The Spectral Radius of Subgraphs of Regular Graphs - Nikiforov, Vladimir (The Electronic Journal of Combinatorics, 2007)
Let μ( G) and μ_( G) be the largest and smallest eigenvalues of the adjacency matrix of a graph G. Our main results are: (i) Let G be a regular graph of order n and finite diameter D. If H is a proper subgraph of G, then μ( G) -μ( H) >1nD. (ii) If G is a regular nonbipartite graph of order n and finite diameter D, then μ( G) +μ_( G) >1nD.