Integral Complete Multipartite Integral Complete Multipartite GraphsGraphs

Ligong WangLigong Wang11 and Xiaodong Liu and Xiaodong Liu22

11Department of Applied Mathematics, Northwestern Department of Applied Mathematics, Northwestern Polytechnical University,Polytechnical University,

E-mail: E-mail: ligongwangnpu@yahoo.com.cnligongwangnpu@yahoo.com.cn

22School of Information, Xi'an University of Finance School of Information, Xi'an University of Finance and Economicsand Economics

Supported by NSFC (N0.70571065), NBSC (No.LX2005-20), SRF for ROCS,Supported by NSFC (N0.70571065), NBSC (No.LX2005-20), SRF for ROCS,SEM (No.2005CJ110002) and DPOP in NPU.}SEM (No.2005CJ110002) and DPOP in NPU.}

ContentsContents

Basic definitions.Basic definitions.

History of integral graphs.History of integral graphs.

Main results on Integral Complete Main results on Integral Complete Multipartite GraphsMultipartite Graphs

Basic definitionsBasic definitions

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

v1

v3v2

v5v4

V(G)={vV(G)={v11,, vv22,, vv33,v,v44,v,v55}, },

E(G)={vE(G)={v11vv22,, vv11vv44,, vv22vv33, v, v22vv44, v, v33vv44, v, v44vv55}.}.

otherwise

vva jiij 0

oadjacent t is 1

nnijaGA ][)(adjacency matrix:adjacency matrix:

Characteristic polynomial:Characteristic polynomial: P(G,x)=det(xIP(G,x)=det(xInn-A(G)).-A(G)).

Integral graph:Integral graph: A graph G is called A graph G is called integralintegral if all the zeros of the characteristic polynomial P(G,x) are if all the zeros of the characteristic polynomial P(G,x) are

integers.integers.

Example 2.Example 2.

Basic definitionsBasic definitions

2

1

3

P(KP(K33,x)=det(xI,x)=det(xI33-A(K-A(K33))=(x+1)))=(x+1)22(x-2)(x-2)

Basic definitionsBasic definitions Our purpose is to determine or characterize:Our purpose is to determine or characterize:

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

Examples of integral graphsExamples of integral graphs

1

2

3

n 4Kn

integralintegral Yes: allYes: all

integralintegral Yes: n=3,4,6Yes: n=3,4,6No: otherwiseNo: otherwise

1

2

4

3

n 5Cn

Basic definitionsBasic definitions

Km,n

1 2

1 2 …. m

…. n

Pn

1

2 3 n-1

n

4

5

Wn

n

2

3

1

Nn

integralintegral Yes: n=2Yes: n=2 No: No:

otherwiseotherwise

integralintegral Yes: Yes:

mn=c mn=c22

No:No:

otherwiseotherwise

integralintegral Yes: n=4Yes: n=4 No: No:

otherwiseotherwise

((Wheel graph)Wheel graph)

integralintegral Yes: allYes: all

((Empty graphEmpty graph))

Basic definitionsBasic definitions integralintegral Yes: m=r=k(k+1)Yes: m=r=k(k+1) or (m,r)=dor (m,r)=d

No: otherwiseNo: otherwise

integralintegral Yes: Yes:

n=k n=k22

No:No:

otherwiseotherwise

integralintegral Yes: t=kYes: t=k2,2,

m+t=(k+s)m+t=(k+s)22

No: No: otherwiseotherwise

integralintegral Yes: t=kYes: t=k2,2,

m+t=(k+s)m+t=(k+s)22

No: No: otherwiseotherwise

K1,n-1 of diameter 2

4

23 n

1

T[m,r] of diameter 3

1

2

rm

21

22

22

)(

)(lk

lk

yy

yy

dr

dm

T(m,t) of diameter 4

t

t

t

m

rtba

rtmba22

22

T(r, m,t) of diameter 6

T(m,t)

T(m,t) T(m,t)r

History of integral graphsHistory of integral graphs

Integral cubic graphs,Integral cubic graphs,

Bussemaker, Cvetković(1975), Schwenk(1978) Bussemaker, Cvetković(1975), Schwenk(1978)

Integral complete multipartite graphsIntegral complete multipartite graphs,,

Roitman, (1984). Wang, Li and Hoede, (2004),Roitman, (1984). Wang, Li and Hoede, (2004),

Integral graphs with maximum degree 4.Integral graphs with maximum degree 4.

Radosavljević,Simić, (1986). Balińska,Simić , Radosavljević,Simić, (1986). Balińska,Simić , (2001). Simić , Zwierzyński, (2004),etc.(2001). Simić , Zwierzyński, (2004),etc.

History of integral graphsHistory of integral graphs

Integral 4-regular graphs,Integral 4-regular graphs, Cvetković, Simić, Stevanović(1998,1999,2003)Cvetković, Simić, Stevanović(1998,1999,2003)

Integral trees.Integral trees. Watanabe, Schwenk, (1979); Li and Lin, Watanabe, Schwenk, (1979); Li and Lin, (1987); Liu, (1988); Cao (1988, 1991) ; P. Hĺc (1987); Liu, (1988); Cao (1988, 1991) ; P. Hĺc and R. Nedela, (1998); Wang, Li and Liu, and R. Nedela, (1998); Wang, Li and Liu, (1999); Wang, Li (2000,2004) ; P. Hĺc and (1999); Wang, Li (2000,2004) ; P. Hĺc and and M. Pokornand M. Pokornўў, (2003),etc., (2003),etc.

Our main resultsOur main resultsIntegral complete multi-partite graphsIntegral complete multi-partite graphs

In 1984, an infinite family of integral complete tripartite graphs was In 1984, an infinite family of integral complete tripartite graphs was constructed by Roitman.constructed by Roitman.

(Roitman, An infinite family of integral graphs, Discrete Math. 52 (1984)(Roitman, An infinite family of integral graphs, Discrete Math. 52 (1984)

In 2001, Balińska and Simić remarked that the general problem seems In 2001, Balińska and Simić remarked that the general problem seems to be intractable.to be intractable.

(Balińska and Simić, The nonregular, bipartite, integral graphs with(Balińska and Simić, The nonregular, bipartite, integral graphs with maximum degree 4. Part I: basic properties, Discrete Math. 236 (2001).maximum degree 4. Part I: basic properties, Discrete Math. 236 (2001).

In 2004, we give a sufficient and necessary condition for complete r-In 2004, we give a sufficient and necessary condition for complete r-partite graphs to be integral, from which we can construct infinitely many partite graphs to be integral, from which we can construct infinitely many new classes of such integral graphs.new classes of such integral graphs.

( Wang, Li and Hoede, Integral complete r-partite graphs, ( Wang, Li and Hoede, Integral complete r-partite graphs, Discrete Discrete MathMath., ., 283283 (2004) (2004)

Our Main ResultsOur Main Results

Our main resultsOur main results

Our main resultsOur main results

Our main resultsOur main results

Our main resultsOur main results

Our main resultsOur main results