Upload
dragance106
View
219
Download
0
Embed Size (px)
Citation preview
8/13/2019 Two shorter and simpler proofs in spectral graph theory
1/14
U n i v . B e o g r a d . P u b l . E l e k t r o t e h n . F a k .
S e r . M a t . 9 1 9 9 8 , 8 9 1 0 2 .
4 - R E G U L A R I N T E G R A L G R A P H S
D . C v e t k o v i c , S . K . S i m i c , D . S t e v a n o v i c
P o s s i b l e s p e c t r a o f 4 - r e g u l a r i n t e g r a l g r a p h s a r e d e t e r m i n e d . S o m e c o n s t r u c t i o n s
a n d a l i s t o f 6 5 k n o w n c o n n e c t e d 4 - r e g u l a r i n t e g r a l g r a p h s a r e g i v e n .
1 . I N T R O D U C T I O N
A g r a p h i s c a l l e d i n t e g r a l i f a l l i t s e i g e n v a l u e s o f t h e a d j a c e n c y m a t r i x
a r e i n t e g e r s . T h e q u e s t f o r i n t e g r a l g r a p h s w a s i n i t i a t e d b y F . H a r a r y a n d A
J . S c h w e n k 1 2 . A l l s u c h c o n n e c t e d c u b i c g r a p h s w e r e o b t a i n e d b y D . C v e t k o -
v i
c a n d F . C . B u s s e m a k e r 6 3 , a n d i n d e p e n d e n t l y b y A . J . S c h w e n k 1 5
T h e r e a r e e x a c t l y t h i r t e e n c o n n e c t e d c u b i c i n t e g r a l g r a p h s . I n f a c t , D . C v e t k o v i
c
6 p r o v e d t h a t t h e s e t o f c o n n e c t e d r e g u l a r i n t e g r a l g r a p h s o f a x e d d e g r e e i s n i t e .
S i m i l a r l y , t h e s e t o f c o n n e c t e d i n t e g r a l g r a p h s w i t h b o u n d e d v e r t e x d e g r e e s i s n i t e .
Z . R a d o s a v l j e v i
c a n d S . S i m i
c 1 6 d e t e r m i n e d a l l 1 3 c o n n e c t e d n o n r e g u l a r n o n -
b i p a r t i t e i n t e g r a l g r a p h s w h o s e m a x i m u m d e g r e e e q u a l s f o u r . T h e c o r r e s p o n d i n g
p r o b l e m f o r b i p a r t i t e g r a p h s i s n o t y e t a n s w e r e d s e e 1 6 f o r s o m e d e t a i l s . R e -
c e n t l y , D . S t e v a n o v i
c 1 7 d e t e r m i n e d a l l 2 4 c o n n e c t e d 4 - r e g u l a r i n t e g r a l g r a p h s
a v o i d i n g 3 i n t h e s p e c t r u m . I n t h i s p a p e r w e a r e i n t e r e s t e d i n c o n n e c t e d 4 - r e g u l a r
i n t e g r a l g r a p h s .
T h e s e a r c h f o r i n t e g r a l g r a p h s b e c o m e s e a s i e r i f w e u s e t h e f o l l o w i n g p r o d u c t
o f g r a p h s . G i v e n t w o g r a p h s G a n d H w i t h v e r t e x s e t s V G a n d V H t h e i r
p r o d u c t G H i s t h e g r a p h w i t h t h e v e r t e x s e t V G V H i n w h i c h t w o v e r t i c e s
x a a n d y b a r e a d j a c e n t i f a n d o n l y i f x i s a d j a c e n t t o y i n G a n d a i s a d j a c e n t
t o b i n H I f
1
; : : : ;
n
a r e t h e e i g e n v a l u e s o f G a n d
1
; : : : ;
m
a r e t h e e i g e n v a l u e s
o f H t h e n t h e e i g e n v a l u e s o f G H a r e
i
j
f o r i = 1 ; : : : ; n ; j = 1 ; : : : ; m s e e 8
I f G i s c o n n e c t e d , n o n b i p a r t i t e , 4 - r e g u l a r a n d i n t e g r a l , t h e n t h e p r o d u c t G K
2
i s
c o n n e c t e d , b i p a r t i t e , 4 - r e g u l a r a n d i n t e g r a l , s i n c e t h e e i g e n v a l u e s o f K
2
a r e 1 a n d
1 T h e r e f o r e , i n d e t e r m i n i n g 4 - r e g u l a r i n t e g r a l g r a p h s w e c a n c o n s i d e r b i p a r t i t e
g r a p h s o n l y , a n d l a t e r e x t r a c t s u c h n o n b i p a r t i t e g r a p h s G f r o m t h e d e c o m p o s i t i o n s
o f b i p a r t i t e o n e s i n t h e f o r m G K
2
O n t h e o t h e r h a n d , i f G i s b i p a r t i t e t h e n
G K
2
= 2 G a n d w e c a n n o t o b t a i n n e w g r a p h s b y i t e r a t i n g t h e p r o d u c t w i t h K
2
I n S e c t i o n 2 w e n d a l l p o s s i b l e s p e c t r a o f c o n n e c t e d b i p a r t i t e 4 - r e g u l a r i n -
t e g r a l g r a p h s . I n S e c t i o n 3 w e g i v e s o m e u p p e r b o u n d s o n t h e n u m b e r o f v e r t i c e s
i n r - r e g u l a r b i p a r t i t e g r a p h s . T o g e t h e r w i t h r e s u l t s a n n o u n c e d i n 1 9 , i t f o l l o w s
t h a t , e x c e p t f o r 5 e x c e p t i o n a l s p e c t r a , e a c h 4 - r e g u l a r b i p a r t i t e i n t e g r a l g r a p h h a s
1 9 9 1 M a t h e m a t i c s S u b j e c t C l a s s i c a t i o n : 0 5 C 5 0
8 9
8/13/2019 Two shorter and simpler proofs in spectral graph theory
2/14
9 0 D . C v e t k o v i c , S . S i m i c , D . S t e v a n o v i c
a t m o s t 1 2 6 0 v e r t i c e s . I n S e c t i o n 4 w e g i v e t h e l i s t o f k n o w n 4 - r e g u l a r i n t e g r a l
g r a p h s . S e c t i o n 5 c o n t a i n s c o n c l u d i n g r e m a r k s .
2 . S P E C T R A O F 4 - R E G U L A R B I P A R T I T E I N T E G R A L G R A P H S
S u p p o s e t h a t G i s a 4 - r e g u l a r b i p a r t i t e i n t e g r a l g r a p h . R e g u l a r b i p a r t i t e
g r a p h s h a v e t h e s a m e n u m b e r o f v e r t i c e s i n e a c h p a r t s o t h a t w e m a y a s s u m e t h a t
t h e y h a v e p = 2 n v e r t i c e s . U s i n g s u p e r s c r i p t s t o r e p r e s e n t m u l t i p l i c i t i e s , w e s h a l l
w r i t e i t s s p e c t r u m i n t h e f o r m
4 3
x
2
y
1
z
0
2 w
1
z
2
y
3
x
4
L e t f u r t h e r q a n d h d e n o t e t h e n u m b e r s o f q u a d r i l a t e r a l s a n d h e x a g o n s i n G I t
i s w e l l k n o w n t h a t t h e s u m o f k
t h
p o w e r s o f t h e e i g e n v a l u e s i s j u s t t h e n u m b e r o f
c l o s e d w a l k s o f l e n g t h k . T h e r e f r o m w e g e t t h e f o l l o w i n g r e s u l t .
