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Integral Complete Multipartite Graphs

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Integral Complete Multipartite Graphs

Ligong Wang1 and Xiaodong Liu2

1Department of Applied Mathematics, Northwestern Polytechnical University,

E-mail: ligongwangnpu@yahoo.com.cn

2School of Information, Xi'an University of Finance and Economics

Supported by NSFC (N0.70571065), NBSC (No.LX2005-20), SRF for ROCS,

SEM (No.2005CJ110002) and DPOP in NPU.}

- Basic definitions.
- History of integral graphs.
- Main results on Integral Complete Multipartite Graphs

V(G)={v1,v2,v3,v4,v5},

E(G)={v1v2,v1v4,v2v3, v2v4, v3v4, v4v5}.

v2

v3

v1

v4

v5

- A simple graph: G:=(V(G),E(G))

- adjacency matrix:

1

3

2

- Characteristic polynomial:P(G,x)=det(xIn-A(G)).
- Integral graph:
A graph G is called integral if all the zeros of the characteristic polynomial P(G,x) are integers.

- Example 2.

P(K3,x)=det(xI3-A(K3))=(x+1)2(x-2)

2

2

3

integralYes: n=3,4,6No: otherwise

integral Yes: all

1

3

4

1

n

5

n

4

Cn

Kn

- Our purpose is to determine or characterize:
Problem: Which graphs are integral? (Harary and Schwenk, 1974).

- Examples of integral graphs

….

n

1

2

n

1

n-1

2

3

….

1

2

m

Km,n

Pn

3

2

4

1

n

5

Nn

Wn

integral

Yes: mn=c2

No:

otherwise

integral

Yes: n=4

No: otherwise

(Wheel graph)

integral

Yes: n=2

No: otherwise

integral

Yes: all

(Empty graph)

1

1

4

2

2

1

n

3

r

m

2

K1,n-1 of diameter 2

T[m,r] of diameter 3

t

T(m,t)

t

m

T(m,t)

T(m,t)

t

r

T(m,t) of diameter 4

T(r,m,t) of diameter 6

integral

Yes: t=k2,

m+t=(k+s)2

No: otherwise

integral

Yes: t=k2,

m+t=(k+s)2

No: otherwise

integral

Yes: n=k2

No:

otherwise

integral

Yes: m=r=k(k+1)

or (m,r)=d

No: otherwise

- Integral cubic graphs,Bussemaker, Cvetković(1975), Schwenk(1978)
- Integral complete multipartite graphs,Roitman, (1984). Wang, Li and Hoede, (2004),
- Integral graphs with maximum degree 4.Radosavljević,Simić, (1986). Balińska,Simić , (2001). Simić , Zwierzyński, (2004),etc.

- Integral 4-regular graphs,Cvetković, Simić, Stevanović(1998,1999,2003)
- Integral trees.Watanabe, Schwenk, (1979); Li and Lin, (1987); Liu, (1988); Cao (1988, 1991) ; P. Hĺc and R. Nedela, (1998); Wang, Li and Liu, (1999); Wang, Li (2000,2004) ; P. Hĺc and and M. Pokornў, (2003),etc.

- In 1984, an infinite family of integral complete tripartite graphs was constructed by Roitman.
(Roitman, An infinite family of integral graphs, Discrete Math. 52 (1984)

- In 2001, Balińska and Simić remarked that the general problem seems to be intractable.
(Balińska and Simić, The nonregular, bipartite, integral graphs with

maximum degree 4. Part I: basic properties, Discrete Math. 236 (2001).

- In 2004, we give a sufficient and necessary condition for complete r-partite graphs to be integral, from which we can construct infinitely many new classes of such integral graphs.
( Wang, Li and Hoede, Integral complete r-partite graphs, Discrete Math., 283 (2004)

Thank you