enumeration of the number of spanning trees in some ... - Toubkal

50 CHAPTER 2. SPANNING TREESIn the following work, we consider only the graphs that do not have loops.Definition 2.4.7 The Laplacian matrix L of an undirected graph G is defined as thedifference between its degree matrix D and its adjacency matrix A.L = D − A.More formally, an undirected graph G = (V G , E G ) of n vertices, weighted by the weightfunction at any edge (v i , v j ) associated weight p(v i , v j ).The Laplacian matrix of G verifies :⎧⎨ deg(v i ) = ∑ nk=1 p(v i, v k ) if i = jL i,j := −p(v i , v j )if i ≠ j and (v i , v j ) ∈ E G⎩0 otherwise.That is, the diagonal elements have values equal to the degree of the corresponding vertices,and the off-diagonal elements are −p(v i , v j ) (the number of edges that connects v iwith v j ) if an edge connects the two vertices, and 0 otherwise.Example 2.4.8 The Laplacian matrix of a labeled graph G shown in Figure 2.7 is givenby:⎛⎞ ⎛⎞ ⎛⎞2 0 0 0 0 1 1 0 2 −1 −1 0L = ⎜ 0 3 0 0⎟⎝ 0 0 4 0 ⎠ − ⎜ 1 0 2 0⎟⎝ 1 2 0 1 ⎠ = ⎜ −1 3 −2 0⎟⎝ −1 −2 4 −1 ⎠0 0 0 1 0 0 1 0 0 0 −1 1Example 2.4.9 The graph G pictured in Figure 2.8 gives:⎡⎤2 −1 0 0 −1 0−1 4 −1 0 −1 −1L =0 −1 3 −1 0 −1⎢ 0 0 −1 2 0 −1⎥⎣−1 −1 0 0 3 −1⎦0 −1 −1 −1 −1 4Remark 2.4.10 Let G be a finite undirected graph without loops. The Laplace matrix ofG is the matrix L indexed by the vertex set of G, with zero row sums, where L vu = −A vufor v ≠ u. If D is the diagonal matrix, indexed by the vertex set of G such that D vv isthe degree (valency) of v, then L = D − A. The matrix Q = D + A is called the signlessLaplace matrix of G. An important property of the Laplace matrix L and the signlessLaplace matrix Q is that they are positive semidefinite. Indeed, one has Q = MM T andL = NN T , if M is the incidence matrix of G and N the directed incidence matrix of thedirected graph obtained by orienting the edges of G in an arbitrary way.