Some Algebraic Properties of Bi-Cayley Graphs

1. Some Algebraic Properties of Bi-Cayley Graphs Hua Zou and Jixiang Meng College of Mathematics and Systems Science,Xinjiang University

2. 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 graph. 1.Definition Circulant Graph When G is a cyclic group, the Cayley digraph(graph) D(G;S)(C(G;S)) is called a circulant digraph(graph).

3. 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

4. Example:

5. 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 2.Main Result We use T(G,S) to denote the number of spanning trees of a Connected Bi-Circulant graph BC(G,S). Theorem 2.1.The adjacency matrix of a Cayley digraph of abelian group is normal.

6. Corollary 2.3.Let be the eigenvalues of C(G,S). Then the eigenvalues of BC(G,S) are Since the eigenvalues of an undirected graph are real, we deduce the following corollary by Theorem 2.2 .

7. (1)The eigenvalues of the Bi-Circulant digraph BC(G,S) are (2)If S=-S, the eigenvalues of the Bi-Circulant graph BC(G,S) are Theorem2.4.Let G be a cyclic group of integers modulo n and be a subset of G.

8. 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

9. 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

10. Lemma 2.7.Let where If , then the roots of f(z) satisfy

11. Theorem2.8. Let BC(G,S) be the connected Bi-Circulant graph of order n. Then

12. Theorem2.9.Let BC(G,S) be the connected Bi-irculant graph of order n.Then

13. Example: 3.Recent Main Result For a digraph D with , we define

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

15. Corollary 3.2 Let D be a graph. Let be the eigenvalues of the adjacency matrix of D.Then the eigenvalues of are

16. Thank You!