An explicit upper bound on disparity for trees of a given diameter. Issue 19 (16th December 2022)

• Record Type: Journal Article
• Authors: Cinzori, Isaac; Johnson, Charles R.; Lang, Hannah
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 19(2022)
• Page Start: 4584
• Page End: 4596
• Publication Date: 2022-12-16
• Subjects: Branch duplication -- distinct eigenvalues -- graph of a matrix -- seed -- rooted seed; 15A18 -- 05B20 -- 05C50; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2021.1887070
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: It is known that the minimum number of distinct eigenvalues c ( T ) of a symmetric matrix whose graph is a given tree T is at least the diameter d ( T ) of that tree. However, the disparity c ( T ) − d ( T ) can be positive. Using branch duplication and rooted seeds, the notion of the 'most complex seed' is introduced, and an explicit upper bound on the disparity is given for any tree of a given diameter.