  Let $G$ be an undirected graph on $n$ vertices and let $S(G)$ be the set of
all $n \times n$ real symmetric matrices whose nonzero off-diagonal entries
occur in exactly the positions corresponding to the edges of $G$. The inverse
eigenvalue problem for a graph $G$ is a problem of determining all possible
lists that can occur as the lists of eigenvalues of matrices in $S(G).$ This
question is, in general, hard to answer and several variations were studied,
most notably the minimum rank problem. In this paper we introduce the problem
of determining for which graphs $G$ there exists a matrix in $S(G)$ whose
characteristic polynomial is a square, i.e. the multiplicities of all its
eigenvalues are even. We solve this question for several families of graphs.
