| | |
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]).
|
