GRAPH SPECTRA AND ISOMORPHISM TESTING
Abstract
The graph isomorphism problem has been heavily researched in the last decade. Recently, approaches from topology and group theory have resulted polynomial time isomorphism tests for graphs of bounded genus and graphs of bounded valence. In this thesis we investigate an algebraic approach the isomorphism problem by studying the spectrum of a graph. The spectrum of a graph is the set of eigenvalues of the graph's adjacency matrix. The eigenvalue multiplicity of a graph is the maximum dimension of the eigenspaces of the graph's adjacency matrix. We give two new classes of graphs for which polynomial time isomorphism tests exist: undirected graphs with bounded eigenvalue multiplicity and the more general class of directed graphs with bounded multiplicity of Jordan blocks. A problem closely related to the isomorphism problem is whether a deterministic certificate exists for a particular class of graphs. A deterministic certificate is a polynomial time computable function mapping graphs to numbers so that two graphs are mapped to the same value if and only if they are isomorphic. We give a deterministic certificate for graphs with bounded eigenvalue multiplicity.
Degree
Ph.D.
Subject Area
Computer science
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