Integral eigen-pair balanced classes of graphs with their ratio, asymptote, area, and involution-complementary aspects
type: Item
instance of: scholarly article
title: Integral eigen-pair balanced classes of graphs with their ratio, asymptote, area, and involution-complementary aspects
zbMATH Open document ID: 1302.05113
DOI: 10.1155/2014/148690
author: Paul August Winter
author: Carol Lynne Jessop
published in: International Journal of Combinatorics
publication date: 2014-11-17T00:00:00Z
review text: Summary: The association of integers, conjugate pairs, and robustness with the eigenvalues of graphs provides the motivation for the following definitions. A class of graphs, with the property that, for each graph (member) of the class, there exists a pair \(a,b\) of nonzero, distinct eigenvalues, whose sum and product are integral, is said to be eigen-bibalanced. If the ratio \((a+b)/(a\cdot b)\) is a function \(f(n)\), of the order \(n\) of the graphs in this class, then we investigate its asymptotic properties. Attaching the average degree to the Riemann integral of this ratio allowed for the evaluation of eigen-balanced areas of classes of graphs. Complete graphs on \(n\) vertices are eigen-bibalanced with the eigen-balanced ratio \((n-2)/(1-2)=f(n)\) which is asymptotic to the constant value of \(-1\). Its eigen-balanced area is \((n-1)(n- \ln (n-1))\) -- we show that this is the maximum area for most known classes of eigen-bibalanced graphs. We also investigate the class of eigen-bibalanced graphs, whose class of complements gives rise to an eigen-balanced asymptote that is an involution and the effect of the asymptotic ratio on the energy of the graph theoretical representation of molecules.
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C75
zbMATH DE Number: 6370089
zbMATH Keywords: asymptotic properties
zbMATH Keywords: eigen-bibalanced complete graphs
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/148690
OpenAlex ID: W2148127706
Wikidata QID: Q59043522
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description: scientific article; zbMATH DE number 6370089