Revisiting Two Classical Results on Graph Spectra

Let (G) and min (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 (G) (H) > 1 2D (G)n : (ii) If G is a connected nonbipartite graph of order n and diameter D, then (G) + min (G) > 2 2D (G)n : These bounds have the correct order of magnitude for large and D. Keywords: smallest eigenvalue, largest eigenvalue, diameter, connected graph, bipartite graph 1 Introduction Our notation is standard (e.g., see [2], [3], and [5]). In particular, unless speci…ed otherwise, all graphs are de…ned on the vertex set [n] = f1; :::; ng and (G) and min (G) stand for the largest and smallest eigenvalues of the adjacency matrix of a graph G. The aim of this note is to re…ne quantitatively two well-known results on graph spectra. The …rst one, following from Frobenius’s theorem on nonnegative matrices, asserts that if H is a proper subgraph of a connected graph G; then (G) > (H) : The second one, due to H. Sachs [7], asserts that if G is a connected nonbipartite graph, then (G) > min (G) : Our main result is the following theorem. Theorem 1 If H is a proper subgraph of a connected graph G of order n and diameter D; then (G) (H) > 1 2D (G)n : (1)