Counting spanning trees

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1 Counting spanning trees Question Given a graph G, howmanyspanningtreesdoesg have? (G) =numberofdistinctspanningtreesofg Definition If G =(V,E) isamultigraphwithe 2 E, theng e (said G contract e ) is the graph obtained from G by contracting/shrinking the edge e until its endpoints are a single vertex. Examples e G e G e G G e e G G e Theorem (Deletion/contraction formula). Let G be a multigraph, and let e be a (non-loop) edge of G. Then (G) = (G e) {z } # of spanning trees that do not use e + (G e). {z } # of spanning trees that use edge e This is the first example of a graph invariant that can be expressed using a recurrence formula involving deletions and/or contractions. More to come down the road... Math - Prof. Kindred - Lecture 5 Page

2 Comment on proof Need to show a one-to-one correspondence between spanning trees of G that include e and spanning trees of G e. spanning tree T of G that uses e! T e, aspanningtreeofg e Example Use deletion-contraction formula to compute (K ). Should get (K )=. Deletion/contraction formula is beautiful but not practically useful (grows exponentially with the size of the graph may be as many as 2 E(G) terms). We consider an alternate computation. Theorem (matrix-tree thm, also called Kirchho s matrix-tree thm). Let G be a graph. Then (G) =( ) i+j det Q ij, where Q ij is the submatrix obtained by removing the ith row and jth column of the Laplacian matrix Q of G. Note that determinants of n n matrices can be computed using fewer than n 3 operations. Math - Prof. Kindred - Lecture 5 Page 2

3 Remark Alternatively, we can express the matrix-tree theorem as (G) = n 2 n where,..., n are n largesteigenvaluesoflaplacianmatrixofg. This calculation was first devised by Gustav Kircho in 8 as a way of obtaining values of current flow in electrical networks. (Matrices were first emerging as a powerful mathematical tool about the same time.) Theorem (Cayley,889). (K n )=n n 2. Proof. We prove as a special case of the matrix-tree theorem. Let A be the adjacency matrix of K n.then 2 3 A =... 5= J I, where J is the n n matrix of all ones and I is the identity matrix. The Laplacian matrix, Q, ofk n is Q = D A =(n )I (J I) =ni J = 2 n - n Note that the vector of all ones is an eigenvector of Q for ,. 5,,. 5, Cayley was interested in representing hydrocarbons by graphs, and in particular, by trees.... n - =,and 3 5 n Math - Prof. Kindred - Lecture 5 Page 3

4 are linearly independent eigenvectors for = n. Thus, (K n )= n (nn )=n n 2. Remark Cayley s formula may also be viewed as the number of possible trees on the vertex set [n] ={, 2,...,n}, i.e.,the#oflabelledtreeson n vtcs. We have now answered the following question: Of the 2 (n 2) simple graphs with vertex set {, 2,...,n}, howmany are trees?? Book gives a proof of Cayley s theorem using Prüfer codes (unique sequence of length n 2assignedtoatreeonn vtcs). Math - Prof. Kindred - Lecture 5 Page

5 Minimum spanning trees Definition A weighted graph is a graph G =(V,E) with weight function w : E! R. Weights usually represent costs, distances, etc. Minimum spanning tree problem given a weighted connected graph, find a spanning tree T with minimum weight w(t )= X w(e). e2e(t ) We may see later that one application of minimum spanning trees is for an approximation algorithm for the Traveling Salesman Problem (TSP). Brute-force: construct all possible spanning trees and find one with minimum weight. Greedy algorithm =) any algorithm that makes a locally optimal choice at each step with the aim of finding the global optimum (Downfall can get locked into certain choices too early which prevent them from finding the best overall solution later.) Kruskal s algorithm (greedy algorithm). Order edges e,...,e m so that w(e i ) apple w(e j )foranyi<j. 2. T ;. 3. For k =tom, ift + e k is acyclic, then T T + e k.. Output T. Math - Prof. Kindred - Lecture 5 Page 5

6 In implementation on a computer, we check if T +e k is acyclic by checking if edge e k has endpoints in two di erent components of T. Comparison sort to sort edges: O(m log m) =O(m log n). Surprise! =) This greedy algorithm gives an optimal solution. Before we prove this, we mention one more result on tree characterizations. Theorem. For a graph G, the following are equivalent: () G is a tree. (2) G is a minimal connected graph (that is, G is connected and if uv 2 E(G), then G uv is disconnected). (3) G is a maximal acyclic graph (that is, G is acyclic and if u, v are nonadjacent vtcs in G, then G + uv contains a cycle. Theorem. The output of Kruskal s algorithm is a minimum weight spanning tree. Proof. Let T be the output of Kruskal s algorithm applied to a connected graph G with n vtcs. Claim: T is a spanning tree. Since T is a maximal acyclic subgraph of G, byourprevioustheorem, T must be a tree, and hence, a spanning tree of G. Claim: T is optimal. Suppose not. Let T be a min spanning tree with the maximum # of edges in common with T (i.e., E(T ) \ E(T ) as large as possible). Note that w(t ) <w(t ). Also, note that T and T both have n edges. Math - Prof. Kindred - Lecture 5 Page

7 Suppose E(T )={e,e 2,...,e n } where w(e ) apple w(e 2 ) apple applew(e n ). Let i =min{k : e k 2 E(T ) E(T )} (so e i is first edge in T that is not in T ). Then T + e i has a cycle. 9 edge e in this cycle that is not in T since T is acyclic. Now T + e i e is a spanning tree (connected and has n edges)withmoreedgesincommonwitht than T. We also know that w(e i ) apple w(e); otherwise, if w(e i ) >w(e), then we should have chosen e,...,e i,e to be in T during Kruskal s algorithm. (These edges do not have a cycle as they are all in T.) Therefore, w(t + e i e)=w(t )+w(e i ) w(e) {z } apple apple w(t ). So T + e i e is a min spanning tree with one more edge in common with T than T. )( Thus, T is optimal. Math - Prof. Kindred - Lecture 5 Page

8 Counting spanning trees Spanning trees Math, Graph Theory Tuesday, February 5, 23 degree matrix D 2 D = 3 v 3 v 3 2 Laplacian matrix of the graph =) adjacency matrix A A = compute the difference D - A 2 D A = Counting spanning trees 2 D A = 3 3 compute (2, 2)-cofactor of matrix D - A 2 (i, j)-cofactor = ( ) i+jh i det of (n )(n ) matrix obtained by removing row i and col j Counting spanning trees 2 D A = (2, 2)-cofactor of matrix D - A = 8 Fact: Every cofactor of this matrix is 8!! ( ) =2 3 2 ( ) 2 = 2( ) + ( 2 ) = 8 What does 8 have to do with spanning trees of this graph?

9 Counting spanning trees Kirchhoff s matrix-tree theorem 8 distinct spanning trees of given graph Theorem: Let G be a graph. Then # of distinct spanning trees of G = the value of any cofactor of the matrix D - A, the Laplacian matrix of G Kruskal s algorithm a b 8 c 5 9 d 5 5 e f 8 9 g

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