝐴2 +
4
𝑛
𝑑
2
𝐴2
2
, if 𝑛 is even and 𝑑 is odd.
, if 𝑛 is even and 𝑑 is even.
ℎΓ ≤
ℎΓ ≤
(𝑛+1)𝐴2 +(𝑑−1)
4𝑑
𝑛
𝑑
2
𝐴2
2
(𝑛+1)𝐴2 +(𝑛−𝑑−1)
4(𝑛−𝑑)
𝑛
𝑑
2
, if 𝑛 is odd and 𝑑 is odd.
𝐴2
2
, if 𝑛 is odd and 𝑑 is even.
. 𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒓𝒆𝒈𝒖𝒍𝒂𝒓 𝒈𝒓𝒂𝒑𝒉
Let Γ be a connected graph. For a vertex 𝑥 ∈ 𝑉 , define Γ𝑖 𝑥 to be the set of
vertices which are at distance precisely 𝑖 from 𝑥 (0 ≤ 𝑖 ≤ 𝑑), where 𝑑 =
max 𝑑 𝑥, 𝑦
𝑥, 𝑦 ∈ 𝑉}. A connected graph Γ with diameter 𝑑 is called
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 if there are integers 𝑏𝑖 , 𝑐𝑖+1 0 ≤ 𝑖 ≤ 𝑑 − 1 such that for
any two vertices 𝑥, 𝑦 with 𝑑 𝑥, 𝑦 = 𝑖, there are precisely 𝑐𝑖 neighbors of 𝑦 in
Γ𝑖−1 𝑥 and 𝑏𝑖 neighbors of 𝑦 in Γ𝑖+1 𝑥 .
. 𝑨𝒔𝒔𝒐𝒄𝒊𝒂𝒕𝒊𝒐𝒏 𝒔𝒄𝒉𝒆𝒎𝒆
Let 𝑋 be a nonempty finite set and 𝑅 = {𝑅0 , 𝑅1 , ⋯ , 𝑅𝑑 } be a family of relations
defined on 𝑋. Let 𝑣 be the order of 𝑋. Then adjacency matrix 𝐴𝑖 of 𝑅𝑖 (𝑖 =
0,1, ⋯ , 𝑑) is the 𝑣 × 𝑣 matrix defined by
1 , if (𝑥, 𝑦) ∈ 𝑅𝑖
(𝐴𝑖 )𝑥,𝑦 = {
0 , otherwise.
We say that the pair 𝔛 = (𝑋, 𝑅) is a symmetric association scheme with 𝑑 classes
if it satisfies the following conditions.
1 𝐴𝑖 is symmetric,
(2) 𝑑𝑖=0 𝐴𝑖 = 𝐽 (the all 1’s matrix),
(3) 𝐴0 = 𝐼,
(4) 𝐴𝑖 𝐴𝑗 = 𝑑𝑘=0 𝑝𝑖,𝑗 𝑘 𝐴𝑘 (𝑖, 𝑗 = 0,1,2, ⋯ , 𝑑).
Let 𝒜 be the algebra spanned by the adjacency matrices 𝐴0 , 𝐴1 , ⋯ , 𝐴𝑑 . Then
𝒜 is called the Bose Mesner algebra of 𝔛 and 𝒜 has two distinguished bases
𝐴𝑖 and 𝐸𝑖 , where the latter consist of primitive idempotent matrices. For
𝐴𝑖 and 𝐸𝑖 , we express one in terms of the other:
1
𝑑
𝐴𝑗 = 𝑑𝑖=0 𝑝𝑗 (𝑖) 𝐸𝑖 , 𝐸𝑗 =
𝑖=0 𝑞𝑗 (𝑖) 𝐴𝑖
for 𝑗 = 0,1, ⋯ , 𝑑.
|𝑋|
The (𝑑 + 1) × 𝑑 + 1 matrix 𝑃 = (𝑝𝑗 𝑖 ) (respectively, Q= (𝑞𝑗 𝑖 ) ) is called the
first eigenmatrix (respectively, the second eigenmatrix) of the association scheme.
Then 𝑃 = (𝑝𝑗 𝑖 ) and Q= (𝑞𝑗 𝑖 ) satisfy
𝑞𝑗 𝑖
𝑚𝑗
=
𝑝𝑖 𝑗
𝑘𝑖
, where 𝑚𝑗 = 𝑟𝑎𝑛𝑘(𝐸𝑗 ), 𝑘𝑖 is the
valency of 𝐴𝑖 ,and 𝑝𝑖 𝑗 is the complex conjugate of 𝑝𝑖 𝑗 .
The adjacency matrices 𝐴𝑖 satisfy
𝐴𝑖 𝐴𝑖 = 𝑑𝑘=0 𝑝𝑖,𝑗 𝑘 𝐴𝑘
for all 𝑖, 𝑗, where for (𝑥, 𝑦) ∈ 𝑅𝑘 , 𝑝𝑖,𝑗 𝑘 is the number of 𝑧 ∈ 𝑋 such that (𝑥, 𝑧) ∈ 𝑅𝑖
and 𝑧, 𝑦 ∈ 𝑅𝑗 . The non-negative integers 𝑝𝑖,𝑗 𝑘 are called the intersection numbers
of 𝔛.
