Tracy Powell

Harvey Mudd College Mathematics 2005

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End Year Report Draft: The Singular Values of the Exponentiated Adjacency Matrices of Broom-Tree Graphs
Poster Draft: Singular Values of the EAM of Broom-Tree Graphs
Mid-Year Report: Bounds on the Ratio of Eigenvalues
Expository Paper: Eigenalues of Exponentiated Adjacency Matrices
Thesis Proposal: Eigenvalues of Adjacency Matrices
Thesis Advisor: Prof. Lesley A. Ward
Second Reader: Estelle Basor
E-Mail: tpowell@hmc.edu

Investigating the Ratio of Eigenvalues for Adjacency Matrices

Search engines, such as Google and Ask.com, use certain algorithms to determine the ranking of pages related to the search input. Usually these algorithms use the adjacency matrix of the graph of the webpages. The ijth entry of this matrix, A, is 1 if there is a link from web page i to web page j and 0 if not. Creating a new matrix, B = exp(A) - I, the dominant eigenvector of BB^T, which is the eigenvector corresponding to the unique leading eigenvalue, will give information on the relevancy of the web pages in the graph. What is of specific interest are the particular graphs where the leading eigenvalue and the second leading eigenvalue become increasingly close, that is, the ratio of the second to the leading eigenvalue tends toward one.