Showing papers on "Spectral graph theory published in 1990"

Partitioning sparse matrices with eigenvectors of graphs

Abstract: The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is, shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorith...

Pairing theorem of graph eigenvalues: Its new proof and a generalization

Abstract: Each undirected graph has its own adjacency matrix, which is real and symmetric. The negative of the adjacency matrix, also real and symmetric, is a well-defined mathematically elementary concept. By this negative adjacency matrix, the negative of a graph can be defined. Then an orthogonal transformation can be readily found that transforms a negative of an alternant graph to that alternant graph: (−G) G. Since the procedure does not involve the edge weights, the pairing theorem holds true for all edge-weighted alternant graphs, including the usual “standard” graphs.

Topological solution of algebraic eigenvalue problem

Abstract: A simple graph theoretical algorithm for simultaneous determination of eigenfunctions, eigenvalues and characteristic polynomials of real symmetric matrices has been developed. The method starts with representing the matrixA−λI, whereI is an unit matrix of the size ofA, by an undirected weighted graph (G) and an assumed set of eigenfunctions. Conditions necessary to disconnect one vertex completely fromG are then developed. The method does not require any property related to the geometrical symmetry group of the graph and is applicable even to matrices containing a number of multiple eigenvalues.

Partially Connected Models of Associative Memory

Abstract: We investigate the role of graph connectivity in a spin glass model of associative memory. Using replica symmetric mean field theory and spectral graph theory we characterize the storage capacity and retrieval error of partially connected neural networks. Our results have the form of topologically invariant upper and lower bounds on the storage capacity and an asymptotic expansion for the retrieval error. We find that partially connected neural networks have storage and retrieval characteristics that are comparable to fully connected networks. This is of some interest for electronic implementations of neural networks which by necessity are partially connected.