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EIGENVALUES OF THE ADJACENCY MATRIX OF CUBIC LATTICE GRAPHS
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by
Renu Laskar
University of Rorth Carolina
Institute of Statistics Mimeo Series Ho. 573
March 1968
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This research was supported by the Rational Science
Foundation Grant Ro. GP-5790, and the Army Research Office
(Durham) Grant 50. DA-ARO-D-3l-12-G9l0.
.
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DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, 5. C. 27514
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ABSTRACT.
A cubic lattice graph is
de~ined
to be a graph G, whose vertices are
the ordered triplets on n symbols, such that two vertices are adjaceBt if
I~
and only if they have two coordinates in cOJlDDOn.
number
o~
1Iertices y, which are at distance 2
adjacency matrix
o~
G, then G has the
of vertices is n3 •
P4 )
~ollowing
properties: PI) the number
P ) n (x)
2
3
the distinct eigenvalues o~ A(G) are -3, n-3, 2n-3, 3(n-l).
shown here that
i~
P )
2
~rom
n (z) denotes the
2
x and A(G) denotes the
n
G is connected and regular.
4
o~
It is
> 7, any graph G (With no loops and multiple edges)
having the properties PI) - P ) must be a cubic lattice graph.
characterization
= 3(n_l)2.
An alternative
cubic lattice graphs has been given by the author (.T.
Comb. Theory, Vol. 3, No.4, December 1967, 386-401).
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1.
Introduetion.
We shall consid.er only finite undirected graph. witbDut loop. or multiple
edges.
A cubic lattice graph with characteristic n is defined to be a graph
whose vertices are identified with the n3 ordered triplets on n symbols, with
two vertices adjacent if and only if their corresponding triplets have two
coordinates in common.
x and y and &(x, y) the number of vertices adjacent to both x and y, then it
has been shown by the author [6] that for n > 7, the following properties
characterize the cubic lattice graph with characteristic nl
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If d(x, y) denotes the distance between two vertices
(b )
l
(b )
2
The nUliber of vertices is n3 •
(b )
3
(b4 )
If d(x, y)
= 1, then A(x, y) = n-2.
If d(x, y)
= 2,
(b )
If d(x, y)
= 2, there exist exactly n-l vertices z, adjacent
5
The graph is connected and regular of degree 3(n-l).
to x such that dey, z)
then A(x, y)
= 2.
= 3.
DOWling [4] ill a note has shown that the property (b ) is implied by
5
properties (b l ) - (b4 ) for n > 7. Hence for n > 7, (b l ) - (b 4 ) characterize
a cubic lattice graph with characteristic n.
The adjacency matrix A(G) of a graph G i8 a square (0, 1) ..trix whose
rows and columns correspond to the vertices of G, and a ij
i and j are adjacent.
=1
if and only if
Let n (x) denote the number of vertices y at distance
2
2 from x.
A cubic lattice graph G with characteristic n has the following properties:
(PI)
The number of vertices is n3 •
(P )
2
G is connected and regular.
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2
2
(P ) n2 (x) = 3(n-l) for all x in G.
3
(P4) The distinct eigenvalues of A(G) are -3, n-3, 2n-3, 3(n-l).
(Pl ), (P ), (P ) are obvious. (P4) is proved in paragraph 2. We go
2
3
on to show that (Pl ), (P ), (P ), (P4 ) characterize a cubic lattice graph
2
3
with characteristic n. Similar characterization for tethrahedral graphs has
been given by Bose and Laskar [lJ.
2.
Determination of the eigenvalues of A(a).
Given v objects, a relation satisfying the following conditions is said
to be an association scheme with m classesl
a) Any two objects are either 1st, 2nd, ••• , or mth associates,
the relation of association being symmetr.lcal.
b)
Each object a has ni ith associates, the number ni being inde-
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pendent of a.
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of objects which are jth associates of
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c)
If any tliO objects a and
~
are ith associates, then the number
i
a, and kth associates of P, is Pjk
and is independent of the pair of ith associates a and /!S.
i
T.he numbers v, n i and Pjk' i,
the association scheme.
j,
k ... 1, 2, •.• , m are the parameters of
The concept of an association scheme was first introduced by Jose and
Shimamoto [3 J•
If we define
lSi = (b:i )
i = 0, 1, 2, ••• ,
m,
=
2
b1l i Dli
2
1
b2i b2i
·..
·..
... ... ·..
l
bvi b 2vi ·..
v
b li
v
b
2i
V
bVl.•
,
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3
where
b:i = 1, if the objects a and ~ are ith associates
= 0, otherwise,
and
Pk
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= (prk)
... ... ...
