The Spectral Radius of Subgraphs of Regular Graphs - Nikiforov, Vladimir (The Electronic Journal of Combinatorics, 2007)
Let $μ\left( G\right) $ and $μ_{\min}\left( G\right) $ 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 $$ μ\left( G\right) -μ\left( H\right) >{1\over nD}. $$ (ii) If $G$ is a regular nonbipartite graph of order $n$ and finite diameter $D$, then $$ μ\left( G\right) +μ_{\min}\left( G\right) >{1\over nD}. $$