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

Abstract

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, sqrt(2 + sqrt(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.