Table of Contents
International Journal of Combinatorics
Volume 2014, Article ID 148690, 16 pages
Research Article

Integral Eigen-Pair Balanced Classes of Graphs with Their Ratio, Asymptote, Area, and Involution-Complementary Aspects

Department of Mathematics, Howard College, University of KwaZulu-Natal, Glenwood, Durban 4041, South Africa

Received 26 May 2014; Accepted 3 September 2014; Published 23 September 2014

Academic Editor: Cai Heng Li

Copyright © 2014 Paul August Winter and Carol Lynne Jessop. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 of nonzero, distinct eigenvalues, whose sum and product are integral, is said to be eigen-bibalanced. If the ratio is a function , of the order 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 vertices are eigen-bibalanced with the eigen-balanced ratio which is asymptotic to the constant value of −1. Its eigen-balanced area is —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.