Let 𝐵𝑖 be a matrix with (𝑗, 𝑘)-entries 𝑝𝑖,𝑗 𝑘 ,and let ℬ be a algebra spanned by 𝐵0 ,
𝐵1 , ⋯ , 𝐵𝑑 . Then 𝐵𝑖 is called the 𝑖-th intersection matrix of 𝔛 and ℬ is called the
intersection algebra of 𝔛 . In fact, the Bose-Mesner algebra 𝒜 of 𝔛 is isomorphic
to ℬ by the map 𝐴𝑖 → 𝐵𝑖 .
Let 𝔛 = 𝑋, 𝑅𝑖 (𝑖 = 0,1, ⋯ , 𝑑) be a symmetric association scheme, and let Γ
be the graph whose set and edge set are 𝑋 and 𝑅1 respectively. Then, it is known
that the following are equivalent.
(𝑎) Γ is distance regular graph.
(𝑏) 𝔛 is a 𝑃–polynomial scheme with respect to 𝑅0 , 𝑅1 , ⋯ , 𝑅𝑑 , that is
𝑣𝑖 𝐴1 (𝑖 = 0,1, ⋯ , 𝑑) for some polynomial 𝑣𝑖 𝑥 of degree 𝑖.
(𝑐) The first eigenmatrix 𝐏 = (𝑝𝑗 𝑖 ) satisfies 𝑝𝑗 𝑖 = 𝑣𝑖 (𝜃𝑗 ) for some
polynomials 𝑣𝑖 𝑥 of degree 𝑖, where 𝜃𝑗 = 𝑝1 𝑗 (𝑖, 𝑗 = 0,1, ⋯ , 𝑑).
(𝑑) The first intersection matrix 𝐵1 is a tridiagonal matrix with non-zero
off diagonal entries,
𝐴𝑖 =
𝑪𝒉𝒆𝒆𝒈𝒆𝒓 𝒊𝒏𝒆𝒒𝒖𝒂𝒍𝒊𝒕𝒚
Let Γ be a graph of order 𝑣, and let λ1 be the smallest positive eigenvalue of the
Laplacian of Γ. Then there are Cheeger inequalities as follows:
(𝑎) [1]
ℎΓ 2
2
≤ 𝜆1 ≤ 2ℎΓ ⟹ ℎΓ ≤
2𝜆1 ;
[1] J. Dodziuk, W.S. Kendall, Combinatorial Laplacians and isoperimetric inequality,
in: K.D. Elworthy (Ed.), From Local Times to Global Geometry, control and Physics,
Research Notes in Mathematics Series, Vol. 150, Pitman, London, 1986, pp.68-74.
𝑏
2 For 𝑣 ≥ 4, ℎΓ ≤
𝜆1 2 − 𝜆1 .
[2] J.Tan, On cheeger inequalities of a graph, Discrete Math. 269(2003) 315-323.
Our Cheeger inequality provides an upper bound of the cheeger constant, which
yields an improvement of the bound (𝑎), (𝑏).
. [1] Let Γ be a distance regular graph. Then its edge-connectivity
equals its valency 𝑘, and the only disconnecting sets of edges are the sets of
𝑘 edges incident with a single vertex.
[1] A.Brouwer, W.Haemers, Eigenvalues and perfect matchings, Linear Algebra and its
Applications. 395 (2005). 155-162.
. [2] Let Γ be a non-complete distance regular graph of valency 𝑘 >
2. Then the vertex-connectivity 𝜅 Γ equals 𝑘, and the only disconnecting sets of
vertices of size not more than 𝑘 are the point neighbourhoods.
[2] A. Brouwer, J. Koolen, the vertex-connectivity of a distance regular graph, European
Journal of Combinatorics. Vol. 30. No. 3. (2009). 668-673.
. [3] Let Γ = (𝑉, 𝐸) be a simple graph with the vertex-connectivity
𝜅 Γ and the edge-connectivity 𝜆(Γ). Then
2𝜅 Γ
|𝑉|
≤
2𝜆(Γ)
|𝑉|
|𝜕𝑆|
|𝑆|
|𝑉|
.
2
≤ 𝑖𝑛𝑓
where 𝑆 is a subset of 𝑉 with |𝑆| ≤
≤ 𝜅 Γ ≤ 𝜆(Γ),
[3] G. Oshikiri, Cheeger constant and connectivity of graphs, Interdisciplinary Information
Sciences, 8 (2) (2002), 147-150.
From Propositions 1,2 and 3 we see that, for a distance regular graph, there are close
connections between the Cheeger constant and vertex and edge connectivity.
Also, we obtain an inequality
2
≤ ℎΓ ≤ 1.
|𝑉|
We are thus interested in finding optimal bounds of ℎΓ for distance regular graph.
For our Cheeger inequality, we use the Green's function, which is defined as the
inverse of the 𝛽 -Laplacian.
Green’s function
Define a transition probability matrix 𝑃 over 𝔛 by 𝑃 =
1
𝐴 , where
𝑘1 1
𝑘1 is the
valency of 𝐴1 . For a function 𝑓 ∶ 𝑋 → ℝ, we define a 𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟 ∆ by
∆𝑓(𝑥) =
Then, we have ∆= 𝐼 − 𝑃 = 𝐼 −
1
𝑘1
𝑓 𝑥 −𝑓 𝑦 .
𝑦~𝑥
1
𝐴
𝑘1 1
and ∆ is a matrix representation of the
Laplacian ℒ. For 𝑗 = 0,1, ⋯ , 𝑑 and orthogonal eigenfunction 𝜙𝑗 ∗ , we have
ℒ=
where 𝜆𝑗 is an eigenvalue of ℒ.
