STOC
2010
ACM

# Approximations for the Isoperimetric and Spectral Profile of Graphs and Related Parameters

13 years 9 months ago
The spectral profile of a graph is a natural generalization of the classical notion of its Rayleigh quotient. Roughly speaking, given a graph G, for each 0 < < 1, the spectral profile G() minimizes the Rayleigh quotient (from the variational characterization) of the spectral gap of the Laplacian matrix of G over vectors with support at most over a suitable probability measure. Formally, the spectral profile G of a graph G is a function G : [0, 1/2] R defined as: G() def = min xRV d(supp(x)) P gij(xi - xj)2 P i dix2 i . where gij is the weight of the edge (i, j) in the graph, di is the degree of vertex i, and d(supp(x)) is the fraction of edges incident on vertices within the support of vector x. While the notion of the spectral profile has numerous applications in Markov chain, it is also is closely tied to its isoperimetric profile of a graph. Specifically, the spectral profile is a relaxation for the problem of approximating edge expansion of small sets in graphs. In this w...