Solutions to decision-making problems in management engineering using molecular computational algorithms and experimentations

制度:新 ; 報告番号:甲3368号 ; 学位の種類:博士(工学) ; 授与年月日:2011/5/23 ; 早大学位記番号:新568

Bipartite Crossing Numbers of Meshes and Hypercubes

. Let G = (V0 ; V1 ; E) be a connected bipartite graph, where V0 ; V1 is the bipartition of the vertex set V (G) into independent sets. A bipartite drawing of G consists of placing the vertices of V0 and V1 into distinct points on two parallel lines x0 ; x1 , respectively, and then drawing each edge with one straight line segment which connects the points of x0 and x1 where the endvertices of the edge were placed. The bipartite crossing number of G, denoted by bcr(G) is the minimum number of crossings of edges over all bipartite drawings of G. We develop a new lower bound method for estimating bcr(G). It relates bipartite crossing numbers to edge isoperimetric inequalities and Laplacian eigenvalues of graphs. We apply the method, which is suitable for "well structured" graphs, to hypercubes and 2-dimensional meshes. E.g. for the n\Gammadimensional hypercube graph we get n4 n\Gamma2 \Gamma O(4 n ) bcr(Qn) n4 n\Gamma1 : We also consider a more general setting of the method whi..