Ordering graphs with index in the interval (2,2+5)

The index of a graph is the largest eigenvalue (or spectral radius) of its adjacency matrix. We consider the problem of ordering graphs by the index in the class of connected graphs with a fixed order n   and index belonging to the interval (2,2+5). For any fixed n (provided that n is not too small), we order a significant portion of graphs whose indices are close to the end points of the above interval.