Spectrally degenerate graphs: Hereditary case

It is well known that the spectral radius of a tree whose maximum degree is∆ cannot exceed 2 √ ∆− 1. Similar upper bound holds for arbitrary planar graphs, whose spectral radius cannot exceed √ 8∆+10, and more generally, for all d-degenerate graphs, where the corresponding upper bound is √ 4d∆. Following this, we say that a graph G is spectrally d-degenerate if every subgraph H of G has spectral radius at most √ d∆(H). In this paper we derive a rough converse of the above-mentioned results by proving that each spectrally d-degenerate graph G contains a vertex whose degree is at most 4d log2(∆(G)/d) (if ∆(G) ≥ 2d). It is shown that the dependence on ∆ in this upper Supported in part by the grant GA201/09/0197 of Czech Science Foundation. Institute for Theoretical Computer Science is supported as project 1M0545 by the Ministry of Education of the Czech Republic. Supported in part by the Research Grant P1–0297 of ARRS (Slovenia), by an NSERC Discovery Grant (Canada) and by the Canada Research Chair program. On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia.