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Integral Complete Multipartite Graphs. Ligong Wang 1 and Xiaodong Liu 2 1 Department of Applied Mathematics, Northwestern Polytechnical University, E-mail: ligongwangnpu@yahoo.com.cn 2 School of Information, Xi'an University of Finance and Economics

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Integral Complete Multipartite Graphs

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Integral complete multipartite graphs
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.}


Contents
Contents

  • Basic definitions.

  • History of integral graphs.

  • Main results on Integral Complete Multipartite Graphs


Basic definitions

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

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

v2

v3

v1

v4

v5

Basic definitions

  • A simple graph: G:=(V(G),E(G))

  • adjacency matrix:


Basic definitions1

1

3

2

Basic definitions

  • 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)


Basic definitions2

2

2

3

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

integral Yes: all

1

3

4

1

n

5

n

4

Cn

Kn

Basic definitions

  • Our purpose is to determine or characterize:

    Problem: Which graphs are integral? (Harary and Schwenk, 1974).

  • Examples of integral graphs


Basic definitions3

….

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

Basic definitions

integral

Yes: n=4

No: otherwise

(Wheel graph)

integral

Yes: n=2

No: otherwise

integral

Yes: all

(Empty graph)


Basic definitions4

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

Basic definitions


History of integral graphs
History of integral graphs

  • 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.


History of integral graphs1
History of integral graphs

  • 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.


Our main results integral complete multi partite graphs
Our main resultsIntegral complete multi-partite graphs

  • 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)









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