Combinatorics and Graph Theory


Integral Eigen-Pair Balanced Classes of Graphs Ratios, Asymptotes, Density and Areas

Authors: Paul August Winter, Carol Lynne Jessop

The association of integers, conjugate pairs and tightness with the eigenvalues of graphs provide 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 non-zero, distinct, eigenvalues, whose sum (or product) yields the same integer, either as a fixed constant, or a function of an inherent aspect of the graph (such as its size), is said to be sum-eigen*(a+b)*pair balanced (or product-eigen*(a.b)*pair balanced, respectively). For example, complete graphs on n vertices, are eigen-bi-balanced with sum-eigen*(n-2)*pair balanced and product-eigen*(1-n)*pair balanced, and since a,b are non-zero their reciprocals (which affect the tightness of a graph ) are defined, so that this class has the eigen-balanced ratio of 1/a+1/b=(a+b)/(a.b)= (n-2)/(1-n) =f(n) which is asymptotic to the constant value of -1. The absolute value of the integral of f(n) multiplied by the average degree yields the area (n-1)(n-ln(n-1)) – we show that this is the maximum area for most known classes of eigen-bi-balanced graphs. We investigate the effect of this asymptotic ratio on the energy of the molecular representation of graphs. Cycles are generally neither sum-eigen-pair, nor product-eigen-pair balanced, while paths are only sum- eigen-pair balanced. In this paper, we introduce a class of graphs, involving q cliques each of size q, and show that this class is eigen-bi-balanced with respect to the sum -1 and product 1-q so that it has ratio 1/(q-1) asymptotic to 0, and has area q(2q+2ln(q-1)), and discuss its eigen-bi-balanced criticality.

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[v1] 2013-05-08 10:15:41

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