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Explore basic definitions, history, and main results on integral complete multipartite graphs. Learn about adjacency matrix, integral graphs, characteristic polynomial, and more. Discover examples and determine integral graphs. Dive into the history of integral graphs and our main results on 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.}
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)
