4,724 research outputs found
Spectral Graph Theory and Research
Our topic of study was Spectral Graph Theory. We studied the algebraic methods used to study the properties of graphs (networks) and became familiar with the applications of network analysis. We spent a significant amount of time studying the way virus’s spread on networks, with particular applications to Covid-19. We also investigated the relationship between graph spectra and structural properties
Unsolved Problems in Spectral Graph Theory
Spectral graph theory is a captivating area of graph theory that employs the
eigenvalues and eigenvectors of matrices associated with graphs to study them.
In this paper, we present a collection of topics in spectral graph theory,
covering a range of open problems and conjectures. Our focus is primarily on
the adjacency matrix of graphs, and for each topic, we provide a brief
historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng
Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper
will be published in Operations Research Transaction
The Grone-Merris Conjecture
In spectral graph theory, Grone and Merris conjecture that the spectrum of
the Laplacian matrix of a finite graph is majorized by the conjugate degree
sequence of this graph. We give a complete proof for this conjecture.Comment: The paper is accepted by Transactions of the American Mathematical
Societ
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