L e m m a 1 . T h e p a r a m e t e r s n ; x ; y ; z ; w ; q ; h s a t i s f y t h e D i o p h a n t i n e e q u a t i o n s
1
2
P
0
i
= 1 + x + y + z + w = n 1
1
2
P
2
i
= 1 6 + 9 x + 4 y + z = 4 n 2
1
2
P
4
i
= 2 5 6 + 8 1 x + 1 6 y + z = 2 8 n + 4 q 3
1
2
P
6
i
= 4 0 9 6 + 7 2 9 x + 6 4 y + z = 2 3 2 n + 7 2 q + 6 h 4
A n o t h e r u s e f u l l e m m a i s d u e t o A . J . H o f f m a n 1 3 . L e t G b e a r e g u l a r
g r a p h w i t h d i s t i n c t e i g e n v a l u e s
1
= r
2
; : : : ;
k
a n d l e t A b e t h e a d j a c e n c y
m a t r i x o f G J s t a n d s f o r a l l 1 - m a t r i x , a n d I i s t h e i d e n t i t y m a t r i x .
L e m m a 2 . T h e a d j a c e n c y m a t r i x A s a t i s e s t h e e q u a t i o n
k
Y
i = 2
r
i
J = p
k
Y
i = 2
A
i
I 5
T h e s e a r c h f o r p o s s i b l e s p e c t r a i s d i v i d e d i n c a s e s d e p e n d i n g o n t h e g r e a t e s t
i n t e g e r l e s s t h a n 4 w h i c h i s a v o i d e d i n t h e s p e c t r u m o f G s e e 1 5 .
1 . C a s e x = 0
B y p u t t i n g x = 0 i n t h e e q u a t i o n s 1 4 a n d t h e n e l i m i n a t i n g n a n d y f r o m
1 a n d 2 w e g e t 3 z + 4 w = 1 2 a n d s i n c e z w 0 t h e o n l y p o s s i b i l i t i e s a r e
z w = 0 3 o r z w = 4 0
8/13/2019 Two shorter and simpler proofs in spectral graph theory
3/14
4 - r e g u l a r i n t e g r a l g r a p h s 9 1
S u b c a s e z w = 0 , 3
B y s u b s t i t u t i n g t h e v a l u e s o f x z a n d w i n 1 4 a n d p u t t i n g y = n 4 w h i c h
f o l l o w s f r o m 1 i n t o 3 a n d 4 w e g e t
3 n + q = 4 8 6
2 8 n + 1 2 q + h = 6 4 0 7
F r o m 6 i t f o l l o w s t h a t q = 3 q
1
a n d t h u s n + q
1
= 1 6 w h i c h s h o w s t h a t n 1 6
F u r t h e r , f r o m 6 a n d 7 w e g e t h = 8 n + 6 4 I f y = 0 t h e n n = 4 a n d t h e
c o r r e s p o n d i n g s p e c t r u m i s s h o w n i n T a b l e 1 . I f y 0 t h e n H o f f m a n ' s i d e n t i t y
5 r e a d s 3 8 4 J = 2 n A
4
4 A
2
+ 8 n A
3
4 A G r a p h G i s b i p a r t i t e , a n d i f w e
t a k e v e r t i c e s u v f r o m d i s t i n c t p a r t s c o l o u r s , t h e n A
2 k
u v
= 0 S o w e h a v e
3 8 4 = 8 n A
3
u v
4 A
u v
w h e r e f r o m 8 n 3 8 4 i . e . n 4 8 S i n c e n 1 6 t h e
o n l y p o s s i b l e v a l u e s f o r n a r e 6 8 1 2 1 6 . T h e c o r r e s p o n d i n g s p e c t r a a r e s h o w n i n
T a b l e 1 . H e r e a n d i n o t h e r t a b l e s o f p o s s i b l e s p e c t r a t h e c o l u m n L a b e l " i n d i c a t e s
t h e e x i s t e n c e o f g r a p h s w i t h t h e c o r r e s p o n d i n g s p e c t r u m b y r e f e r r i n g t o t h e l i s t o f
k n o w n 4 - r e g u l a r i n t e g r a l g r a p h s i n S e c t i o n 4 . I f t h e g r a p h w i t h t h e c o r r e s p o n d i n g
s p e c t r u m d o e s n o t e x i s t , t h e n t h i s c o l u m n c o n t a i n s s y m b o l " . I f a l l t h e g r a p h s
w i t h t h e g i v e n s p e c t r u m a r e k n o w n , t h e n t h e c o l u m n A l l " c o n t a i n s s y m b o l + " .
n x y z w q h L a b e l A l l
4 0 0 0 3 3 6 9 6 I
8 1
+
6 0 2 0 3 3 0 1 1 2 I
1 2 4
+
8 0 4 0 3 2 4 1 2 8 I
1 6 1 2
+
1 2 0 8 0 3 1 2 1 6 0 I
2 4 1 2
+
1 6 0 1 2 0 3 0 1 9 2 I
3 2 1
+
T a b l e 1 : T h e p o s s i b l e i n t e g r a l g r a p h s p e c t r a w i t h x ; z ; w = 0 0 3
S u b c a s e z w = 4 , 0
B y s u b s t i t u t i n g t h e v a l u e s o f x z a n d w i n 1 4 a n d p u t t i n g y = n 5
w h i c h f o l l o w s f r o m 1 i n t o 3 a n d 4 w e g e t
3 n + q = 4 5 8
2 8 n + 1 2 q + h = 6 3 0 9
F r o m 8 i t f o l l o w s q = 3 q
1
a n d t h u s n + q
1
= 1 5 w h i c h s h o w s t h a t n 1 5
F u r t h e r , f r o m 8 a n d 9 w e g e t h = 8 n + 9 0 I f y = 0 t h e n n = 5 a n d t h e
c o r r e s p o n d i n g s p e c t r u m i s s h o w n i n T a b l e 2 . I f y 0 H o f f m a n ' s i d e n t i t y 5
r e a d s 1 4 4 0 J = 2 n A
5
5 A
3
+ 4 A + 8 n A
4
5 A
2
+ 4 I t h e r e f o r e n 1 8 0 a n d s i n c e
n 1 5 t h e o n l y p o s s i b l e v a l u e s f o r n a r e 6 9 1 0 1 2 1 5 . T h e c o r r e s p o n d i n g s p e c t r a
a r e s h o w n i n T a b l e 2 .
8/13/2019 Two shorter and simpler proofs in spectral graph theory
4/14
9 2 D . C v e t k o v i c , S . S i m i c , D . S t e v a n o v i c
n x y z w q h L a b e l A l l
5 0 0 4 0 3 0 1 3 0 I
1 0 1
+
6 0 1 4 0 2 7 1 3 8 I
1 2 2
+
9 0 4 4 0 1 8 1 6 2 I
1 8 1 3
+
1 0 0 5 4 0 1 5 1 7 0 I
2 0 1 3
+
1 2 0 7 4 0 9 1 8 6 |
1 5 0 1 0 4 0 0 2 1 0 I
3 0 1
+
T a b l e 2 : T h e p o s s i b l e i n t e g r a l g r a p h s p e c t r a w i t h x ; z ; w = 0 4 0
2 . C a s e x 0 y = 0
B y p u t t i n g y = 0 i n t h e e q u a t i o n s 1 4 a n d s u b t r a c t i n g 2 f r o m 3 w e
o b t a i n 2 4 0 + 7 2 x = 2 4 n + 4 q f r o m w h i c h q = 6 q a n d
3 x = n + q 1 0 1 0
O n t h e o t h e r h a n d , s u b t r a c t i n g 1 f r o m 2 y i e l d s
8 x = 3 n + w 1 5 1 1
E l i m i n a t i n g x f r o m 1 0 a n d 1 1 w e o b t a i n n = 8 q 3 w 3 5 , a n d s i n c e x ; z ; h 0
w e g e t t h e f o l l o w i n g c o n d i t i o n s :
x = 3 q w 1 5 0 1 2
z = 5 q 3 w 2 1 0 1 3
h = 2 1 0 1 6 q 6 w 0 1 4
H o f f m a n ' s i d e n t i t y 5 n o w r e a d s 3 3 6 0 J = 2 n A
6
1 0 A
4
+ 9 A
2
+ 8 n A
5
1 0 A
3
+ 9 A T h u s , w e g e t n 4 2 0 B y s o l v i n g t h e s y s t e m 1 2 1 4 f o r q a n d w
a n d e l i m i n a t i n g t h e s p e c t r a n o t s a t i s f y i n g n 4 2 0 w e g e t t h e p o s s i b i l i t i e s s h o w n i n
T a b l e 3 . N o n e x i s t e n c e o f g r a p h s w i t h s p e c t r a i n d i c a t e d b y " i s s h o w n i n 1 9
1
3 . C a s e x 0 y 0 z = 0
B y p u t t i n g z = 0 i n t h e e q u a t i o n s 1 4 a n d e l i m i n a t i n g n f r o m 1 a n d 2
w e o b t a i n 4 w = 1 2 + 5 x s o t h a t x = 4 x
1
S u b s t i t u t i n g t h e v a l u e o f n f r o m 2 i n t o
3 w e g e t q = 3 6 + 1 8 x
1
3 y s o t h a t q = 3 q
1
a n d
q
1
= 1 2 + 6 x
1
y 1 5
S u b s t i t u t i n g t h e v a l u e o f n f r o m 2 a n d t h e v a l u e o f q
1
f r o m 1 5 i n t o 4 w e g e t
1
2
h = 4 8 3 9 x
1
+ 4 y 0 a n d t h e r e f o r e
y
3 9
4
x
1
1 2 1 6
1
T h i s w i l l b e p r o v e n e l s e w h e r e .
