We state here a few noteworthy results on graph energy that did not fit into the previous sections.
E(G) ≥ 4 holds for all connected graphs, except for K1, K2, K2,1, and K3,1 .
The rank of a graph is the rank of its adjacency matrix. For a connected bipartite graph G of rank ,
For any graph, E ≥ .
Let χ(G) be the chromatic number of graph G. For any n-vertex graph G, E(G) ≥ 2(n – χ()) .
The inequality E(G) + E() ≥ 2n is satisfied by all n-vertex graphs, n ≥ 5, except by Kn and Kn – e .
As an immediate special case of the Koolen–Moulton upper bound (7.2), for an n-vertex regular graph of degree r, we have E(G) ≤ E0, where
Balakrishnan  showed that for any ε > 0, there exist infinitely many n, for which there are n-vertex regular graphs of degree ...