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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 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 • 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 Basic definitions • A simple graph: G:=(V(G),E(G)) • adjacency matrix:
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)
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
…. 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)
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 • 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 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 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)