8/13/2019 Two shorter and simpler proofs in spectral graph theory
5/14
4 - r e g u l a r i n t e g r a l g r a p h s 9 3
n x y z w q h L a b e l A l l
7 1 0 3 2 3 6 1 0 2 I
1 4 1
+
1 0 2 0 6 1 3 6 1 0 8
1 2 3 0 5 3 4 2 8 0
1 4 4 0 4 5 4 8 5 2
1 5 4 0 8 2 4 2 8 6
2 0 6 0 1 0 3 4 8 6 4 |
2 1 6 0 1 4 0 4 2 9 8
2 8 9 0 1 5 3 5 4 8 4 |
4 2 1 4 0 2 6 1 6 0 4 4 |
T a b l e 3 : T h e p o s s i b l e i n t e g r a l g r a p h s p e c t r a w i t h x 0 y = 0
F r o m 1 5 a n d 1 6 w e g e t
0 q
1
2 4
1 5
4
x
1
1 7
w h i c h g i v e s t h a t x
1
6 E l i m i n a t i n g y f r o m 2 a n d 1 5 y i e l d s n = 1 5 x
1
q
1
+ 1 6
a n d f r o m 1 7 i t f o l l o w s
7 5
4
x
1
8 n 1 5 x
1
+ 1 6 H o f f m a n ' s i d e n t i t y 5 n o w
r e a d s 2 6 8 8 J = 2 n A
6
1 3 A
4
+ 3 6 A
2
+ 8 n A
5
1 3 A
3
+ 3 6 A a n d t h e r e f o r e n 3 3 6
T h e c o r r e s p o n d i n g s p e c t r a a r e s h o w n i n T a b l e 4 .
n x y z w q h L a b e l
1 4 4 1 0 8 5 1 2 6
1 6 4 3 0 8 4 5 4 2
2 1 4 8 0 8 3 0 8 2
2 4 4 1 1 0 8 2 1 1 0 6
2 8 4 1 5 0 8 9 1 3 8
4 2 8 2 0 0 1 3 1 2 1 0 0
5 6 1 2 2 5 0 1 8 1 5 6 2
T a b l e 4 : T h e p o s s i b l e i n t e g r a l g r a p h s p e c t r a w i t h x ; y 0 z = 0
4 . C a s e x 0 y 0 z 0 w = 0
E l i m i n a t i n g y a n d z f r o m 1 a n d 2 w e g e t y = n
8
3
x 5 a n d z =
5
3
x + 4
f r o m w h i c h x = 3 x
1
. E l i m i n a t i n g t h e n q a n d h f r o m 3 a n d 4 , w e o b t a i n q =
3 n + 3 0 x
1
+ 4 5 a n d h = 8 n 8 0 x
1
+ 9 0 S i n c e y 1 a n d q h 0 w e g e t
m a x
1
n
1 0
3
2
x
1
m i n
n
1 0
+
9
8
n
8
3
4
8/13/2019 Two shorter and simpler proofs in spectral graph theory
6/14
9 4 D . C v e t k o v i c , S . S i m i c , D . S t e v a n o v i c
n x y z w q h L a b e l
1 4 3 1 9 0 3 3 1 2 2
1 5 3 2 9 0 3 0 1 3 0
1 8 3 5 9 0 2 1 1 5 4
2 0 3 7 9 0 1 5 1 7 0 |
2 1 3 8 9 0 1 2 1 7 8
2 8 6 7 1 4 0 2 1 1 5 4 |
3 0 6 9 1 4 0 1 5 1 7 0
3 0 9 1 1 9 0 4 5 9 0 I
6 0 3
3 5 6 1 4 1 4 0 0 2 1 0 I
7 0 1
3 5 9 6 1 9 0 3 0 1 3 0
3 6 9 7 1 9 0 2 7 1 3 8 |
4 2 9 1 3 1 9 0 9 1 8 6
4 2 1 2 5 2 4 0 3 9 1 0 6
4 5 9 1 6 1 9 0 0 2 1 0 I
9 0 1
4 5 1 2 8 2 4 0 3 0 1 3 0
6 0 1 5 1 5 2 9 0 1 5 1 7 0 |
6 0 1 8 7 3 4 0 4 5 9 0 |
6 3 1 5 1 8 2 9 0 6 1 9 4
6 3 1 8 1 0 3 4 0 3 6 1 1 4
6 3 2 1 2 3 9 0 6 6 3 4
7 0 1 8 1 7 3 4 0 1 5 1 7 0
7 0 2 1 9 3 9 0 4 5 9 0
7 0 2 4 1 4 4 0 7 5 1 0
8 4 2 1 2 3 3 9 0 3 2 0 2 |
8 4 2 4 1 5 4 4 0 3 3 1 2 2 |
8 4 2 7 7 4 9 0 6 3 4 2 |
9 0 2 4 2 1 4 4 0 1 5 1 7 0
9 0 2 7 1 3 4 9 0 4 5 9 0
9 0 3 0 5 5 4 0 7 5 1 0
n x y z w q h L a b e l
1 0 5 2 7 2 8 4 9 0 0 2 1 0
1 0 5 3 0 2 0 5 4 0 3 0 1 3 0
1 0 5 3 3 1 2 5 9 0 6 0 5 0
1 2 6 3 6 2 5 6 4 0 2 7 1 3 8
1 2 6 3 9 1 7 6 9 0 5 7 5 8
1 4 0 3 9 3 1 6 9 0 1 5 1 7 0 |
1 4 0 4 2 2 3 7 4 0 4 5 9 0 |
1 4 0 4 5 1 5 7 9 0 7 5 1 0 |
1 8 0 5 1 3 9 8 9 0 1 5 1 7 0 |
1 8 0 5 4 3 1 9 4 0 4 5 9 0 |
1 8 0 5 7 2 3 9 9 0 7 5 1 0 |
2 1 0 6 0 4 5 1 0 4 0 1 5 1 7 0
2 1 0 6 3 3 7 1 0 9 0 4 5 9 0
2 1 0 6 6 2 9 1 1 4 0 7 5 1 0
2 5 2 7 2 5 5 1 2 4 0 9 1 8 6 |
2 5 2 7 5 4 7 1 2 9 0 3 9 1 0 6 |
2 5 2 7 8 3 9 1 3 4 0 6 9 2 6 |
3 1 5 9 0 7 0 1 5 4 0 0 2 1 0
3 1 5 9 3 6 2 1 5 9 0 3 0 1 3 0
3 1 5 9 6 5 4 1 6 4 0 6 0 5 0
4 2 0 1 2 3 8 7 2 0 9 0 1 5 1 7 0 |
4 2 0 1 2 6 7 9 2 1 4 0 4 5 9 0 |
4 2 0 1 2 9 7 1 2 1 9 0 7 5 1 0 |
6 3 0 1 8 6 1 2 9 3 1 4 0 1 5 1 7 0 |
6 3 0 1 8 9 1 2 1 3 1 9 0 4 5 9 0 |
6 3 0 1 9 2 1 1 3 3 2 4 0 7 5 1 0 |
1 2 6 0 3 7 5 2 5 5 6 2 9 0 1 5 1 7 0 |
1 2 6 0 3 7 8 2 4 7 6 3 4 0 4 5 9 0 |
1 2 6 0 3 8 1 2 3 9 6 3 9 0 7 5 1 0 |
T a b l e 5 : T h e p o s s i b l e i n t e g r a l g r a p h s p e c t r a w i t h x ; y ; z 0 w = 0
H o f f m a n ' s i d e n t i t y 5 n o w r e a d s
1 0 0 8 0 J = 2 n A
7
1 4 A
5
+ 4 9 A
3
3 6 A + 8 n A
6
1 4 A
4
+ 4 9 A
2
3 6 I
t h e r e f o r e n 1 2 6 0 . T h e p o s s i b l e s p e c t r a a r e s h o w n i n T a b l e 5 . N o n e x i s t e n c e o f
g r a p h s w i t h s p e c t r a i n d i c a t e d b y " i s s h o w n i n 1 9
8/13/2019 Two shorter and simpler proofs in spectral graph theory
7/14
4 - r e g u l a r i n t e g r a l g r a p h s 9 5
5 . C a s e x 0 y 0 z 0 w 0
E l i m i n a t i n g z ; w ; q ; h f r o m 1 4 , w e g e t
