An n-ary k-radius sequence is a finite sequence of elements taken from an alphabet of size n such that any two distinct elements of the alphabet occur within distance k of each other somewhere in the sequence. These sequences were introduced by Jaromczyk and Lonc to model a caching strategy for computing certain functions on large data sets such as medical images. Let fk(n) be the shortest leng...

It is well known that the spectral radius of a tree whose maximum degree is D cannot exceed 2 √ D − 1. In this paper we derive similar bounds for arbitrary planar graphs and for graphs of bounded genus. It is proved that a the spectral radius ρ(G) of a planar graph G of maximum vertex degree D ≥ 4 satisfies √D ≤ ρ(G) ≤ √8D − 16 + 7.75. This result is best possible up to the additive constant—we...