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A lower bound for the spectral radius of graphs with fixed diameter
Cioaba, S.M.,van Dam, E.R.,Koolen, J.H.,Lee, J.H. Academic Press 2010 European journal of combinatorics : Journal europ& Vol.31 No.6
We determine a lower bound for the spectral radius of a graph in terms of the number of vertices and the diameter of the graph. For the specific case of graphs with diameter three we give a slightly better bound. We also construct families of graphs with small spectral radius, thus obtaining asymptotic results showing that the bound is of the right order. We also relate these results to the extremal degree/diameter problem.
Asymptotic results on the spectral radius and the diameter of graphs
Cioaba, S.M.,van Dam, E.R.,Koolen, J.H.,Lee, J.H. North Holland [etc.] 2010 Linear Algebra and its Applications Vol. No.
We study graphs with spectral radius at most 322 and refine results by Woo and Neumaier [R. Woo, A. Neumaier, On graphs whose spectral radius is bounded by 322, Graphs Combinatorics 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.