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)2D n . (ii) If G is a connected nonbipartite graph of order n and diameter D, then µ (G) + µmin (G) > 2 µ (G)2D n . For large µ and D these bounds are close to the best possible ones.

• Type: Journal
• Year: 2007
• Where: COMBINATORICS
• Authors: Vladimir Nikiforov
• Keywords: Adjacency Matrix, COMBINATORICS 2007, Graph G, Smallest Eigenvalues