𝑑
𝑗=0 𝜆𝑗
𝜙𝑗 ∗ 𝜙𝑗 ,
Let ℒ𝛽 be the 𝛽–normalized Laplacian by 𝛽𝐼 + ℒ. For 𝛽 > 0, let a Green’s
function 𝒢𝛽 denote the symmetric matrix satisfying ℒ𝛽 𝒢𝛽 = 𝐼. Then we have
𝒢𝛽 =
1
𝑑
𝑗=0 𝛽+𝜆
𝑗
𝜙𝑗 ∗ 𝜙𝑗 ,
For 𝛽 > 0 we have 𝒢𝛽 (𝛽𝐼 + 𝐼 − 𝑃) = 𝐼. Thus, this implies that
ℒ𝛽 =(𝛽 + 1)𝐼 − 𝑃 =
𝑑
𝑗=0(𝛽
+1−
1
𝑝
𝑘1 1
𝑗 )𝐸𝑗 .
Hence, a Green’s function 𝒢𝛽 can be expressed by
𝑘1
𝑑
𝑗=0( 𝛽+1 𝑘 −𝑝 𝑗
1
1
𝒢𝛽 =
Since 𝐸𝑗 =
1
|𝑋|
) 𝐸𝑗 .
𝑞𝑗 𝑖 𝐴𝑖 , 𝒢𝛽 is a linear combination of adjacency matrices 𝐴𝑖
as follows:
𝒢𝛽 = 𝑟0
where 𝑟𝑖
(𝛽)
=
1
𝑋
1
𝛽
(𝛽)
( + 𝑞1 𝑖
𝐴0 + 𝑟1
1
𝛽+𝜆1
(𝛽)
𝐴1 + ⋯ + 𝑟𝑑
+ ⋯ + 𝑞𝑑 (𝑖)
1
)
𝛽+𝜆𝑑
𝛽
𝐴𝑑 ,
(𝑖 = 0,1, ⋯ , 𝑑)
Let 𝐿 be a ( 𝑋 − 1) × |𝑋| matrix obtained by the removal of the first row of
1
ℒ𝛽 =(𝛽 + 1)𝐼 − 𝐴1 .
𝑘1
Since the Bose-Mesner algebra 𝒜 is isomorphic to the intersection algebra 𝔅 of
1
1
𝔛 then that isomorphism of algebra takes (𝛽 + 1)𝐼 − 𝐴1 to (𝛽 + 1)𝐼 − 𝐵1 .
Let 𝐿′ = −𝑘1 ( 𝛽 + 1 𝐼 −
matrix. Let 𝐿𝑠𝑢𝑏
(𝛽)
𝑘1
− 𝑘1 𝛽 + 1 𝐼, which is a 𝑑 + 1 × 𝑑 + 1
be a matrix obtained by the removal of the first row of 𝐿′ .
Then, we obtain 𝐿𝑠𝑢𝑏
𝐿𝑠𝑢𝑏
1
𝐵 )= 𝐵1
𝑘1 1
𝑘1
(𝛽)
as follows:
(𝛽)
where 𝑠𝑖 = 𝑎𝑖 − 𝑘1 (𝛽 + 1) for 𝑖 = 1,2, ⋯ , 𝑑 and 𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 are the same entries as
in 𝐵1 .
. For 𝛽 > 0, let 𝒢𝛽 be a Green’s function for a 𝑃–polynomial
scheme. Then we have the following:
(𝑎) A Green’s function can be expressed as
𝒢𝛽 = 𝑡𝑢0
(𝛽)
𝐴0 + 𝑡𝑢1
for some nonzero 𝑡 ∈ ℝ, where (𝑢0
basis
(𝛽)
(𝛽)
(𝛽)
𝐴1 + ⋯ + 𝑡𝑢𝑑
, 𝑢1
(𝛽)
𝛽
, ⋯ , 𝑢𝑑
(𝛽)
(𝛽)
of the nullspace 𝒩(𝐿(𝛽)
) of(𝛽)
𝐿𝑠𝑢𝑏
with 𝑢𝑑𝛽 =1 1.
𝑠𝑢𝑏
(𝑏) For 𝛽 > 0, 𝑘0 𝑟0
+ 𝑘1 𝑟1
+ ⋯ + 𝑘𝑑 𝑟𝑑
= ,
where 𝑘𝑗 is the valency of 𝐴𝑗 for 𝑗 = 0, 1, ⋯ , 𝑑.
(𝑐) 𝑟0
(𝛽)
> 𝑟1
(𝛽)
(𝑑) As 𝛽 → 0+ , 𝑟0
> ⋯ > 𝑟𝑑
(𝛽)
≈ 𝑟1
𝛽
(𝛽)
> 0.
≈ ⋯ ≈ 𝑟𝑑
𝛽
.
𝛽
𝐴𝑑
(𝛽)
) is the unique
For 𝛽 > 0, let 𝒢𝛽 be a Green’s function of the distance regular graph of order 𝑣
We denote a subset 𝑪𝛽 of 0,1,2, ⋯ , 𝑑 by
𝑪𝛽 ≔{ 𝑖 |
1
𝛽
− 𝑣𝑟𝑖
𝛽
> 0}.
It is clear that 𝑪𝛽 is a non-empty set.
