18.312: Algebraic Combinatorics Lionel Levine. Lecture 19

Size: px
Start display at page:

Download "18.312: Algebraic Combinatorics Lionel Levine. Lecture 19"

Transcription

1 832: Algebraic Combinatorics Lionel Levine Lecture date: April 2, 20 Lecture 9 Notes by: David Witmer Matrix-Tree Theorem Undirected Graphs Let G = (V, E) be a connected, undirected graph with n vertices, and let κ(g) be the number of spanning trees of G Definition (Laplacian matrix of undirected graph) The Laplacian matrix L of G is equal to D A, where d 0 D = 0 d n such that d i is the degree of vertex i, ie the number of edges incident to vertex i, and A is the adjacency matrix of G such that A = (a ij ), { if (i, j) E a ij = 0 else Theorem 2 (Matrix-Tree Theorem, Version ) κ(g) = n λ λ 2 λ n, where λ, λ 2,, λ n are non-zero eigenvalues of the Laplacian matrix L of G 2 Directed Graphs We can give another version of the Matrix-Tree Theorem for directed graphs First, we need to define spanning trees and Laplacian matrices for directed graphs Let Γ = (V, E) be a directed graph 9-

2 Definition 3 (Oriented spanning tree) An oriented spanning tree of Γ rooted at r V is a spanning subgraph T = (V, A) such that Every vertex v r has out degree 2 r has out degree 0 3 T has no oriented cycles Example 4 Consider the following directed graph: It has three oriented spanning trees: Definition 5 (Laplacian matrix of directed graph) The Laplacian matrix L of Γ is equal to D A, where d 0 D = 0 d n such that d i is the out degree of vertex i, ie #{j V (i, j) E}, and A is the adjacency matrix of Γ Theorem 6 (Matrix-Tree Theorem, Version 2) Let κ(γ, r) = #{oriented spanning trees of Γ rooted at r} 9-2

3 and L r be the Laplacian matrix of Γ with the row and column corresponding to vertex r crossed out Then κ(γ, r) = det L r where L r is the Laplacian matrix L with row and column r removed Example 7 Consider the directed graph from the previous example: Then we see that and so and Then D = A = L = L r = det L r = = 3 which matches what we found in the previous example We will prove this version of the Matrix-Tree Theorem and then show that it implies the version for undirected graphs Proof: Reorder the vertices of Γ so that r is the nth vertex Then det L r = d d 2 d n (other terms), since L r has the d i s on the diagonal and either or 0 for the off-diagonal 9-3

4 entries d d 2 d n counts the number of subgraphs H of Γ such that each vertex v r has out-degree So we have that H = T C C k, where T is an oriented tree rooted at r and each C i is an oriented cycle Then det L r = σ S n sgn(σ)l,σ() L n,σ(n ) Let fix(σ) = {i σ(i) = i} Then we have det L r = sgn(σ) d i L i,σ(i) σ S n i fix(σ) i/ fix(σ) i/ fix(σ) L i,σ(i) is only non-zero when (i, σ(i)) E for all i / fix(σ) In this case, L i,σ(i) = ( ) n fix(σ) We wish to write i/ fix(σ) det L r = subgraphs H Γ where C H is if H is an oriented spanning tree and 0 otherwise Any permutation σ consists of fixed points and cycles A subgraph H = T C C k arises from σ if and only if the union of all cycles C i of H contains all vertices not fixed by H, which, in turn, is true if and only if T fix(σ) We can then conclude that C H = {σ S n T fix(σ)} C H, sgn(σ)( ) n fix(σ) Our goal is then to show that C H is when H is a tree and 0 otherwise When H is a tree, H = T and there are no cycles Then all vertices are in fix(σ) and σ is the identity permutation The sign of the identity permutation is and n points are fixed, so C H = Lastly, we need to show that C H = 0 if k, ie if H has a cycle For each C i, we can either choose C i fix(σ) or C i to be a cycle of σ Let i,, i l be the indices of the C i s that are formed from vertices in cycles of σ All other points must be fixed by σ, so This means that C H = {i,,i l } [k] sgn(σ) = ( ) ( C i )++( C il ) ( ) ( C i )++( C il ) ( ) C i ++ C il 9-4

5 So, C H = ( ) S = S [k] k l=0 ( ) k ( ) k l = 0 if k 3 Proof of the Matrix Tree Theorem, Version Now we will show that Version 2 of the Matrix Tree Theorem implies the version for undirected graphs Proof: Given undirected graph G, let Γ be the directed graph with edges (i, j) and (j, i) for every edge of G We first observe that there is a bijection between the set of oriented spanning trees of Γ rooted at r and the set of spanning trees of G We can take any oriented spanning tree of Γ rooted at r and get a spanning tree of G by disregarding the root and the orientation of the edges For any spanning tree T of G, we can get an oriented spanning tree of Γ by orienting edges along the unique path from each vertex to r Such a path exists because T is connected and is unique because T has no cycles Then n nκ(g) = κ(γ, r) Let L be the Laplacian matrix of Γ Then the characteristic polynomial of L is χ(t) = det (ti L) It is true that n det L r = ( ) n [t]χ(t), r= where [t]χ(t) is the coefficient of t in χ(t) r= 9-5

6 So, we have that n nκ(g) = det L r = ( ) n [t]χ(t) r n = ( ) n [t] (t λ i ), where the λ i s are eigenvalues of L and λ n = 0 i= = ( ) n ( ) n λ λ n = λ λ n Therefore, κ(g) = n λ λ n 2 Cayley s Theorem Theorem 8 (Cayley s Theorem) The number of trees on n labeled vertices is n n 2 Example 9 Consider trees containing 4 vertices There are 6 = total, 4 of the form and 2 of the form 4 Proof: Any tree on n vertices is a spanning tree of the complete graph K n, so we can apply Version 2 of the Matrix-Tree Theorem So, κ(k n ) = n λ λ n, where λ,, λ n 9-6

