If is a digraph (say, irreflexive binary relation on a set ), we then defined the incidence matrix as the matrix whose entry was iff for vertices NOTE: This definition is not the traditional one; I accidentally wrote it incorrectly in a previous post and have made the correction to that definition. The correct definition of is the matrix where the entry is iff or . That is, the entry is iff is a vertex on the edge We can then define the Laplacian of as
We will simply write as the Laplacian when it is clear what digraph we are discussing.
Now consider a map and the corresponding matrix whose th row is We will denote this matrix The authors () call the map a representation of the graph (or digraph in our case) in As the -vectors aren’t operators on it seems counterintuitive to call the map a representation, but I will discuss an matrix they mention in a bit.
Definition 1. Let We define the energy with respect to of the graph by
Furthermore we can define each edge to have a weight and define the weighted energy of with respect to and by
Now let be the diagonal matrix whose diagonal entry is for a chosen weight function
Proposition 2. Let be an antisymmetric () digraph on , and be a weight function on Then
One may call the matrix the weighted Laplacian of Now the matrix is an matrix which depends upon a triple with an antisymmetric digraph on and So we could define a map where
Unfortunately this map isn’t a homomorphism, and hence not technically a representation; I couldn’t find a simple structure on compatible with addition of matrices.
 Godsil, Chris and Gordon Royle. Algebraic Graph Theory. Graduate Texts in Mathematics. Vol. 207. Springer Science and Business Media. 2004.