Matrix analysis
The complexity of regular subgraph recognition
Discrete Applied Mathematics - Computational combinatiorics
Dense graphs without 3-regular subgraphs
Journal of Combinatorial Theory Series B
On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors
Journal of the ACM (JACM)
A simple condition implying rapid mixing of single-site dynamics on spin systems
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Spectral radius of finite and infinite planar graphs and of graphs of bounded genus
Journal of Combinatorial Theory Series B
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It is well known that the spectral radius of a tree whose maximum degree is @D cannot exceed 2@D-1. A similar upper bound holds for arbitrary planar graphs, whose spectral radius cannot exceed 8@D+10, and more generally, for all d-degenerate graphs, where the corresponding upper bound is 4d@D. Following this, we say that a graph G is spectrally d-degenerate if every subgraph H of G has spectral radius at most d@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 4dlog"2(@D(G)/d) (if @D(G)=2d). It is shown that the dependence on @D in this upper bound cannot be eliminated, as long as the dependence on d is subexponential. It is also proved that the problem of deciding if a graph is spectrally d-degenerate is co-NP-complete.