Some Algebraic Properties of Bi-Cayley Graphs

Hua Zou and Jixiang Meng

College of Mathematics and Systems Science,Xinjiang University

Circulant Graph When G is a cyclic group, the Cayley digraph(graph) D(G;S)(C(G;S)) is called a circulant digraph(graph).

Cayley Graph For a group G and a subset S of G, the Cayley digraph D(G; S) is a graph with vertex set G and arc set

. When ,D(G,S) corresponds to an undirected graph C(G,S), which is called a Cayley gr

aph.

SsGgsgg ,|,1SS

1.Definition

Bi-Cayley Graph For a finite group G and a subset T of G, the Bi-Cayley graph X=BC(G,T) is defined as the bipartite graph with vertex set and edge set

}1,0{G},|)}1,(),0,{{( SsGgsgg

Example:

Theorem2.2. Let G be an abelian group and let be the eigenvalues of the Cayley digraph D(G,S). Then the eigenvalues of BC(G,S) are

n ,,, 21

.,,, 21 n

Theorem 2.1. The adjacency matrix of a Cayley digraph of abelian group is normal.

We use T(G,S) to denote the number of spanning trees of a Connected Bi-Circulant graph BC(G,S).

2.Main Result

Since the eigenvalues of an undirected graph are real, we deduce the following corollary by Theorem 2.2 .

Corollary 2.3. Let be the eigenvalues of C(G,S). Then the eigenvalues of BC(G,S) are

n ,,, 21

n ,,, 21

Theorem2.4. Let G be a cyclic group of integers modulo n and

be a subset of G.

ksssS ,,, 21

)1,,2,1)((, 21 njk jsjsjs k

(2)If S=-S, the eigenvalues of the Bi-Circulant

graph BC(G,S) are

)1,,2,1(||, 21 njk jsjsjs k

(1)The eigenvalues of the Bi-Circulant digraph BC(G,S) are

Theorem2.5. Let G be a cyclic group of integers modulo n and S be a subset of G.If S is a union of some , then BC(G,S) is integral. In particular, if S=-S, then BC(G,S) is integral if and only if S is a union of some

)(dGn

)(dGn

Lemma 2.6. Let G be a cyclic group of integers modulo n. Let be a subset of G with S=-S. If the polynomial

have the roots ,then

where

1 2, , , kS s s s

ml

ss

i

is

i

is

i

is

i

imlk

zzzzzf1

0

12

0

12

0

12

0

2)(21

,1221 ,,, ks

)1(

)1()1(

),(

12

1

)12)(1(

f

k

SGT

k

k

s

j

nj

sn

ml

mlk ssssf )(222)1( 1

）（ kss 11

Lemma 2.7. Let

where If , then the roots of f(z) satisfy

ml

ss

i

is

i

is

i

is

i

imlk

zzzzzf1

0

12

0

12

0

12

0

2)(21

.1 21 ksss 1),,,gcd( 21 ksss

1, 1,2, , 2 1.i ki s

Theorem 2.8. Let BC(G,S) be the connected Bi-Circulant graph of order n. Then

nf

kSGT

n

,)1(

~),(12

Theorem 2.9.Let BC(G,S) be the connected Bi-irculant graph of order n.Then

1),(1

lim 2

1

np

SGTk

3.Recent Main Result For a digraph D with , we define nvvvDV ,,,)( 21 D

)}(|)}1,(),0,{{()(

)1,(,),1,()0,(,),0,()( 11

DEvvvvDE

vvvvDV

jiji

nn

Example:

Theorem 3.1 Let D be a digraph and A be its adjacency matrix. Let be the eigenvalues of A. If A is normal,the eigenvalues of the adjacency matrix of are

n ,,, 21

||,|,||,| 21 n

D

For a graph X with ,we define graph of X where is the associated digraph of X.

},,,{)( 21 nvvvXV

X

X

n ,,, 21

Corollary 3.2 Let D be a graph. Let

be the eigenvalues

of the adjacency matrix of D.Then the

eigenvalues of are

n ,,, 21

D

Thank You!