adjacency matrix A, that is, its set of eigenvalues together with their multiplic- ities The Laplace cency matrix of a bipartite graph has the form A = [ 0 B B⊤ 0 ]
thought of as the adjacency matrix of a bipartite graph B(G) of order 2n, where the rows and columns correspond to the bipartition of B(G) For agraph H, let k(H)
Solution (#1027) Let A be the adjacency matrix of a bipartite graph with vertices v1, ,vn As the graph is bipartite we can partition the vertex set into disjoint
A regular graph is a graph in which all vertices have the same degree Note that Adjacency Matrix of an A bipartite graph (or bigraph) is a network whose
The formula for the inverse of a bipartite graph with a unique perfect matching ( and the inverse of a nonsingular tree) follow as special cases Page 50 Conclusion
A powerful and widespread class of network analysis methods is based on algebraic graph theory, i e , representing graphs as square adjacency matrices
adjacency matrix A, that is, its set of eigenvalues together with their multiplic- ities The Laplace cency matrix of a bipartite graph has the form A = [ 0 B B⊤ 0 ]
13 mai 2013 · An example Bipartite graph on 6 vertices: 1 2 3 4 5 6 Figure 2 4 Bipartite Graph The general form for the adjacency matrix of a bipartite
10 sept 2007 · The reduced adjacency matrix of a bipartite graph G = (A,B,E) (having A ∪ B = { a1, , am}∪{b1, , bn} as a vertex set, and E as an edge set), is
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Spectra of graphs Andries E. Brouwer Willem H. Haemers 2
Contents1 Graph spectrum11 1.1 Matrices associated to a graph . . . . . . . . . . . . . . . . . . . 11 1.2 The spectrum of a graph . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 Characteristic polynomial . . . . . . . . . . . . . . . . . . 13 1.3 The spectrum of an undirected graph . . . . . . . . . . . . . . . 13 1.3.1 Regular graphs . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.2 Complements . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.3 Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.4 Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.5 Spanning trees . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.6 Bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.7 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Spectrum of some graphs . . . . . . . . . . . . . . . . . . . . . . 17 1.4.1 The complete graph . . . . . . . . . . . . . . . . . . . . . 17 1.4.2 The complete bipartite graph . . . . . . . . . . . . . . . . 17 1.4.3 The cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.4 The path . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.5 Line graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.6 Cartesian products . . . . . . . . . . . . . . . . . . . . . . 19 1.4.7 Kronecker products and bipartite double . . . . . . . . . . 19 1.4.8 Strong products . . . . . . . . . . . . . . . . . . . . . . . 19 1.4.9 Cayley graphs . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5.1 DecomposingK10into Petersen graphs . . . . . . . . . . 20 1.5.2 DecomposingKninto complete bipartite graphs . . . . . 20 1.6 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.7 Algebraic connectivity . . . . . . . . . . . . . . . . . . . . . . . . 22 1.8 Cospectral graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.8.1 The 4-cube . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.8.2 Seidel switching . . . . . . . . . . . . . . . . . . . . . . . . 23 1.8.3 Godsil-McKay switching . . . . . . . . . . . . . . . . . . . 24 1.8.4 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 24 1.9 Very small graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3
4CONTENTS 2 Linear algebra29 2.1 Simultaneous diagonalization . . . . . . . . . . . . . . . . . . . . 29 2.2 Perron-Frobenius Theory . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Equitable partitions . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 Equitable and almost equitable partitions of graphs .. . 32 2.4 The Rayleigh quotient . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5 Interlacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6 Schur"s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.7 Schur complements . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.8 The Courant-Weyl inequalities . . . . . . . . . . . . . . . . . . . 36 2.9 Gram matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.10 Diagonally dominant matrices . . . . . . . . . . . . . . . . . . . 38 2.10.1 Gersgorin circles . . . . . . . . . . . . . . . . . . . . . . . 38 2.11 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Eigenvalues and eigenvectors of graphs 41 3.1 The largest eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.1 Graphs with largest eigenvalue at most 2 . . . . . . . . . 42 3.1.2 Subdividing an edge . . . . . . . . . . . . . . . . . . . . . 43 3.1.3 The Kelmans operation . . . . . . . . . . . . . . . . . . . 44 3.1.4 Spectral radius of a graph with a given number of edges .44 3.2 Interlacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Regular graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Bipartite graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Cliques and cocliques . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5.1 Using weighted adjacency matrices . . . . . . . . . . . . . 47 3.6 Chromatic number . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.6.1 Using weighted adjacency matrices . . . . . . . . . . . . . 50 3.6.2 Rank and chromatic number . . . . . . . . . . . . . . . . 50 3.7 Shannon capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.7.1 Lov´asz"?-function . . . . . . . . . . . . . . . . . . . . . . 51 3.7.2 The Haemers bound on the Shannon capacity . . . . . . . 53 quotesdbs_dbs7.pdfusesText_5
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