Diameter minimal trees. Issue 3 (3rd March 2016)

• Record Type: Journal Article
• Authors: Johnson, Charles R.; Saiago, Carlos M.
• Is Part Of: Linear & multilinear algebra. Volume 64:Issue 3(2016)
• Page Start: 557
• Page End: 571
• Publication Date: 2016-03-03
• Subjects: branch duplication -- diameter -- distinct eigenvalues -- Hermitian matrix -- multiplicities -- tree; 15A18 -- O5CO5 -- O5C50; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2015.1057097
• Languages: English
• ISSNs: 0308-1087

Abstract

Abstract : Using the method of seeds and branch duplication, it is shown that for every tree of diameter, there is an Hermitian matrix with as few as distinct eigenvalues (a known lower bound). For diameter 7, some trees require 8 distinct eigenvalues, but no more; the seeds for which 7 and 8 are the worst case are classified. For trees of diameter, it is shown that, in general, the minimum number of distinct eigenvalues is bounded by a function of . Many trees of high diameter permit as few of distinct eigenvalues as the diameter and a conjecture is made that all linear trees are of this type. Several other specific, related observations are made.