The Graphs for which the Maximum Multiplicity of an Eigenvalue is Two

Details The Graphs for which the Maximum Multiplicity of an Eigenvalue is Two The Graphs for which the Maximum Multiplicity of an Eigenvalue is Two

2007-01-20

arxiv.org/abs/math/0701562v1

Mathematics

Combinatorics

Scientific paper

Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general fields), but only certain partial 2-trees guarantee maximum multiplicity 2. Among partial linear 2-trees, they are only those whose vertices can be covered by two "parallel" induced paths. The remaining graphs that guarantee maximum multiplicity 2 are comprised by certain identified families of "exceptional" partial 2-trees that are not linear.

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