z = 9 x 4 y + 4 n 1 6
w = 8 x + 3 y 3 n + 1 5
q = 1 8 x + 3 y 6 n + 6 0
h = 9 6 x 2 6 y + 3 4 n 4 0
H o f f m a n ' s i d e n t i t y 5 n o w r e a d s
4 0 3 2 0 J = 2 n A
8
1 4 A
6
+ 4 9 A
4
3 6 A
2
+ 8 n A
7
1 4 A
5
+ 4 9 A
3
3 6 A
t h e r e f o r e n 5 0 4 0 S i n c e z w 1 a n d q h 0 w e c a n e a s i l y u s i n g a c o m p u t e r
p r o g r a m o b t a i n t h e p o s s i b l e s p e c t r a . T h e r e a r e 1 8 0 3 s u c h s p e c t r a . I n S u b s e c t i o n 3
w e s h o w t h a t t h e u p p e r b o u n d f o r n i s 3 2 8 0 , h e n c e t h e c a s e n = 5 0 4 0 i s i m p o s s i b l e .
I n 1 9 g r a p h a n g l e s a r e e x p l o i t e d t o s h o w t h e n o n e x i s t e n c e o f g r a p h s w i t h s o m e o f
t h e s e s p e c t r a , r e d u c i n g t h e t o t a l n u m b e r o f p o t e n t i a l s p e c t r a i n t h i s c a s e t o 1 2 5 9 .
I t i s a l s o s h o w n t h e r e t h a t t h e r e a r e o n l y 5 p o t e n t i a l s p e c t r a w i t h 6 3 0 n 2 5 2 0
I n T a b l e 7 w e h a v e s h o w n t h o s e w i t h n 2 0 a n d t h e s e 5 e x c e p t i o n a l s p e c t r a . I n
T a b l e 6 f o r e a c h n 5 0 4 0 n 6 3 0 w e g i v e t h e n u m b e r S P o f p o t e n t i a l s p e c t r a .
n S P n S P n S P n S P n S P n S P n S P
2 1 1 2 4 0 2 4 6 3 3 8 9 0 4 4 1 4 0 4 0 2 4 0 2 8 3 6 0 1 7
2 4 1 4 4 2 2 7 7 0 4 0 1 0 5 4 4 1 4 4 4 2 2 5 2 3 2 4 2 0 2 2
2 8 1 7 4 5 2 9 7 2 3 5 1 1 2 3 9 1 6 8 3 2 2 8 0 4 0 5 0 4 3 3
3 0 1 8 4 8 3 1 8 0 3 5 1 2 0 4 4 1 8 0 3 7 3 1 5 4 7 5 6 0 4 0
3 5 2 2 5 6 3 5 8 4 3 9 1 2 6 4 7 2 1 0 4 7 3 3 6 1 4 6 3 0 4 7
3 6 2 3 6 0 3 3
T a b l e 6 : T h e n u m b e r o f p o t e n t i a l i n t e g r a l g r a p h s p e c t r a w i t h x ; y ; z ; w 0 a n d
2 1 n 6 3 0
3 . U P P E R B O U N D O N T H E N U M B E R O F V E R T I C E S
H e r e w e g i v e t h e u p p e r b o u n d o n t h e n u m b e r o f v e r t i c e s i n r e g u l a r b i p a r t i t e
g r a p h , f r o m w h i c h f o l l o w s t h a t t h e r e a r e n o 4 - r e g u l a r b i p a r t i t e i n t e g r a l g r a p h s w i t h
n = 5 0 4 0 . T h i s u p p e r b o u n d p r e s e n t s i m p r o v e m e n t i n c a s e o f b i p a r t i t e g r a p h s u p o n
p r e v i o u s l y k n o w n u p p e r b o u n d 1 1 f o r r e g u l a r g r a p h s o f g i v e n d i a m e t e r .
T h e o r e m 3 . L e t p b e t h e n u m b e r o f v e r t i c e s o f a c o n n e c t e d r - r e g u l a r b i p a r t i t e
g r a p h G = V E w i t h r a d i u s R T h e n
p
2 r 1
R
2
r 2
8/13/2019 Two shorter and simpler proofs in spectral graph theory
8/14
9 6 D . C v e t k o v i c , S . S i m i c , D . S t e v a n o v i c
n x y z w q h L a b e l
8 1 1 3 2 3 3 1 1 0
9 1 2 3 2 3 0 1 1 8
1 0 1 3 3 2 2 7 1 2 6 I
2 0 5
1 0 2 1 2 4 3 9 8 2
1 2 1 5 3 2 2 1 1 4 2
1 2 2 2 6 1 3 0 1 2 4 I
2 4 3
1 2 2 3 2 4 3 3 9 8
1 2 3 1 1 6 4 5 5 4
1 4 1 7 3 2 1 5 1 5 8
1 4 2 4 6 1 2 4 1 4 0
1 4 2 5 2 4 2 7 1 1 4
1 4 3 2 5 3 3 6 9 6
1 4 3 3 1 6 3 9 7 0
1 5 1 8 3 2 1 2 1 6 6 I
3 0 2 3
1 5 2 5 6 1 2 1 1 4 8
1 5 2 6 2 4 2 4 1 2 2
1 5 3 3 5 3 3 3 1 0 4
1 5 3 4 1 6 3 6 7 8
1 5 4 1 4 5 4 5 6 0
1 6 1 9 3 2 9 1 7 4
1 6 2 6 6 1 1 8 1 5 6
1 6 2 7 2 4 2 1 1 3 0
1 6 3 4 5 3 3 0 1 1 2
1 6 3 5 1 6 3 3 8 6
1 6 4 1 8 2 3 9 9 4
1 6 4 2 4 5 4 2 6 8
n x y z w q h L a b e l
1 8 1 1 1 3 2 3 1 9 0
1 8 2 8 6 1 1 2 1 7 2 I
3 6 1
1 8 2 9 2 4 1 5 1 4 6
1 8 3 6 5 3 2 4 1 2 8 I
3 6 2
1 8 3 7 1 6 2 7 1 0 2
1 8 4 3 8 2 3 3 1 1 0
1 8 4 4 4 5 3 6 8 4 I
3 6 3
1 8 5 1 7 4 4 5 6 6
1 8 5 2 3 7 4 8 4 0
2 0 2 1 0 6 1 6 1 8 8
2 0 2 1 1 2 4 9 1 6 2
2 0 3 8 5 3 1 8 1 4 4
2 0 3 9 1 6 2 1 1 1 8
2 0 4 5 8 2 2 7 1 2 6
2 0 4 6 4 5 3 0 1 0 0 I
4 0 1 2
2 0 5 2 1 1 1 3 6 1 0 8
2 0 5 3 7 4 3 9 8 2
2 0 5 4 3 7 4 2 5 6
2 0 6 1 6 6 5 1 3 8
2 0 6 2 2 9 5 4 1 2
7 2 0 2 0 8 1 7 2 3 0 4 3 5 0 0
8 4 0 2 4 4 1 9 6 3 6 4 3 5 0 0
1 2 6 0 3 7 0 2 8 0 5 7 4 3 5 0 0
1 6 8 0 4 9 6 3 6 4 7 8 4 3 5 0 0
2 5 2 0 7 4 8 5 3 2 1 2 0 4 3 5 0 0
T a b l e 7 : T h e p o s s i b l e i n t e g r a l g r a p h s p e c t r a w i t h x ; y ; z ; w 0 a n d n 2 0 a n d
5 e x c e p t i o n a l s p e c t r a .
