General Distance Energies and General Distance Estrada Index of Random Graphs

Abstract

In 2000s, Gutman and Güngör introduced the concept of distance energy and the distance Estrada index for a simple graph <i>G</i> respectively. Moreover, many researchers established a large number of upper and lower bounds for these two invariants. But there are only a few graphs attaining the equalities of those bounds. In this paper, however, the exact estimates to general distance energy are formulated for almost all graphs by probabilistic and algebraic approaches. The bounds to general distance Estrada index are also established for almost all graphs by probabilistic and algebraic approaches. The results of this paper generalize the results of the distance energy and distance Estrada of random graph.