Revisiting two classical results on graph spectra
Let μ (G) and μmin (G) be the largest and smallest eigenvalues of the adjacency matrix of a graph G. Our main results are: (i) If H is a proper subgraph of a connected graph G of order n and diameter D, then μ (G) - μ (H) > 1/μ (G)2dn. (ii) If G is a connected nonbipartite graph of order n and diameter D, then μ (G) + μmin (G) > 2/μ (G)2dn. For large μ and D these bounds are close to the best possible ones.
Electronic Journal of Combinatorics
Nikiforov, V. (2007). Revisiting two classical results on graph spectra. Electronic Journal of Combinatorics, 14 (1 R), 1-7. https://doi.org/10.37236/932