0
Pmk
pl
mk
...
m
POk
m
Plk
, k
= 0,
1, ••• , m,
m
Pmk
then it has been shown by Bose and Mesner [2], that the materices Pi'
i = 0, 1, ••• , m are linearly independent and combine in the same way as
the B's under addition as well as lJIIlltiplication.
It was further shown that
if
m
B
ICiB
=I
=
i
i=O
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0
1
POk POk
1
0
Plk Plk
m
p
CiP i ,
i=O
then B and P have the same distinct eigenvalues.
Co = 0,
cl
= 1,
c2
= c3 = ••• = cm = 0,
If in particular we take
it follows that the .distinct eigen-
values of B are the sue as those of Pl.
l
Consider a cubic lattice graph G with characteristic n.
If a relation
of association on the vertices of G is defined, such that Wwo vertices are
1st, 2nd, or 3rd associates if they are at distances 1, 2 or 3 respectively,
then it can be easily checked that G yields a three-class association scheme.
It may be pointed out that the matrix A(G) is the matrix Bl and thus the
distinct eigenvalues of A(G) are given by those of the matrix
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4
1
0
=
Pl
n
1
l
0
Pll
1
P12
1
0
0
2
3
Pll
Pll
2
P12
p3
12
.;
p2
13
13
i
The parameters Pjk of the association scheme corresponding to G are easily
0
calculated.
P13
lfhey are given by
nl
= 3(n-l),
1
= n-2,
1
= 2(n-l),
Pll
P
12
1
P
= 0,
13
Substituting these values in the matrix P l' the eigaavalues are easily
calculated.
They are found to be
-3, n-3, 2n-3, 3(n-l).
Thus, we have the following lemma:
Lemma 2.1.
If G is a cubic lattice graph with characteristic n and if A(O)
is the adjacency matrix of G, then the distinct eigenvalues of A(G) are
(2.1)
3.
-3, n-3, 2n-3, 3 (n-l) .
Some Preliminaries on Matrices.
Before stating the next lemma, we need the concept of the polynomial of
a graph introduced by Hoffman [5].
are unity.
~en
for any graph
Q.
a polynomial p(x) such that peA)
Let J be the matrix all of whose entries
with adjacency matrix A = A(G), there exists
= J
if and only if g. is regular and connected.
The unique polynomial of least degree satisfying this equation is called the
polynomial of G, and is calculated as follows: if G has v vertices, it is
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regular of degree d, and the other distinct eigenvalues of A(G) are
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5
v
p(x)
i=l
Consider a regular connected graph H (with no loops and multiple edges)
on v • n3 vertices such that the adjacency matrix A
= A(H)
has the distinct
eigenvalues -3, n-3, 2n-3, 3(n-l).
Lemma 3.1.
The matrix A satisf'ies the equation
where J is a v
X
v matrix all of' whose entries are 1, and I is the v
X
v
identity matrix.
Proof':
It f'ollows immediately by calculating the polynomial of' the graph as
given in (3.1).
For any two vertices x, y in H, d(x, y) < 3.
Lemma 3.2.
Proof':
· 2 3
If' in (3.2) we set Aij = 0, Aij = 0, then Aij
= 6,
but this implies
that d(i,j) < 3 for all vertices i, j in H.
LelllJl8.
3.3.
Consider the matrix
B
= t[A2
- (n-2)A - 3(n-l) I].
Let ~(i) denote the number of' vertices j, such that d(i,j)
denote the number of' vertices k, such that d(i,k) = 3.
= 2,
and n,(i)
If' n (i) = 3(n-l)2
2
f'or all vertices i in H, then
i)
B is a (0,1) matrix,
ii) A(x,y)
I.
I
I
=
t
n (x-(li)
iii) A(x,y)
= n-2, f'or all vertices x,y in H, such that d(x,y) = 1,
= 2, for all .vertices x,y in H, such that d(x,y) = 2.
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6
~:
Since H is regular and 3(n-l) is the dominant eigenvalue, it follows
H is regular of degree ~
= 3(n-l).
Divide the set of vertices of H, with respect to a particular vertex i
into four subsets SO' Sl' S2' S3 as follows:
SO:
i
Sl
jl' j2' ••. , jt .•• , jn ' such that d(i,jt) ~ 1, t-l,2,.:.,nl
l
kl,k2,···,ks1 ••• ,kn2(i)' such that d(i,k s )=2,S=1,2, •.. ,n2 (i)
S2:
S3:
Jl,J2,···,Jr,···,Jn3(i)' such that d(i,Jr )=3,r=1,2, .•. ,n3 (i).
Thus the vertices in St are tth associates of the vertex 1. The following
relations can be deduced easily from (3.2) by noting that AJ = JA.
n
(3.3)
A~i
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=
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tel
Aij
= 3 (n-l)(n-2) •
n
l
4
\
3
Aii = L Aijt
tel
= 3(n-l) (n2 +3n-3).