P r o o f . L e t u b e a v e r t e x o f G w i t h e c c e n t r i c i t y R L e t N
k
u = f v 2 V d u v = k g
b e t h e k
t h
n e i g h b o r h o o d o f u a n d d
k
u = N
k
u t h e n u m b e r o f v e r t i c e s i n i t .
T h e n d
0
u = 1 a n d d
1
u = r W h e n 2 k R 1 i t h o l d s d
k
u r 1 d
k 1
u
s i n c e e a c h v e r t e x o f N
k 1
u m a y b e a d j a c e n t t o a t m o s t r 1 v e r t i c e s o f N
k
u
T h e r e f r o m b y i n d u c t i o n d
k
u r r 1
k 1
f o r 2 k R 1 O n t h e o t h e r h a n d ,
e a c h v e r t e x o f N
R
u i s a d j a c e n t t o r v e r t i c e s o f N
R 1
u s i n c e G i s b i p a r t i t e n o
t w o v e r t i c e s o f N
R
u m a y b e a d j a c e n t s o d
R
u
r 1
r
d
R 1
u r 1
R 1
N o w
p =
P
R
k = 0
d
k
u
2 r 1
R
2
r 2
a n d t h e t h e o r e m i s p r o v e d . 2
A s n o t e d i n 5 , f o r t h e d i a m e t e r D o f a c o n n e c t e d g r a p h G t h e i n e q u a l i t y
D s 1 h o l d s , w h e r e s i s t h e n u m b e r o f d i s t i n c t e i g e n v a l u e s o f G W h e n G i s
8/13/2019 Two shorter and simpler proofs in spectral graph theory
9/14
4 - r e g u l a r i n t e g r a l g r a p h s 9 7
c o n n e c t e d r e g u l a r i n t e g r a l g r a p h o f d e g r e e r w e g e t D 2 r F r o m T h e o r e m 3 t h e n
f o l l o w s t h a t c o n n e c t e d b i p a r t i t e 4 - r e g u l a r i n t e g r a l g r a p h h a s d i a m e t e r a t m o s t 8
a n d i t m a y h a v e a t m o s t p = 2 n 6 5 6 0 v e r t i c e s . B u t f r o m t h e p r e v i o u s s e c t i o n i t
c a n b e s e e n t h a t t h e h i g h e s t p o s s i b l e v a l u e f o r n n o t g r e a t e r t h a n 3 2 8 0 i s 2 5 2 0 T h i s
b o u n d c a n n o t b e l o w e r e d , b u t u s i n g g r a p h a n g l e s i t i s s h o w n i n 1 9 t h a t t h e r e a r e
o n l y 5 p o s s i b l e s p e c t r a w i t h 6 3 0 n 2 5 2 0
4 . A L I S T O F K N O W N 4 - R E G U L A R I N T E G R A L G R A P H S
I n t h i s s e c t i o n w e g i v e a l i s t o f 6 5 c o n n e c t e d 4 - r e g u l a r i n t e g r a l g r a p h s . A
p a r t o f t h e s e g r a p h s a r e k n o w n i n t h e l i t e r a t u r e . T h e o t h e r s a p p e a r h e r e f o r t h e
r s t t i m e a n d a r e d e r i v e d f r o m t h e k n o w n g r a p h s b y s o m e c o n s t r u c t i o n p r o c e d u r e s
d e s c r i b e d b e l o w . W e g i v e t o e a c h o f t h e s e g r a p h s a l a b e l o f t h e f o r m I
p a
w h e r e
p i s t h e n u m b e r o f v e r t i c e s , a n d a i s t h e o r d e r i n g n u m b e r o f g r a p h i n t h e g r o u p
o f g r a p h s w i t h t h e s a m e n u m b e r o f v e r t i c e s . F o r e a c h g r a p h w e g i v e i t s s p e c t r u m
a n d a s h o r t d e s c r i p t i o n o f t h e g r a p h , s o m e t i m e s w i t h r e f e r e n c e s t o t h e l i t e r a t u r e .
G r a p h s i n t h e l i s t a r e s o r t e d b y t h e n u m b e r o f v e r t i c e s a n d t h e n b y t h e s p e c t r a .
O p e r a t i o n s + a n d d e n o t e t h e u s u a l g r a p h s u m a n d p r o d u c t . O p e r a t i o n
d e n o t e s t h e s t r o n g s u m : g i v e n t w o g r a p h s G a n d H w i t h v e r t e x s e t s V G a n d
V H t h e i r s t r o n g s u m G H i s t h e g r a p h w i t h t h e v e r t e x s e t V G V H i n
w h i c h t w o v e r t i c e s x a a n d y b a r e a d j a c e n t i f a n d o n l y i f a i s a d j a c e n t t o b
i n H a n d e i t h e r x i s a d j a c e n t t o y i n G o r x = y L G a n d S G a r e t h e l i n e g r a p h
a n d t h e s u b d i v i s i o n g r a p h o f G , r e s p e c t i v e l y , w h i l e L
2
G = L S G
G r a p h s G
1
; : : : ; G
1 3
a r e c o n n e c t e d c u b i c i n t e g r a l g r a p h s f r o m 1 5 G
1
; : : : ; G
8
a r e b i p a r t i t e , w h i l e G
9
; : : : ; G
1 3
a r e n o t . I n d i e r e n t o r d e r t h e y a r e a l s o g i v e n i n
6 3
I n t h e f o l l o w i n g l i s t w e g i v e t h e i r s p e c t r a .
