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Gersgorin variations I: on a theme of Pupkov and Solov'ev

✍ Scribed by Alan J. Hoffman

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 43 KB
Volume: 304
Category: Article
ISSN: 0024-3795
DOI: 10.1016/s0024-3795(99)00218-9

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✦ Synopsis

We unify and extend various "additive" sufficient conditions of Pupkov and of Solov'ev for the nonsingularity of a complex matrix. This paper is intended to be the first in a sequcnce of "variations" on theorems of the Gersgorin genre. The whole sequence is dedicated to the memory of Olga Taussky-Todd, whose lovely paper [O. Taussky, Amer. Math. Monthly 56 (1949) 672-676] inspired an interest in matrix theory for a generation of mathematicians.

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## Abstract A graph __G__ is degree‐bounded‐colorable (briefly, db‐colorable) if it can be properly vertex‐colored with colors 1,2, …, k ≤ Δ(__G__) such that each vertex __v__ is assigned a color __c__(__v__) ≤ __v__. We first prove that if a connected graph __G__ has a block which is neither a com