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

Charles R. Johnson·Raphael Loewy·Paul Anthony Smith
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Abstract

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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Charles R. Johnson, Raphael Loewy, Paul Anthony Smith. 2007-01-20. The Graphs for which the Maximum Multiplicity of an Eigenvalue is Two. https://arxiv.org/abs/math/0701562

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