G
1
= K
3 3
3 0
4
3
G
2
= K
2
+ K
2
+ K
2
= G
9
K
2
t h e c u b e g r a p h :
3 1
3
1
3
3 G
3
T u t t e ' s 8 - c a g e : 3 2
9
0
1 0
2
9
3
G
4
= G
1 0
K
2
= G
1 1
K
2
t h e D e s a r g u e s g r a p h : 3 2
4
1
5
1
5
2
4
3
G
5
3 2
4
1
5
1
5
2
4
3
G
6
3 2 1
2
0
2
1
2
2 3
G
7
= K
2
+ C
6
= G
1 2
K
2
t h e 6 - s i d e d p r i s m : 3 2
2
1 0
4
1 2
2
3
G
8
= G
1 3
K
2
3 2
6
1
3
0
4
1
3
2
6
3
G
9
= K
4
3 1
3
G
1 0
t h e P e t e r s e n g r a p h : 3 1
5
2
4
G
1 1
3 2 1
3
1
2
2
3
G
1 2
= K
2
+ K
3
t h e 3 - s i d e d p r i s m : 3 1 0
2
2
2
G
1 3
= L
2
K
4
: 3 2
3
0
2
1
3
2
3
8/13/2019 Two shorter and simpler proofs in spectral graph theory
10/14
9 8 D . C v e t k o v i c , S . S i m i c , D . S t e v a n o v i c
F o r e a c h G
i
i = 1 ; : : : ; 1 3 t h e g r a p h s L G
i
L G
i
K
2
G
i
+ K
2
a n d G
i
K
2
a r e c o n n e c t e d 4 - r e g u l a r i n t e g r a l g r a p h s . I n t h e T a b l e 8 w e g i v e t h e i r p o s i t i o n s i n
t h e l i s t o f 4 - r e g u l a r i n t e g r a l g r a p h s . I f G
i
i s b i p a r t i t e , t h e n G
i
+ K
2
= G
i
K
2
s o
w e d o n o t d o u b l e t h e c o l u m n f o r t h e s e g r a p h s .
i L G
i
L G
i
K
2
G
i
+ K
2
i L G
i
L G
i
K
2
G
i
+ K
2
G
i
K
2
1 I
9 2
I
1 8 1
I
1 2 2
9 I
6 1
I
1 2 4
I
8 2
I
8 1
2 I
1 2 7
I
2 4 2
I
1 6 1
1 0 I
1 5 2
I
3 0 1
I
2 0 4
I
2 0 1
3 I
4 5 1
I
9 0 1
I
6 0 3
1 1 I
1 5 4
I
3 0 3
I
2 0 7
I
2 0 6
4 I
3 0 4
I
6 0 1
I
4 0 1
1 2 I
9 4
I
1 8 1
I
1 2 5
I
1 2 2
5 I
3 0 5
I
6 0 2
I
4 0 2
1 3 I
1 8 6
I
3 6 2
I
2 4 5
I
2 4 4
6 I
1 5 3
I
3 0 2
I
2 0 5
7 I
1 8 5
I
3 6 1
I
2 4 3
8 I
3 6 4
I
7 2 1
I
4 8 1
T a b l e 8 : T h e p o s i t i o n s o f g r a p h s L G
i
L G
i
K
2
G
i
+ K
2
a n d G
i
K
2
i n t h e
l i s t .
L e t G
i
b e n o n b i p a r t i t e , i . e . i = 9 1 0 ; : : : ; 1 3 G r a p h s G
i
+ K
2
K
2
a n d
G
i
K
2
+ K
2
a r e c o n n e c t e d , 4 - r e g u l a r i n t e g r a l g r a p h s w h i c h , i n a d d i t i o n , a r e
c o s p e c t r a l . H o w e v e r , w e h a v e t h e f o l l o w i n g p r o p o s i t i o n .
P r o p o s i t i o n 1 . L e t G b e a n o n b i p a r t i t e g r a p h . G r a p h s G + K
2
K
2
a n d G
K
2
+ K
2
a r e i s o m o r p h i c .
P r o o f . F o r v e r t i c e s o f G + K
2
K
2
a n d G K
2
+ K
2
w e w i l l s h o r t l y w r i t e u a b
i n s t e a d o f u ; a ; b T h e n u
0
a
0
b
0
u
1
a
1
b
1
2 E G + K
2
K
2
, u
0
a
0
u
1
a
1
2
E G + K
2
a n d b
0
6= b
1
, u
0
u
1
2 E G a
0
= a
1
b
0
6= b
1
o r u
0
= u
1
a
0
6=
a
1
b
0
6= b
1
; a n d u
0
a
0
b
0
u
1
a
1
b
1
2 E G K
2
+ K
2
, u
0
a
0
u
1
a
1
2 E G
K
2
b
0
= b
1
o r u
0
a
0
= u
1
a
1
b
0
6= b
1
, u
0
u
1
2 E G a
0
6= a
1
b
0
= b
1
o r
u
0
= u
1
a
0
= a
1
b
0
6= b
1
N o w i t i s n o t h a r d t o s e e t h a t t h e m a p p i n g f V G +
K
2
K
2
! V G K
2
+ K
2
g i v e n b y f u ; a ; b = u a + b a w h e r e a d d i t i o n
i s m o d u l o 2 , i s a c t u a l l y t h e i s o m o r p h i s m o f t h e s e g r a p h s . 2
H e n c e , a l l g r a p h s G
i
+ K
2
K
2
a n d G
i
K
2
+ K
2
f o r i = 9 1 0 ; : : : ; 1 3
a r e c o n t a i n e d i n t h e l a s t r o w o f T a b l e 8 .
G r a p h s D
1
; : : : ; D
1 6
a n d E
1
; : : : ; E
8
a r e c o n n e c t e d , 4 - r e g u l a r a n d i n t e g r a l , a s
f o u n d i n 1 7 ; t h e y a r e l a b e l l e d i n t h e l i s t b y I
8 1
I
1 0 1
I
1 2 4
I
1 2 2
I
1 6 1
I
1 6 2
I
2 4 1
I
2 4 2
I
3 2 1
I
1 8 1
I
1 8 2
I
1 8 3
I
2 0 1
I
2 0 2
I
2 0 3
I
3 0 1
. G r a p h s E
1
; : : : ; E
8
a r e
l a b e l l e d b y I
5 1
I
6 1
I
8 2
I
1 2 6
I
1 2 7
I
9 4
I
9 2
I
1 5 2
A l l g r a p h s i n t h e l i s t u p t o 1 2 v e r t i c e s h a v e a l s o b e e n g e n e r a t e d i n 1 s e e
F i g . 2 1 ; n o t e t h a t E
5
, i . e . t h e g r a p h 1 7 , i s n o t w e l l d r a w n - s e e a l s o 1 7
8/13/2019 Two shorter and simpler proofs in spectral graph theory
11/14
4 - r e g u l a r i n t e g r a l g r a p h s 9 9
T h e l i s t :
I
5 1
= E
1
= K
5
4 1
4
I
6 1
= E
2
= L K
4
= L G
9
: 4 0
3
2
2
a l s o N E P S o f C
3
a n d K
2
w i t h b a s i s
f 1 1 1 0 g c f . 1 8
I
7 1
= C
4
C
3
4 1 0
2
1
2
3
c f . g r a p h 7 4 0 2 i n t a b l e s o f 1 0
I
8 1
= D
1
= G
9
K
2
= K
4 4
4 0
6
4
c f . g r a p h 8 4 0 6 i n t a b l e s o f 1 0
I
8 2
= E
3
= G
9
+ K
2
= L K
2 4
: 4 2 0
3
2
3
c f . g r a p h 8 4 0 1 i n t a b l e s o f 1 0
I
9 1
4 1
3
0
2
2
2
3 c f . g r a p h 9 4 1 4
i n t a b l e s o f 1 0
I
9 2
= E
7
= L G
1
= C
3
+ C
3
= C
3
C
3
4 1
4
2
4