When 𝛽 is sufficiently close to 0+ , we consider a set
𝑪𝛽 ′ = 𝑖
Since
𝛽𝑣𝑟𝑖
𝛽𝑣𝑟𝑖
𝛽
𝛽
𝛽 + 𝜆1 < 𝜆1
𝛽
𝛽+𝜆𝑑
= 1 + 𝑞1 𝑖
lim+ 𝛽𝑣𝑟𝑖
𝛽→0
Thus, we have
𝛽
1
𝛽
⇒
𝛽𝑣𝑟𝑖
𝛽𝑣𝑟𝑖
𝛽
𝛽 + 𝜆1 < 𝜆1 }.
𝛽
+ ⋯ + 𝑞𝑑 (𝑖)
1
𝛽
< 𝜆1 ( − 𝑣𝑟𝑖
𝛽
𝛽+𝜆𝑑
𝛽
)
,
= 1.
− 𝑣𝑟𝑖
𝛽
> 0. That is, 𝑪𝛽 ′ is a subset of 𝑪𝛽 .
and
. For 𝛽 > 0, let Γ be a distance regular graph of order 𝑣 and let
(𝛽)
(𝛽)
𝒢𝛽 = 𝑟0
𝐴0 + 𝑟1
for 𝑖 ∈ 𝑪𝛽 ′ , we have
(a)
𝛽 2 𝑣𝑟𝑖
1−𝛽𝑣𝑟𝑖
𝛽
𝐴1 + ⋯ + 𝑟𝑑
𝛽
𝐴𝑑 be a Green’s function of Γ. Then,
is decreasing in 𝑖 ∈ 𝑪𝛽 ′
𝛽
′
(𝑏) For 𝑖 ∈ 𝑪𝛽 , lim+
𝛽 2 𝑣𝑟𝑖
𝛽→0 1−𝛽𝑣𝑟
𝑖
𝛽
𝛽
= 𝛼𝑖 , where 𝛼𝑖 =
Moreover, for some 𝑖 ∈ 𝑪𝛽 ′ , 𝛼𝑖 < 𝜆1 .
(𝑐)
𝛽 2 𝑣𝑟𝑖
1−𝛽𝑣𝑟𝑖
𝛽
𝛽
is decreasing in 𝛽 > 0.
1
−𝑞1 𝑖
1
−
𝜆1
⋯ −𝑞𝑑 𝑖
1
𝜆𝑑
.
- A Cheeger inequality of a distance regular graph . Let Γ be a distance regular graph of diameter 𝑑 and let 𝜆1 be the
smallest positive eigenvalue of the Laplacian. Then
𝜆 1 ℎΓ < 𝛼𝑑 < 𝛼𝑑−1 < ⋯ < 𝜆1 .
. Let 𝑆 be a subset of vertices of Γ with 𝑣𝑜𝑙(𝑆) ≤ 𝑣𝑜𝑙(Γ)/2.
Let
1−𝛽𝑣𝑟𝑖
𝛽𝑣𝑟𝑖
𝛽
=
𝛽
1
.
𝑐
Then we have lim+
𝛽→0
number such that 𝐴 <
𝑣𝑜𝑙(𝑆)𝛽𝑣𝑟𝑖
1−𝛽𝑣𝑟𝑖
𝛽
𝛽
1−𝛽𝑣𝑟𝑖
𝛽𝑣𝑟𝑖
𝛽
= 0+ . Let 𝐴 be a positive
𝛽
. Then
𝛽
𝛽
1 − 𝛽𝑣𝑟𝑖𝛽
𝐴 < 𝑣𝑜𝑙(𝑆)𝛽𝑣𝑟𝑖
⇒
1−𝛽𝑣𝑟𝑖
′
+
Let 𝛽 =
𝛽 . Since lim+ 𝛽 = 0 , we have
′
<
𝛽→0
𝛽𝑣𝑟𝑖
𝐴
𝑣𝑜𝑙(𝑆)
⇒
𝐴
𝑣𝑜𝑙(𝑆)
<
𝐴
1
𝑣𝑜𝑙(𝑆)( ′ )
𝛽
𝛽 ′ 𝑣𝑟𝑖
𝛽′
1−𝛽 ′ 𝑣𝑟𝑖
<
⇒
𝛽′
2
𝛽 ′ 𝑣𝑟𝑖
1−𝛽 ′ 𝑣𝑟𝑖
𝐴
𝑣𝑜𝑙(𝑆)
𝛽′
𝛽′
⇒
<
𝐴
𝑣𝑜𝑙 𝑆 𝑐
2
𝛽 ′ 𝑣𝑟𝑖
1−𝛽 ′ 𝑣𝑟𝑖
<
𝛽′
𝛽′
2
𝛽 ′ 𝑣𝑟𝑖
1−𝛽 ′ 𝑣𝑟𝑖
𝛽𝑣𝑟𝑖
1−𝛽𝑣𝑟𝑖
1
𝛽′
𝛽′
𝛽′
𝛽
𝛽
.
