Ordered multiplicity inverse eigenvalue problem for graphs on six vertices

For a graph $G$, we associate family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location nonzero off-diagonal entries $M$ is governed by adjacency structure $G$. The ordered multiplicity Inverse Eigenvalue Problem Graph (IEPG) concerned with finding all attainable lists eigenvalue multiplicities matrices in $\mathcal{S}(G)$. connected graphs order six, offer significant progress on IEPG, as well complete solution to IEPG. We also show that while $K_{m,n}$ $\min(m,n)\ge 3$ attains particular list, it cannot do so arbitrary spectrum.