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

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).