Since 𝜕𝑆 𝜆1 < 𝑣𝑜𝑙(𝑆) and 𝐴 <
𝜕𝑆 𝜆1 𝑐. Therefore
2
𝛽 ′ 𝑣𝑟𝑖
Since 𝛼𝑖 >
2
𝛽 ′ 𝑣𝑟𝑖
1−𝛽 ′ 𝑣𝑟𝑖
𝛽′
1−𝛽 ′ 𝑣𝑟𝑖
𝛽′
𝛽′
𝛽′
>
𝑣𝑜𝑙(𝑆)𝛽𝑣𝑟𝑖
1−𝛽𝑣𝑟𝑖
𝜕𝑆 𝜆1 𝑐
𝑣𝑜𝑙 𝑆 𝑐
=
𝛽
𝛽
= 𝑣𝑜𝑙 𝑆 𝑐, we can choose 𝐴 =
𝜕𝑆 𝜆1
𝑣𝑜𝑙(𝑆)
≥ ℎΓ 𝜆1 .
and 𝛼𝑑 < 𝛼𝑑−1 < ⋯ < 𝜆1 , we have
𝜆 1 ℎΓ < 𝛼𝑑 < 𝛼𝑑−1 < ⋯ < 𝜆1 .
∎
. Let Γ be a distance regular graph and let ℎΓ be a Cheeger
constant of Γ . Then we have
𝛼𝑑
ℎΓ <
≤ 𝜆1 2 − 𝜆1 ≤ 2𝜆1 ,
𝜆1
where 𝜆1 is a smallest positive eigenvalue of the Laplacian.
Let Γ be the Hamming graph 𝐻(𝑑, 𝑞). Then Γ is a distance regular
graph with 𝑞 vertices, valency 𝑑(𝑞 − 1) and 𝑑 diameter.
𝑑
We consider two cases: 𝑎 𝑑 = 3, 𝑞 = 2 and 𝑏 𝑑 = 5, 𝑞 = 2. Then
2
3
𝑎 𝐻 3,2 ∶ 𝑣 = 8, 𝑘1 = 3, 𝜆1 = , 𝛼𝑑 =
ℎΓ <
𝛼𝑑
𝜆1
≈ 0.545455 <
4
11
𝜆1 2 − 𝜆1 = 0.942809 <
2
5
𝑏 𝐻 5,2 ∶ 𝑣 = 32, 𝑘1 = 5, 𝜆1 = , 𝛼𝑑 =
ℎΓ <
𝛼𝑑
𝜆1
≈ 0.437956 <
.
24
137
𝜆1 2 − 𝜆1 = 0.8 <
2𝜆1 = 1.1547.
.
2𝜆1 = 0.894427.
graph with
𝑛
𝑑
Let Γ be the Johnson graph 𝐽(𝑛, 𝑑). Then Γ is a distance regular
vertices, valency 𝑑(𝑛 − 𝑑) and 𝑑 diameter.
We consider two cases: 𝑎 𝑛 = 10, 𝑑 = 4 and 𝑏 𝑛 = 11, 𝑑 = 5. Then
𝑎 𝐽 10,4 ∶ 𝑣 = 210, 𝑘1 = 24, 𝜆1 =
ℎΓ <
𝛼𝑑
𝜆1
≈ 0.429302 <
𝛼𝑑
𝜆1
≈ 0.382344 <
𝛼𝑑 =
11088
79091
𝜆1 2 − 𝜆1 = 0.773879 <
𝑏 𝐽 11,5 ∶ 𝑣 = 462, 𝑘1 = 30, 𝜆1 =
ℎΓ <
11
,
30
10
,
24
𝛼𝑑 =
105
587
𝜆1 2 − 𝜆1 = 0.812233 <
.
2𝜆1 = 0.856349.
.
2𝜆1 = 0.912871.
In fact, 𝛼𝑑 is
1
−𝑞1 𝑑
1
−
𝜆1
⋯ −𝑞𝑑 𝑑
1
𝜆𝑑
,
where 𝑞1 𝑑 , ⋯ , 𝑞𝑑 𝑑 are the 𝑞-numbers of a 𝑃-polynomial scheme.
However, in general, the 𝑞 -numbers are not easy to compute. We thus
find an approximated value 𝛼𝑑 of 𝛼𝑑 with some error 𝛽 > 0 in the following
theorem.
If 𝛽 is sufficiently close to 0 then we obtain the estimate 𝛼𝑑 which is close to
𝛼𝑑 and easier to compute.
𝑟1
(𝛽)
. Let Γ be a distance regular graph of order 𝑣, and let 𝒢𝛽 = 𝑟0
𝐴1 + ⋯ + 𝑟𝑑
𝛽
(𝛽)
𝐴𝑑 be a Green’s function of Γ for 𝛽 > 0. Then we have
𝛽𝑣𝑟𝑑
𝛼𝑑 − 1
𝛽
−𝑣𝑟𝑑
𝛽
𝛽
< 𝛽.
𝐴0 +
𝐿𝑠𝑢𝑏
(𝛽)
For 𝛽 > 0, let 𝐿0
(𝛽)
be the 𝑑 × 𝑑 matrix obtained by the removal of the first
(𝛽)
column of 𝐿𝑠𝑢𝑏 .
Let 𝐿𝑗 (𝛽) be the (𝑑 − 𝑗) × 𝑑 − 𝑗 matrix obtained by the removal from the
first row(respectively, column) to the 𝑗–th row(respectively, column) of 𝐿0
(𝛽)
In the following proposition 5, we find an expression for 𝛼𝑑 in terms of a
basis (𝑢0
(𝛽)
, 𝑢1
(𝛽)
, ⋯ , 𝑢𝑑
(𝛽)
) of 𝒩(𝐿𝑠𝑢𝑏
We also find an explicit expression for 𝑢𝑗
𝐿𝑗 (𝛽) of 𝐿𝑠𝑢𝑏
(𝛽)
.
