  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.
