On a class of polynomials and its relation with the spectra and diameters of graphs
type: Item
instance of: scholarly article
title: On a class of polynomials and its relation with the spectra and diameters of graphs
zbMATH Open document ID: 0857.05101
author: E. Garriga
author: Miquel Angel Fiol
author: J. Luis A. Yebra
published in: Journal of Combinatorial Theory. Series B
publication date: 1996-10-14T00:00:00Z
review text: Let \(\lambda_1,\lambda_2,\dots,\lambda_d\) be reals with \(\lambda_1>\lambda_2>\cdots>\lambda_d\). For every \(k=1,2,\dots,d\), the \(k\)-alternating polynomial \(P_k\) is the polynomial of degree \(k\) and norm \(|P_k|_\infty= \max_{1\leq i\leq d}\{|P_k(\lambda_i)|\}\leq 1\) that attains maximum absolute value at any point \(\lambda\not\in[\lambda_d,\lambda_1]\). These polynomials can be considered as the discrete version of Chebychev polynomials. Some basic properties of \(P_k\) are studied, and it is shown how to compute them in general. The results are applied to the study of the relationship between the (standard or Laplacian) spectrum of a graph or bipartite graph and its diameter, improving previous results.
Mathematics Subject Classification ID: 05E35
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C12
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 934804
zbMATH Keywords: \(k\)-alternating polynomial
zbMATH Keywords: Chebychev polynomials
zbMATH Keywords: spectrum
zbMATH Keywords: diameter
reviewed by: Hans-Dietrich O. F. Gronau
MaRDI profile type: Publication
full work available at URL: jctb.1996.0033
OpenAlex ID: W2001247565
DOI: 10.1006/JCTB.1996.0033
description: scientific article; zbMATH DE number 934804