(𝛽)
(𝛽)
) and the valencies 𝑘𝑗 ’s. And
by a determinant of a submatrix
.
(𝛽)
Let Γ be a distance regular graph of order 𝑣. For 𝛽 > 0, let
(𝛽)
(𝛽)
(𝛽)
(𝑢0 , 𝑢1
, ⋯ , 𝑢𝑑 ) be a basis of 𝒩(𝐿𝑠𝑢𝑏 ) with 𝑢𝑑
𝒢𝛽
be a Green’s function of Γ. Then we have the following:
𝛽𝑣
𝑎 𝛼𝑑 = lim+
,
(𝛽)
𝛽→0
𝑏 𝑢𝑗
(𝛽)
where 𝐿𝑗
𝑑 𝑘 𝑢
𝑗=0 𝑗 𝑗
= (−1)𝑑−𝑗
𝛽
(𝛽)
= 1, and let
−𝑣
det(𝐿𝑗
𝛽
)
𝑐𝑗+1 𝑐𝑗+2 ⋯𝑐𝑑
, 𝑗 = 0,1, ⋯ , 𝑑 − 1,
are submatrices of 𝐿𝑠𝑢𝑏
(𝛽)
and det 𝐿𝑑
𝛽
= 1.
We consider a Johnson scheme 𝐽(8,4) . Let Γ be a distance regular
graph with respect to 𝐴1 . Then Γ is a distance regular graph with 70 vertices and
valency 16 . Also, the valencies of 𝐽(8,4) are 1, 16, 36, 16, 1 and
𝐿𝑠𝑢𝑏
(𝛽)
Since
𝛼𝑑 =
1
−𝑞1 𝑑
1
−
𝜆1
⋯ −𝑞𝑑 𝑑
1
𝜆𝑑
,
8 14 18 20
, , ),
16 16 16 16
(𝑞1 4 , ⋯ ,𝑞4 4 ) = (−7,20, −28,14) and 𝜆1 , ⋯ , 𝜆4 = ( ,
have 𝛼𝑑 =
Let 𝛽 =
(𝑢0
315
1522
1
.
1000
(𝛽)
, 𝑢1
we
≈ 0.206965.
By proposition 5, a basis for 𝒩(𝐿𝑠𝑢𝑏
(𝛽)
,𝑢2
(𝛽)
, 𝑢3
(𝛽)
, 𝑢4
(𝛽)
)=
(𝛽)
) is
263196691 15762389 94021 1001
,
,
,
,1
244140625 15625000 93750 1000
.
Thus, we have
(70)
1 𝑢0
(𝛽)
+ 16 𝑢1
(𝛽)
+ 36 𝑢2
1
1000
(𝛽)
+ 16 𝑢3
(𝛽)
+ 1 𝑢4
(𝛽)
− 70
≈ 0.206609.
Also, by Theorem 4, we have 𝛼𝑑 − 𝛼𝑑 < 0.001 and 𝛼𝑑 < 𝛼𝑑 ≈ 0.206609 + 0.001 =
0.207609.
Let 𝑋 be the set of 𝑑 × 𝑑 matrices over 𝐺𝐹(𝑝𝑡 ) (𝑑 ≤ 𝑛). Define the
𝑖–th relation on 𝑋 by 𝑥, 𝑦 ∈ 𝑅𝑖 ⇔ 𝑟𝑎𝑛𝑘 𝑥 − 𝑦 = 𝑖.
Then 𝔛 = 𝑋, 𝑅𝑖 (0 ≤ 𝑖 ≤ 𝑑) is a 𝑃-polynomial scheme with respect to the
ordering 𝑅0 , 𝑅1 , ⋯ , 𝑅𝑑 .
Let 𝑝 = 2, 𝑡 = 1, 𝑑 = 4, 𝑛 = 5. Then, 𝐿𝑠𝑢𝑏
(𝛽)
is as follows:
Also, we have 𝑋 = 𝑣 = 1048576, 𝑘0 = 1, 𝑘1 = 465, 𝑘2 = 32550, 𝑘3 = 390600
𝑘 𝑏 𝑏 ⋯𝑏
and 𝑘4 = 624960 by using 𝑘𝑖 = 1 1 2 𝑖−1 𝑖 = 2,3, ⋯ , 𝑑 .
𝑐2 𝑐3 ⋯𝑐𝑖
1
Let 𝛽 =
. Then by Proposition 5, we obtain the unique basis of𝒩(𝐿𝑠𝑢𝑏
100
as follows:
(𝑢0
(𝛽)
, 𝑢1
(𝛽)
,𝑢2
(𝛽)
, 𝑢3
(𝛽)
, 𝑢4
(𝛽)
)=
3921317781669 486743013 661683 831
,
,
,
,1
358400000
17920000 448000 800
.
(𝛽)
)
Thus, we have
(1048576)
1 𝑢0
(𝛽)
+ 465 𝑢1
(𝛽)
+ 32550 𝑢2
(𝛽)
1
100
+ 390600 𝑢3
(𝛽)
+ 624960 𝑢4
(𝛽)
− 1048576
≈ 0.195023.
Also, by Theorem 4, we have 𝛼𝑑 < 𝛼𝑑 ≈ 0.195023 + 0.01 = 0.205023.
