Integral eigen-pair balanced classes of graphs with their ratio, asymptote, area, and involution-complementary aspects
Summary: 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 \((a+b)/(a\cdot b)\) 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 \((n-2)/(1-2)=f(n)\) 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.
scientific article; zbMATH DE number 6370089