Abstract
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·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-n)=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.
Highlights
Integers, Conjugate Pairs, and Eigenvalues of a GraphThe graphs in this paper are simple and connected and all graph properties mentioned will be according to definitions in Harris et al [1]
In Sarkar and Mukherjee [2], graphs are considered with reciprocal pairs of eigenvalues (λ, 1/λ) whose product is the integer 1
(iii) Summing the eigenvalues of the adjacency matrix of a graph is connected to the energy of physical structures; see Aimei et al [4]
Summary
The graphs in this paper are simple and connected and all graph properties mentioned will be according to definitions in Harris et al [1]. There often exist two eigenvalues (associated with the adjacency matrix of a graph) whose sum or product is integral It is possible to get the same integer when adding or multiplying two distinct, nonzero eigenvalues. This integer is either a fixed constant or a function of an inherent property of the graph, for all graphs belonging to a certain class of graphs. The class of complete bipartite graphs Kp,p on n = 2p vertices has as its associated eigenvalues p, −p, and 0 (as per Brouwer and Haemers [6]), so it is exact sum∗(0)∗eigen-pair balanced.
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