The minimal spectral radius of graphs of order n with diameter n-4

In this paper we determine the graphs which have the minimal spectral radius (i.e., the largest eigenvalue of its corresponding adjacency matrix) among all the graphs of order n with the diameter D=n-4. This result settles a problem proposed in [E.R. van Dam, R.E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra Appl. 423 (2007) 408–419], which is also the special case D=n-4 of the Conjecture 8 in van Dam and Kooij (2007).