Elsevier

Discrete Mathematics

Volume 310, Issue 24, 28 December 2010, Pages 3455-3461
Discrete Mathematics

Path spectra for trees

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Abstract

The path spectrum of a graph is the set of lengths of all maximal paths in the graph. A set S of positive integers is spectral if it is the path spectrum of a tree. We characterize the spectral sets containing at most two odd integers (and arbitrarily many even ones) and obtain several necessary conditions for a set to be spectral. We show that for each even integer s2 at least 1/4 of all subsets of the set {2,3,,s} are spectral and conjecture that all the subsets with at least 3s/4 integers are spectral.

Keywords

Maximal path
Tree
Path spectrum

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