Let Γ be a distance regular graph with respect to 𝐴1 . Since an eigenvalue 𝜆1 of
the Laplacian is
ℎΓ <
256
,
465
we obtain ℎΓ <
𝛼𝑑
≈ 0.372405 <
𝜆1
𝛼𝑑
𝜆1
<
𝛼𝑑
𝜆1
≈ 0.372405. Moreover,
𝜆1 2 − 𝜆1 = 0.893299 <
2𝜆1 = 1.04932.
- Explicit expression of the krawtchouk polynomial For 𝛽 > 0, let 𝒢𝛽 be a Green’s function for a 𝑃–polynomial scheme.
Then we have
A Green’s function can be expressed as
𝒢𝛽 = 𝑡𝑢0
(𝛽)
𝐴0 + 𝑡𝑢1
(𝛽)
𝐴1 + ⋯ + 𝑡𝑢𝑑
for some nonzero 𝑡 ∈ ℝ, where is the (𝑢0
the nullspace 𝒩(𝐿𝑠𝑢𝑏
𝐿𝑠𝑢𝑏
(𝛽)
) of 𝐿𝑠𝑢𝑏
(𝛽)
(𝛽)
with 𝑢𝑑
, 𝑢1
(𝛽)
(𝛽)
𝛽
𝐴𝑑
, ⋯ , 𝑢𝑑
(𝛽)
) unique basis of
= 1.
(𝛽)
where 𝑠𝑖 = 𝑎𝑖 − 𝑘1 (𝛽 + 1) for 𝑖 = 1,2, ⋯ , 𝑑 and 𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 are the same entries as
in 𝐵1 .
A Green's function 𝒢𝛽 is defined only for 𝛽 > 0, and 𝒢𝛽 is expressed as a linear
combination of adjacency matrices 𝐴𝑖 . But, for 𝛽 ≤ 0, 𝒢𝛽 may be a singular matrix,
so there is no Green's function notion for this case.
We however still have 𝑟𝑎𝑛𝑘(𝐿𝑠𝑢𝑏
(𝑢0
(𝛽)
, 𝑢1
(𝛽)
, ⋯ , 𝑢𝑑
(𝛽)
(𝛽)
) = 𝑑 for 𝛽 ∈ ℝ, so we can obtain a unique basis
) of 𝒩(𝐿𝑠𝑢𝑏
(𝛽)
) with 𝑢𝑑
(𝛽)
= 1.
We extend a notion of a Green's function associated with any real number 𝛽 as
follows. The following definition plays an important role for computation of the 𝑝number and the 𝑞-number.
with 𝑢𝑑
(𝛽)
. For 𝛽 ∈ ℝ , let (𝑢0
(𝛽)
, 𝑢1
(𝛽)
= 1 and let
(𝛽)
(𝛽)
, ⋯ , 𝑢𝑑
(𝛽)
)∈ 𝒩(𝐿𝑠𝑢𝑏
𝛽
(𝛽)
)
𝒢𝛽,𝒩 = 𝑡𝑢0
𝐴0 + 𝑡𝑢1
𝐴1 + ⋯ + 𝑡𝑢𝑑 𝐴𝑑 ,
where 𝑡 is some nonzero ∈ ℝ if 𝛽 > 0 and 𝑡 = 1 if 𝛽 ≤ 0. Then 𝒢𝛽,𝒩 is called
The normalized Green’s function.
Let 𝐻(5,3) be a Hamming scheme . Choosing 𝛽 = −
obtain a 5 × 6 matrix 𝐿𝑠𝑢𝑏
(𝛽)
as follows:
40
(𝛽)
1
,
10
26
1
1 4
Also, a basis of 𝒩(𝐿𝑠𝑢𝑏 ) is (− , − , − , , , 1).
3
15
15 2 5
Thus we obtain a normalized Green’s function 𝒢𝛽,𝒩 as follows:
𝒢𝛽,𝒩 = −
40
𝐴
3 0
−
26
𝐴
15 1
−
1
𝐴
15 2
1
2
4
5
+ 𝐴3 + 𝐴4 + 𝐴5 ,
where 𝐴𝑖 (𝑖 = 0,1,2, ⋯ , 5) are the adjacency matrices of 𝐻 5,3 .
then we
For some 𝛽𝑗 ∈ ℝ, let (𝑢0
(𝑗)
, 𝑢1
(𝑗)
, ⋯ , 𝑢𝑑
(𝑗)
) be a basis of 𝐿𝑠𝑢𝑏
(𝛽𝑗 )
Let 𝔛 = 𝑋, 𝑅𝑖 (𝑖 = 0,1, ⋯ , 𝑑) be a 𝑃-polynomial scheme, and let 𝑃 = (𝑝𝑗 𝑖 )
(respectively, Q= (𝑞𝑗 𝑖 ) ) be the first eigenmatrix (respectively, the second
eigenmatrix) of 𝔛.
In the following proposition 6, we show that the 𝑗-th column vector of the second
eigenmatrix Q belongs to 𝒩(𝐿𝑠𝑢𝑏
(𝛽𝑗 )
𝑝1 𝑗
𝑘1
(𝑗)
) for 𝛽𝑗 =
− 1 (j = 0,1,2, ⋯ , 𝑑) .
That is, we show the relationship between 𝑢𝑖
and component 𝑞𝑗 𝑖 of the second
eigenmatrix Q= (𝑞𝑗 𝑖 ) over the 𝑃 - polynomial scheme.
