Generalizing some results to the normalized Laplacian
نویسنده
چکیده
The use of spectral methods in graph theory has allowed for some amazing results where an arithmetic invariant (i.e., diameter, chromatic number, and so on) has been bounded and analyzed using analytic tools. The key has been to examine the spectrum of various matrices associated with graphs and to try to “hear the shape” of the graph from the spectrum. The three most widely used spectrums are those of the adjacency matrix, the combinatorial Laplacian, and the normalized Laplacian. The entries of the adjacency matrix A = (aij) act as indicator variables for an edge, i.e., aij = 1 if i and j are joined by an edge in the graph and 0 otherwise. The combinatorial Laplacian is L = D − A where A is the adjacency matrix and D is the diagonal matrix with the degree of the vertices along the diagonal. Finally, the normalized Laplacian is given by L = D−1/2LD−1/2 = I −D−1/2AD−1/2, so that its entries are defined by
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