The recently conceived Randić energy (RE) is examined, and its relation to the (earlier much studied) total π-electron energy (E) is investigated. Within classes of molecular graphs, there exists a relatively good (increasing) linear correlation between RE and E. However, several significant differences between the structure-dependencies of RE and E have been discovered, the most striking of which is their dependence on the number m of edges of the underlying graph. Whereas, with increasing m, the average value of E increases, reaches a maximum and then decreases, the average value of RE monotonically decreases. The structure of the connected graph with a fixed number of vertices and maximal RE value was established.
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Furtula et al. (2013) studied this question.