Spanning Trees Recall the definitions of: - graphs, the vertex set - - PowerPoint PPT Presentation

Spanning Trees

Recall the definitions of:

• graphs, the vertex set V={0,1,2,…,n-1}, the edge set E
• n = |V|, m = |E|
• adjacency matrix
• spanning tree

[Section 1.1]

Spanning Trees

Thm 1.1 (Kirchhoff): Let A be the adjacency matrix of an undirected graph and D be the diagonal matrix with the degrees of vertices on its

• diagonal. Then, for every i∈ {0,1,2,…,n-1}:

# spanning trees of G = det (D-A)ii Where (D-A)ii is an (n-1)x(n-1) submatrix obtained from D-A by removing the i-th row and the i-th column.

[Section 1.1]

Spanning Trees

Thm (Kirchhoff): Let A be the adjacency matrix of an undirected graph and D be the diagonal matrix with the degrees of vertices on its

• diagonal. Then, for every i∈ {0,1,2,…,n-1}:

# spanning trees of G = det (D-A)ii Where (D-A)ii is an (n-1)x(n-1) submatrix obtained from D-A by removing the i-th row and the i-th column. Note: determinant can be computed in time O(n3)

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Spanning Trees

Lemma (Binet-Cauchy): If A is an (rxm) matrix and B is an (mxr) matrix, then: det(A.B) = ∑S⊆ [m], |S|=r det(A*S).det(BS*) Where A*S is the (rxr) matrix obtained from A by keeping only the columns in S, and BS* is the (rxr) matrix obtained from B by keeping only the rows in S.

[Section 1.1]

Spanning Trees

Def: Incidence matrix of a directed graph H: an (nxm) matrix N=(νve) such that: +1 if e starts at v νve =

• 1

if e ends at v

• \w

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Spanning Trees

Fact 1.6: rank N = n - # weakly connected components of H

[Section 1.1]

Spanning Trees

Fact 1.6: rank N = n - # weakly connected components of H

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Spanning Trees

Fact 1.7: If B is a square matrix with entries in {+1,-1,0} and every column has at most one +1 and at most one -1. Then, det(B) is +1, -1, or 0.

[Section 1.1]

Spanning Trees

Thm 1.1 (Kirchhoff): # spanning trees = det (D-A)ii Proof:

[Section 1.1]

Spanning Trees

Thm 1.1 (Kirchhoff): # spanning trees = det (D-A)ii Proof:

[Section 1.1]