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 *K*_{1}, *K*_{2}, *K*_{2,1}, and *K*_{3,1} [126].

The rank of a graph is the rank of its adjacency matrix. For a connected bipartite graph *G* of rank [126],

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()) [126].

The inequality *E*(*G*) + *E*() ≥ 2*n* is satisfied by all *n*-vertex graphs, _{n} ≥ 5, except by *K _{n}* and

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*) ≤ *E*_{0}, where

Balakrishnan [59] showed that for any ε > 0, there exist infinitely many *n*, for which there are *n*-vertex regular graphs of degree ...

Start Free Trial

No credit card required