Let G be a graph of order n and let μ be an eigenvalue of multiplic- ity m. A star complement for μ in G is an induced subgraph of G of order n-m with no eigenvalue μ. Here we first identify among Cactus graphs, complete three partite graphs Kn,n,n, and bicyclic graphs which can be star complement for 1 as the second largest eigenvalue. Using the graphs obtained, we next search for their maximal extensions, either by theoretical means, or by computer aided search.