Spectral Graph Theory via Higher Order Eigenvalues and Applications to the Analysis of Random Walks | Zendy

Shayan Oveis Gharan | Zendy

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Spectral Graph Theory via Higher Order Eigenvalues and Applications to the Analysis of Random Walks

Author(s) -

Shayan Oveis Gharan

Publication year - 2015

Publication title -

annales de la faculté des sciences de toulouse mathématiques

Language(s) - English

Resource type - Journals

eISSN - 2258-7519

pISSN - 0240-2963

DOI - 10.5802/afst.1465

Subject(s) - adjacency matrix , spectral graph theory , eigenvalues and eigenvectors , laplacian matrix , mathematics , graph energy , random matrix , spectral properties , random walk , spectral theory , adjacency list , combinatorics , matrix (chemical analysis) , spectral clustering , graph theory , graph , pure mathematics , line graph , voltage graph , physics , statistics , quantum mechanics , materials science , hilbert space , astrophysics , composite material , cluster analysis

— A basic goal in the field of spectral theory is to relate eigenvalues of matrices associated to a graph, namely the adjacency matrix, the Laplacian matrix or the random walk matrix, to the combinatorial properties of that graph. Classical results in this area mostly study the properties of first, second or the last eigenvalues of these matrices [4, 3, 21, 2]. In the last several years many of these results are extended and the bounds are improved using higher order eigenvalues. In this short monologue we overview several of these recent advances, and we describe one of the fundamental tools employed in these results, namely, the spectral embedding of graphs. (1) University of Washington

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