Let 𝔛 = 𝑋, 𝑅𝑖
𝛽𝑗 =
𝑝1 𝑗
𝑘1
(𝑖 = 0,1, ⋯ , 𝑑) be a 𝑃-polynomial scheme. For
− 1 j = 0,1,2, ⋯ , 𝑑 , let 𝒢𝛽𝑗,𝒩 = 𝑢0
𝑗
a normalized Green’s function with 𝑢𝑑
(𝑗)
𝐴0 + 𝑢1
(𝑗)
𝐴1 + ⋯ + 𝑢𝑑
𝑗
= 1. Then 𝑞𝑗 𝑖 satisfy 𝑞𝑗 𝑖 = 𝑚𝑗
𝐴𝑑 be
𝑢𝑖
𝑢0
(𝑗)
(𝑗)
(𝑖 = 0,1, ⋯ , 𝑑). That is, the 𝑗-th column of the second eigenmatrix Q= (𝑞𝑗 𝑖 ) is
equal to
𝑚𝑗
𝑢0
(𝑗)
(𝑢0
(𝑗)
, 𝑢1
(𝑗)
, ⋯ , 𝑢𝑑
Moreover, we have 𝑞𝑗 𝑖 = 𝑞𝑗 𝑑 𝑢𝑖
(𝑗)
)𝑇 , where 𝑚𝑗 is the 𝑗-th multiplicity of 𝔛.
(𝑗)
.
By Proposition 5 (b),
𝑢𝑖
(𝑗)
= (−1)𝑑−𝑖
where 𝐿𝑖
𝛽𝑗
det(𝐿𝑖
𝛽𝑗
)
𝑐𝑖+1 𝑐𝑖+2 ⋯𝑐𝑑
, 𝑖 = 0,1, ⋯ , 𝑑 − 1,
are submatrices of 𝐿𝑠𝑢𝑏
(𝛽𝑗 )
and det 𝐿𝑑
𝛽𝑗
= 1.
Let 𝐻(𝑑, 𝑞) be a Hamming scheme. Since the Hamming scheme is a self-dual
scheme, 𝑃 = (𝑝𝑗 𝑖 ) is equal to Q= (𝑞𝑗 𝑖 ) . The 𝑝-number 𝑝𝑗 𝑖 of a Hamming
scheme 𝐻(𝑑, 𝑞) is defined by the Krawtchouk polynomial. Thus, we have
𝑝1 𝑗 = 𝑑 𝑞 − 1 − 𝑞𝑗.
And, 𝑘1 = 𝑑 𝑞 − 1 . Thus 𝛽𝑗 =
𝑝1 𝑗
𝑘1
−1=
𝑑 𝑞−1 −𝑞𝑗−𝑑 𝑞−1
𝑑 𝑞−1
=−
𝑞𝑗
𝑑(𝑞−1)
.
For 𝛽𝑗 = −
𝑞𝑗
,
𝑑(𝑞−1)
we have a matrix 𝐿𝑠𝑢𝑏
(𝛽𝑗 )
as follows:
where 𝑠𝑖 = 𝑖 𝑞 − 2 − 𝑑(𝑞 − 1)(𝛽𝑗 + 1) for 𝑖 = 1,2, ⋯ , 𝑑, and 𝑡𝑘 = (𝑑 − 𝑘)(𝑞 −
1) for 𝑘 = 1,2, ⋯ , 𝑑 − 1.
. Let 𝐻 𝑑, 𝑞 a Hamming scheme. Let 𝐿𝑖
submatrix of 𝐿𝑠𝑢𝑏
(𝛽𝑗 )
for 𝛽𝑗 = −
𝑞𝑗
𝑑 𝑞−1
𝑝𝑗 𝑖 = 𝑝𝑗 𝑑 𝑢𝑖
(𝑗)
𝛽𝑗
be an (𝑑 − 𝑖) × (𝑑 − 𝑖)
(j = 0,1,2, ⋯ , 𝑑). Then we have
=
𝑖!det(𝐿𝑖
(−1)𝑑+𝑗−𝑖 𝑑𝑗
𝑑!
𝛽𝑗
)
.
. Let 𝐻(5,3) be a Hamming scheme. Then, the first eigenmatrix 𝑃 =
(𝑝𝑗 𝑖 ) is as follows:
Let 𝑗 = 1, then we obtain the 𝑝-number 𝑝1 𝑖 as the entries of the second
column of the first eigenmatrix 𝑃.
Let 𝐿𝑠𝑢𝑏
(𝛽1 )
be a 5 × 6 matrix for 𝛽1 = −
3 1
5 3−1
=−
3
10
as follows:
Then, the matrices 𝐿0
Also, det 𝐿0
− 4, det 𝐿4
𝛽1
𝛽1
𝛽1
, 𝐿1
𝛽1
, 𝐿2
= 240 , det 𝐿1
𝛽1
𝛽1
, 𝐿3
𝛽1
and 𝐿4
𝛽1
= −168, det 𝐿2
are
𝛽1
= −2. Therefore, 𝑝1 𝑖 (𝑖 = 0,1,2,3,4) are
= 48, det 𝐿3
𝛽1
=
respectively. Thus, (10,7,4,1, −2, −5)𝑇 is the first column vector of the first
eigenmatrix 𝑃.
Thank you