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

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.