  Let $G$ be a graph. For a subset $X$ of $V(G)$, the switching $\sigma$ of $G$
is the signed graph $G^{\sigma}$ obtained from $G$ by reversing the signs of
all edges between $X$ and $V(G)\setminus X$. Let $A(G^{\sigma})$ be the
adjacency matrix of $G^{\sigma}$. An eigenvalue of $A(G^{\sigma})$ is called a
main eigenvalue if it has an eigenvector the sum of whose entries is not equal
to zero. Let $S_{n,k}$ be the graph obtained from the complete graph $K_{n-r}$
by attaching $r$ pendent edges at some vertex of $K_{n-r}$. In this paper we
prove that there exists a switching $\sigma$ such that all eigenvalues of
$G^{\sigma}$ are main when $G$ is a complete multipartite graph, or $G$ is a
harmonic tree, or $G$ is $S_{n,k}$. These results partly confirm a conjecture
of Akbari et al.
