Laplacian matrix of a graph

The incidence matrix of an oriented graph -
Orientation of a graph X is the assignment of direction to each edge.
An arc of a graph is an ordered pair of adjacent vertices.
Formal definition -
       An orientation of X can be defined as a function sigma from the arcs of X to {-1,1} such that, if (u,v) is an arc, then sigma(u,v) = -sigma(v,u).

Let D be the incidence matrix of the graph X.
The laplacian of X is the matrix Q(X) = D*D'
Laplacian does not depend on the orientation.

Q is symmetric => its eigen values are real.
Its positive semidefinite => evas are all non-negative.

TREES - The number of spanning trees is determined by the laplacian.
[Spanning tree - The spanning tree of a connected undirected graph G is a tree that includes all the vertices and some or all of the edges of G.]

Let e = uv be an edge of X.
The graph obtained by deleting the edge e is denoted by X \ e. Its edge set is denoted by E(X) \ e.
Vertex set remains the same.
contracting the edge e => X / e.