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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.

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