Also, since AtJ
= (3(n-l)}t J,
we get
2
(A J)ii
(3.5)
= 9(n-l)2,
n
(3.6)
Aii
=
l
I
tel
Aij
= 3(n-l)
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t
Also
t
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7
~(i)
L
(3.7)
r=l
A~"r· O.
Hence it follows from (3.3), (3.5), (3.6), (3.7) that
110
6(n-1) 2 •
Consider
~(i)
- 2
L
biks +
~(i).
8=1
We first
abow
that
Xi = n2(i) - 3(n-1)2.
Since
B • t[A2_(n-~ A-3(n-1)I], we get
(3.11)
Substituting values fram (3.3), (3.4), (3.6) in (3.11) we get
B~1
But
Hence
I:t
3(n_1)2.
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8
2
= 3(n-l) •
Also from (3.10)
~(i)
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sal
b ik
s
2
= 3(n-l) •
SUbstituting values from (3.12), (3.13) in (3.9) we get
Xi ..
Bow if ~(i)
= 3(n-l)2
~(i) - 3(n-l)2.
for all i in H, then Xi
=0
tar all i in H.
!hen it
follows from (:3.9) that :B is a (0,1) matrix which proves i).
To prove ii), we note that ifAij
it follows
n
l
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tal
But since b
ij
= 0 or 1, this implies b
follows that A2ij
=1, then from (3.10),(3.3) and (3.6)
t
b ij
ijt
=
o.
t
==
0, and hence from (3.10) it
= n-2.
2
To prove iii) we note that if A = 0, Aij
ij
4.
'fheorem.
'f
0, then b ij
F0
and hence
If H is a graph satisfying the following properties:
Iftle number of vertices is n3 .
H is connected and regular.
~(x)
= 3(n-l) 2
for all x in H.
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9
P4)
Then, for n
Proof:
The distinct eigenvalues of A{H) are -3, n-3, 2.n-3, 3{n-l).
> 7,
H is cubic lattice.
From lemmas (3.l) - (3.3) and ilhe bJpothesis H clearly satisfies
the following contitions:
(b )
l
!he number of vertices is n3 •
(b )
2
H is connected and regular of degree 3{n-l).
(b ) A{x,y) = n-2 for d(x,y) ... 1.
3
(b4) A(x,y)'" 2, for d(x,y) • 2.
> 7, H is cubic lattice (6], (4].
Hence if n
It is conjectured that the property P ) of the theorem is implied by
3
other properties P ), P ), P ).
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2
l
1.2.t§.:
It :may be pointed out that the main purpose of assuming P ) is to prove
3
that B is a (0,1) JllLtrix. If we replace P3) by P~) and P,) as follows:
3).
P
B is edge-regular, Le., A{x,y)
all x,y, such that
A(x,y) = 1,
p,) A{x,y) ... even, for all x,y, such that d(x,y)
then it can be shown tbat B is a (0,1) mtrix.
3) and
From P
(3.3) it follows that A = n-2.
(3.10) and noting p;) we get b
A.
ij
... O.
ij
=0
= 2,
The proof 80es like this:
Substituting value for A in
if A ... 1, and b
ij
ij
= an
integer if
Again from (3.10) and (3.12) it follows that
v
I
v
b ij =
.1-1
I b~j
.1=1
Thus B is a matrix whose entries are either 0 or integer such that for any
row, sum of the elements is equal to the
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I
= A for
8UJa
but this implies that :B is a (0,1) Datrix.
of the squares of the elements,
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10
3),
Hence we can also state that for n > 7, P ), P2 ), P
l
characterize a cubic lattice graph with characteristic n.
p;), P )
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1-.
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11
L
R. C. BOSE and R. IASKAR, Eigenvalues of the Adjacency Matrix of
Tetrahedral Graphs, Institute of Statistics Mimeo Series 10.
57~,
University of North Carolina ~ch 1968.
2.
R. C. BOSE and D. M. MESBER, On Linear Assooiative Algebras Corresponding
to Association Schemes of Partially Balanoed Designs, Ann. Math.
Statist, 30 (1959), 21-36.
3.
R. C. BOSE and T. SHIMIJI)'fO, Classification and Analysis of Partially
Balanced Incomplete
B~ck
Designs with !WO Associate Classes, J,
Amer. Stat. Assn., Vol 47 (1952), 151-184.
4.
T. A. DOWLIIG, Bote on "A Characterization of Cubio Lattice Graphs",
J. Combinatorial Theory (to appear)
5. A. J. HOF.FMAIf,
On the Polynomial of
ti
Gr:"ph, Am. Math. Monthly, 70
(1963), 30-36.
6.
R. IASKAR,
A Characterization of Cubic Lattice Graphs, J. Combinatorial
!beory, Vol. 3, No 4, December 1967, 386-401.