A note on degenerate and spectrally degenerate graphs A graph G is called spectrally d-degenerate if the largest eigenvalue of each subgraph of it with Summary: A note on degenerate and spectrally degenerate graphs Noga Alon Abstract A graph G is called spectrally d-degenerate if the largest eigenvalue of each subgraph of it with maximum degree D is at most dD. We prove that for every constant M there is a graph with minimum degree M which is spectrally 50-degenerate. This settles a problem of DvorŽak and Mohar. 1 Introduction The spectral radius (G) of a (finite, simple) graph G is the largest eigenvalue of its adjacency matrix. A graph is d-degenerate if any subgraph of it contains a vertex of degree at most d. A result of Hayes [5] asserts that any d-degenerate graph with maximum degree at most D has spectral radius at most 2 dD. In fact, the result is a bit stronger, as follows. Proposition 1.1 ([5]) Let G be a graph having an orientation in which every outdegree is at most d and every indegree is at most D. Then (G) 2 dD. For completeness we include a simple proof (which is somewhat shorter than the one given in [5]). Collections: Mathematics