Ordered Multiplicity Inverse Eigenvalue Problem For Graphs On Six Vertices

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