c f . g r a p h 9 4 1 0 i n t a b l e s o f 1 0
I
9 3
4 2 1 0
2
1
2
2 3 c f . g r a p h
9 4 0 7 i n t a b l e s o f 1 0
I
9 4
= E
6
= L G
1 2
: 4 2 1
2
1
2
2
3
c f . g r a p h 9 4 0 4 i n t a b l e s o f 1 0
I
1 0 1
= I
5 1
K
2
= D
2
= 5 K
2
4 1
4
1
4
4
I
1 2 1
4 1
6
0 2
2
3
2
N E P S o f C
3
K
2
a n d K
2
w i t h b a s i s f 1 1 1 0 0 1
0 1 0 g 1 8
I
1 2 2
= D
4
= G
1
+ K
2
= G
1 2
K
2
4 2 1
4
1
4
2 4 a l s o N E P S o f C
3
K
2
a n d K
2
w i t h b a s i s f 1 1 1 0 1 1
0 0 1 g c f . 1 8
I
1 2 3
4 2 1
4
0 1
2
2 3
2
c f . g r a p h
1 2 i n F i g . 2 1 o f 1
I
1 2 4
= I
6 1
K
2
= D
3
= L G
9
K
2
=
C
3
C
4
4 2
2
0
6
2
2
4
I
1 2 5
= G
1 2
+ K
2
= C
3
+ C
4
4 2
2
1
2
0
1
4
3
2
I
1 2 6
= E
4
4 2
3
0
3
2
5
c f . g r a p h
9 i n T a b l e 9 . 1 i n 4
I
1 2 7
= E
5
= L G
2
: 4 2
3
0
3
2
5
I
1 2 8
4 3 1
3
0 1
3
2
2
3 c f . g r a p h
1 8 i n F i g . 2 1 o f 1
I
1 4 1
= I
7 1
K
2
4 3 1
3
0
4
1
3
3 4
I
1 5 1
4 2
4
1 0
2
1
2
2
4
3
c f . g r a p h X f r o m 1 7
I
1 5 2
= E
8
= L G
1 0
: 4 2
5
1
4
2
5
I
1 5 3
= L G
6
: 4 3 2
2
1
2
0
2
1 2
6
I
1 5 4
= L G
1 1
: 4 3 2
3
0
2
1
3
2
5
I
1 6 1
= I
8 2
K
2
= D
5
= G
2
+ K
2
=
C
4
+ C
4
4 2
4
0
6
2
4
4
I
1 6 2
= D
6
4 2
4
0
6
2
4
4
I
1 8 1
= I
9 2
K
2
= D
1 0
= L G
1
K
2
=
L G
1 2
K
2
= C
3
+ C
3
K
2
4 2
4
1
4
1
4
2
4
4
I
1 8 2
= D
1 1
4 2
4
1
4
1
4
2
4
4
I
1 8 3
= D
1 2
4 2
4
1
4
1
4
2
4
4
I
1 8 4
= C
3
+ C
6
4 3
2
1
4
0
5
2
4
3
2
I
1 8 5
= L G
7
: 4 3
2
2 1
4
0 1
2
2
7
I
1 8 6
= L G
1 3
: 4 3
3
1
2
0
3
1
3
2
6
I
2 0 1
= D
1 3
= G
1 0
K
2
4 2
5
1
4
1
4
2
5
4
I
2 0 2
= D
1 4
4 2
5
1
4
1
4
2
5
4
I
2 0 3
= D
1 5
4 2
5
1
4
1
4
2
5
4
I
2 0 4
= G
1 0
+ K
2
4 2
6
0
5
1
4
3
4
I
2 0 5
= G
6
+ K
2
4 3 2
3
1
3
0
4
1
3
2
3
3 4
8/13/2019 Two shorter and simpler proofs in spectral graph theory
12/14
1 0 0 D . C v e t k o v i c , S . S i m i c , D . S t e v a n o v i c
I
2 0 6
= G
1 1
K
2
4 3 2
3
1
3
0
4
1
3
2
3
3 4
I
2 0 7
= G
1 1
+ K
2
4 3 2
4
1 0
5
1
3
2
2
3
3
I
2 0 8
= L
2
K
5
: 4 3
4
0
5
1
4
2
6
I t i s
p r o v e d i n 7 t h a t L
2
G G c o n n e c t e d
w i t h a t l e a s t t w o v e r t i c e s i s i n t e g r a l i f
a n d o n l y i f G i s a c o m p l e t e g r a p h .
I
2 4 1
= D
7
4 2
8
0
6
2
8
4
I
2 4 2
= D
8
= L G
2
K
2
4 2
8
0
6
2
8
4
I
2 4 3
= G
7
+ K
2
= C
4
+ C
6
4 3
2
2
2
1
6
0
2
1
6
2
2
3
2
4
I
2 4 4
= G
1 3
K
2
4 3
3
1
5
0
6
1
5
3
3
4
I
2 4 5
= G
1 3
+ K
2
4 3
3
2 1
5
0
3
1
5
2
3
3
3
I
3 0 1
= D
1 6
= L G
1 0
K
2
4 2
1 0
1
4
1
4
2
1 0
4
I
3 0 2
= L G
6
K
2
= I
1 5 1
K
2
4 3 2
8
1
3
0
4
1
3
2
8
3 4
I
3 0 3
= L G
1 1
K
2
4 3 2
8
1
3
0
4
1
3
2
8
3 4
I
3 0 4
= L G
4
: 4 3
4
2
5
0
5
1
4
2
1 1
I
3 0 5
= L G
5
: 4 3
4
2
5
0
5
1
4
2
1 1
I
3 2 1
= D
9
4 2
1 2
0
6
2
1 2
4
I
3 5 1
4 2
1 4
1
1 4
3
6
t h e o d d g r a p h
O
4
1 4
I
3 6 1
= L G
7
K
2
4 3
2
2
8
1
6
0
2
1
6
2
8
3
2
4
I
3 6 2
= L G
1 3
K
2
4 3
3
2
6
1
5
0
6
1
5
2
6
3
3
4
I
3 6 3
= C
6
+ C
6
= C
3
+ C
6
K
2
4 3
4
2
4
1
4
0
1 0
1
4
2
4
3
4
4
I
3 6 4
= L G
8
: 4 3
6
2
3
1
4
0
3
1
6
2
1 3
I
4 0 1
= G
4
+ K
2
4 3
4
2
6
1
4
0
1 0
1
4
2
6
3
4
4
I
4 0 2
= G
5
+ K
2
= G
1 0
+ K
2
K
2
=
G
1 1
+ K
2
K
2
= L
2
K
5
K
2
4 3
4
2
6
1
4
0
1 0
1
4
2
6
3
4
4
I t i s i n t e r e s t i n g t o n o t e t h a t t h e g r a p h s
G
1 0
+ K
2
G
1 1
+ K
2
a n d L
2
K
5
a l l h a v e
m u t u a l l y d i e r e n t s p e c t r a .
I
4 5 1
= L G
3
: 4 3
9
1
1 0
1
9
2
1 6
I
4 8 1
= G
8
+ K
2
4 3
6
2
4
1
1 0
0
6
1
1 0
2
4
3
6
4
I
6 0 1
= L G
4
K
2
4 3
4
2
1 6
1
4
0
1 0
1
4
2
1 6
3
4
4
I
6 0 2
= L G
5
K
2
4 3
4
2
1 6
1
4
0
1 0
1
4
2
1 6
3
4
4
I
6 0 3
= G
3
+ K
2
4 3
9
2 1
1 9
1
1 9
2
3
9
4
I
7 0 1
= I
3 5 1
K
2
4 3
6
2
1 4
1
1 4
1
1 4
2
1 4
3
6
4
I
7 2 1
= L G
8
K
2
4 3
6
2
1 6
1
1 0
0
6
1
1 0
2
1 6
3
6
4
I
9 0 1
= L G
3
K
2
4 3
9
2
1 6
1
1 9
1
1 9
2
1 6
3
9
4
8/13/2019 Two shorter and simpler proofs in spectral graph theory
13/14
4 - r e g u l a r i n t e g r a l g r a p h s 1 0 1
5 . C O N C L U D I N G R E M A R K S
P o s s i b l e s p e c t r a o f c o n n e c t e d 4 - r e g u l a r b i p a r t i t e i n t e g r a l g r a p h s , d e t e r m i n e d
i n S e c t i o n 2 , a r e q u i t e n u m e r o u s a n d w e c a n n o t e x p e c t t h a t a l l 4 - r e g u l a r i n t e g r a l
g r a p h s w i l l b e d e t e r m i n e d i n n e a r f u t u r e . N o n e x i s t e n c e r e s u l t s f o r m a n y o f t h e s e
p o t e n t i a l s p e c t r a w i l l b e o b t a i n e d b y c o n s i d e r i n g a p p r o p r i a t e g r a p h p r o p e r t i e s b u t
m a n y o t h e r s w i l l r e m a i n t o b e c o n s i d e r e d .
