### Abstract:

The energy of a graph is the sum of the absolute values of the eigenvalues
of its adjacency matrix. The concept is related to the energy of a class of
molecules in chemistry and was first brought to mathematics by Gutman
in 1978 ([8]). In this thesis, we do a comprehensive study on the energy
of graphs and digraphs.
In Chapter 3, we review some existing upper and lower bounds for
the energy of a graph. We come up with some new results in this chapter.
A graph with n vertices is hyper-energetic if its energy is greater than
2n−2. Some classes of graphs are proved to be hyper-energetic. We find
a new class of hyper-energetic graphs which is introduced and proved to
be hyper-energetic in Section 3.3.
The energy of a digraph is the sum of the absolute values of the real
part of the eigenvalues of its adjacency matrix. In Chapter 4, we study
the energy of digraphs in a way that Pe˜na and Rada in [19] have defined.
Some known upper and lower bounds for the energy of digraphs are reviewed.
In Section 4.5, we bring examples of some classes of digraphs
in which we find their energy.
Keywords. Energy of a graph, hyper-energetic graph, energy of a digraph.