The graphs for which the maximum multiplicity of an eigenvalue is two

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 composed of certain identified families of ‘exceptional’ partial 2-trees that are not linear.