P o t e n t i a l s p e c t r a f r o m S u b s e c t i o n 1 c a s e x = 0 , i . e . t h o s e f r o m T a b l e s 1
a n d 2 , h a v e b e e n c o n s i d e r e d i n 1 7 a n d a l l t h e c o r r e s p o n d i n g g r a p h s d e t e r m i n e d .
T h e y a r e g r a p h s D
1
; : : : ; D
1 6
. O t h e r k n o w n 4 - r e g u l a r i n t e g r a l g r a p h s a r e p r e s e n t e d
i n S e c t i o n 4 .
R E F E R E N C E S
1 K . T . B a l i
n s k a , M . K u p c z y k , K . Z w i e r z y
n s k i : M e t h o d s o f g e n e r a t i n g i n -
t e g r a l g r a p h s , C o m p u t e r S c i e n c e C e n t e r R e p o r t 4 5 7 , T e c h n i c a l U n i v e r s i t y o f
P o z n a n , 1 9 9 7 .
2 K . T . B a l i
n s k a , S . K . S i m i
c : T h e n o n r e g u l a r , b i p a r t i t e , i n t e g r a l g r a p h s w i t h
m a x i m u m d e g r e e f o u r , t o a p p e a r .
3 F . C . B u s s e m a k e r , D . C v e t k o v i
c D : T h e r e a r e e x a c t l y 1 3 c o n n e c t e d , c u -
b i c , i n t e g r a l g r a p h s , U n i v . B e o g r a d , P u b l . E l e k t r o t e h n . F a k . , S e r . M a t . F i z . ,
5 4 4 5 7 6 1 9 7 6 , 4 3 4 8 .
4 F . C . B u s s e m a k e r , D . C v e t k o v i
c , J . J . S e i d e l : G r a p h s r e l a t e d t o e x c e p -
t i o n a l r o o t s y s t e m s , T . H . - R e p o r t 7 6 - W S K - 0 5 , T e c h n o l o g i c a l U n i v e r s i t y E i n d -
h o v e n , E i n d h o v e n , 1 9 7 6 .
5 D . C v e t k o v i
c : G r a p h s a n d t h e i r s p e c t r a . U n i v . B e o g r a d , P u b l . E l e k t r o t e h n .
F a k . S e r . M a t . F i z . , 3 5 4 3 5 6 1 9 7 1 , 1 - 5 0 .
6 D . C v e t k o v i
c : C u b i c i n t e g r a l g r a p h s , U n i v . B e o g r a d , P u b l . E l e k t r o t e h n . F a k .
S e r . M a t . F i z . , 4 9 8 5 4 1 1 9 7 5 , 1 0 7 1 1 3 .
7 D . C v e t k o v i
c : S p e c t r a o f g r a p h s f o r m e d b y s o m e u n a r y o p e r a t i o n s , P u b l . I n s t .
M a t h . B e o g r a d 1 9 3 3 , 1 9 7 5 , 3 7 4 1 .
8 D . C v e t k o v i
c , M . D o o b , H . S a c h s : S p e c t r a o f g r a p h s - T h e o r y a n d A p -
p l i c a t i o n , D e u t s c h e r V e r l a g d e r W i s s e n s c h a f t e n - A c a d e m i c P r e s s , B e r l i n - N e w
Y o r k , 1 9 8 0 ; s e c o n d e d i t i o n 1 9 8 2 ; t h i r d e d i t i o n , J o h a n n A m b r o s i u s B a r t h V e r l a g ,
H e i d e l b e r g - L e i p z i g , 1 9 9 5 .
9 D . C v e t k o v i
c , I . G u t m a n , N . T r i n a j s t i
c : C o n j u g a t e d m o l e c u l e s h a v i n g
i n t e g r a l g r a p h s p e c t r a , C h e m . P h y s . L e t t e r s 2 9 1 9 7 4 , 6 5 6 8 .
1 0 D . C v e t k o v i
c , Z . R a d o s a v l j e v i
c : A t a b l e o f r e g u l a r g r a p h s o n a t m o s t t e n
v e r t i c e s , P r o c e e d i n g s o f t h e S i x t h Y u g o s l a v S e m i n a r o n G r a p h T h e o r y , D u b r o v n i k
1 9 8 5 , 7 1 1 0 5 .
1 1 H . D . F r i e d m a n : O n t h e i m p o s s i b i l i t y o f c e r t a i n M o o r e g r a p h s , J . C o m b . T h e -
o r y 1 0 1 9 7 1 , 3 , 2 4 5 2 5 2 .
8/13/2019 Two shorter and simpler proofs in spectral graph theory
14/14
1 0 2 D . C v e t k o v i c , S . S i m i c , D . S t e v a n o v i c
1 2 F . H a r a r y , A . J . S c h w e n k : W h i c h g r a p h s h a v e i n t e g r a l s p e c t r a ? , G r a p h s a n d
C o m b i n a t o r i c s P r o c . C a p i t . C o n f . G r a p h T h e o r y a n d C o m b i n a t o r i c s , G e o r g e
W a s h i n g t o n U n i v . , J u n 1 8 2 2 , 1 9 7 3 , e d s . R . B a r i , F . H a r a r y , S p r i n g e r - V e r l a g ,
B e r l i n , 1 9 7 4 , 4 5 5 1 .
1 3 A . J . H o f f m a n : O n t h e p o l y n o m i a l o f a g r a p h , A m e r . M a t h . M o n t h l y 7 0
1 9 6 3 , 3 0 3 6 .
1 4 T . Y . H u a n g : S p e c t r a l c h a r a c t e r i z a t i o n o f o d d g r a p h s O
k
k 6 G r a p h s a n d
C o m b i n a t o r i c s 1 0 1 9 9 4 , 3 , 2 3 5 2 4 0 .
1 5 A . J . S c h w e n k : E x a c t l y t h i r t e e n c o n n e c t e d c u b i c g r a p h s h a v e i n t e g r a l s p e c t r a ,
T h e o r y a n d A p p l i c a t i o n s o f G r a p h s P r o c . I n t e r n a t . C o n f . W e s t e r n M i c h i g a n
U n i v . , K a l a m a z o o , M i c h . M a y 1 1 1 5 , 1 9 7 6 , e d s . Y . A l a v i , D . L i c k , S p r i n g e r -
V e r l a g , B e r l i n - H e i d e l b e r g - N e w Y o r k 1 9 7 8 , 5 1 6 5 3 3 .
1 6 S . S i m i
c , Z . R a d o s a v l j e v i
c : T h e n o n r e g u l a r , n o n b i p a r t i t e , i n t e g r a l g r a p h s
w i t h m a x i m u m d e g r e e f o u r , J . C o m b . , I n f . S y s t . S c i . 2 0 1 4 1 9 9 5 , 9 2 6 .
1 7 D . S t e v a n o v i
c : 4 - R e g u l a r i n t e g r a l g r a p h s a v o i d i n g 3 i n t h e s p e c t r u m , t o a p -
p e a r .
1 8 D . S t e v a n o v i
c : R e g u l a r i n t e g r a l N E P S o f g r a p h s , t o a p p e a r .
1 9 D . S t e v a n o v i
c : N o n e x i s t e n c e o f s o m e 4 - r e g u l a r i n t e g r a l g r a p h s , t o a p p e a r .
U n i v e r s i t y o f B e l g r a d e , R e c e i v e d O c t o b e r 7 , 1 9 9 8
F a c u l t y o f E l e c t r i c a l E n g i n e e r i n g ,
P . O . B o x 3 5 - 5 4 , 1 1 1 2 0 B e l g r a d e ,
Y u g o s l a v i a
e c v e t k o d @ u b b g . e t f . b g . a c . y u
e s i m i c s @ u b b g . e t f . b g . a c . y u
U n i v e r s i t y o f N i s ,
D e p a r t m e n t o f M a t h e m a t i c s ,
F a c u l t y o f P h i l o s o p h y ,
C i r i l a i M e t o d i j a 2 , 1 8 0 0 0 N i s ,
Y u g o s l a v i a
d r a g a n c e @ n i . a c . y u