Integral complete multipartite graphs
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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: [email protected] 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: [email protected]

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)


Our main results

Our Main Results


Our main results1

Our main results


Our main results2

Our main results


Our main results3

Our main results


Our main results4

Our main results


Our main results5

Our main results


Integral complete multipartite graphs

Thank you


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