Asymptotic results on the spectral radius and the diameter of graphs

We study graphswith spectral radius atmost 3/2√2 and refine results byWoo and Neumaier [R.Woo, A. Neumaier, On graphs whose spectral radius is bounded by 3/2√2, Graphs Combin. 23 (2007) 713–726]. We study the limit points of the spectral radii of certain families of graphs, and apply the results to the problem of minimizing the spectral radius among the graphs with a given number of vertices and diameter. In particular, we consider the cases when the diameter is about half the number of vertices, and when the diameter is near the number of vertices. We prove certain instances of a conjecture posed by Van Dam and Kooij [E.R. Van Dam, R.E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra Appl. 423 (2007) 408–419] and show that the conjecture is false for the other instances