7 are the non-zero eigenvalues of the Laplacian matrix n n L = = ni J, n where J is the n n matrix of ones J has the ones vector as one of its eigenvectors The remaining n eigenvectors are of the form, so J has eigenvalues n, 0,, 0, with 0 having multiplicity n This implies that L has eigenvalues 0, n,, n, with n having multiplicity n So, κ(k n ) = nn n = nn 2 3 Eigenvalues of the Adjacency Matrix Let G be an undirected, connected graph with n vertices Let P l be the number of closed paths in G of length l: P l = #{(v 0, v,, v l, v l = v 0 ) (v i, v i+ ) E for i = 0,,, l } Theorem 0 P l = φ l + + φ l n, where φ,, φ n are the eigenvalues of the adjacency matrix A of G Proof: We observe that (A l ) ij = #{paths of length l from i to j} So, P l = (A l ) + (A l ) (A l ) nn = Tr(A l ) 9-7

8 Note that this holds for both directed and undirected graphs Since G is undirected, A is symmetric, which means that A is diagonalizable so there exists some S such that φ 0 0 SAS 0 φ 2 0 = 0 0 φ n So, P l = Tr(A l ) = Tr(SA l S ) = Tr((SAS ) l ) φ l φ l 2 0 = Tr 0 0 φ l n = φ l + + φ l n 9-8

Spectral Graph Theory and You: Matrix Tree Theorem and Centrality Metrics

Spectral Graph Theory and You: Matrix Tree Theorem and Centrality Metrics Spectral Graph Theory and You: and Centrality Metrics Jonathan Gootenberg March 11, 2013 1 / 19 Outline of Topics 1 Motivation Basics of Spectral Graph Theory Understanding the characteristic polynomial

More information

18.312: Algebraic Combinatorics Lionel Levine. Lecture (u, v) is a horizontal edge K u,v = i (u, v) is a vertical edge 0 else

18.312: Algebraic Combinatorics Lionel Levine. Lecture (u, v) is a horizontal edge K u,v = i (u, v) is a vertical edge 0 else 18.312: Algebraic Combinatorics Lionel Levine Lecture date: Apr 14, 2011 Lecture 18 Notes by: Taoran Chen 1 Kasteleyn s theorem Theorem 1 (Kasteleyn) Let G be a finite induced subgraph of Z 2. Define the

More information

The Matrix-Tree Theorem

The Matrix-Tree Theorem The Matrix-Tree Theorem Christopher Eur March 22, 2015 Abstract: We give a brief introduction to graph theory in light of linear algebra. Our results culminates in the proof of Matrix-Tree Theorem. 1 Preliminaries

More information

An Introduction to Spectral Graph Theory

An Introduction to Spectral Graph Theory An Introduction to Spectral Graph Theory Mackenzie Wheeler Supervisor: Dr. Gary MacGillivray University of Victoria WheelerM@uvic.ca Outline Outline 1. How many walks are there from vertices v i to v j

More information

1.10 Matrix Representation of Graphs

1.10 Matrix Representation of Graphs 42 Basic Concepts of Graphs 1.10 Matrix Representation of Graphs Definitions: In this section, we introduce two kinds of matrix representations of a graph, that is, the adjacency matrix and incidence matrix

More information

1 Counting spanning trees: A determinantal formula

1 Counting spanning trees: A determinantal formula Math 374 Matrix Tree Theorem Counting spanning trees: A determinantal formula Recall that a spanning tree of a graph G is a subgraph T so that T is a tree and V (G) = V (T ) Question How many distinct

More information

Notes on the Matrix-Tree theorem and Cayley s tree enumerator

Notes on the Matrix-Tree theorem and Cayley s tree enumerator Notes on the Matrix-Tree theorem and Cayley s tree enumerator 1 Cayley s tree enumerator Recall that the degree of a vertex in a tree (or in any graph) is the number of edges emanating from it We will

More information

Graph fundamentals. Matrices associated with a graph

Graph fundamentals. Matrices associated with a graph Graph fundamentals Matrices associated with a graph Drawing a picture of a graph is one way to represent it. Another type of representation is via a matrix. Let G be a graph with V (G) ={v 1,v,...,v n

More information

Linear algebra and applications to graphs Part 1

Linear algebra and applications to graphs Part 1 Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

More information

Laplacian Integral Graphs with Maximum Degree 3

Laplacian Integral Graphs with Maximum Degree 3 Laplacian Integral Graphs with Maximum Degree Steve Kirkland Department of Mathematics and Statistics University of Regina Regina, Saskatchewan, Canada S4S 0A kirkland@math.uregina.ca Submitted: Nov 5,

More information

Eigenvalue and Eigenvector Homework

Eigenvalue and Eigenvector Homework Eigenvalue and Eigenvector Homework Olena Bormashenko November 4, 2 For each of the matrices A below, do the following:. Find the characteristic polynomial of A, and use it to find all the eigenvalues

More information

An Algorithmist s Toolkit September 10, Lecture 1

An Algorithmist s Toolkit September 10, Lecture 1 18.409 An Algorithmist s Toolkit September 10, 2009 Lecture 1 Lecturer: Jonathan Kelner Scribe: Jesse Geneson (2009) 1 Overview The class s goals, requirements, and policies were introduced, and topics

More information

Linear Algebra II Lecture 13

Linear Algebra II Lecture 13 Linear Algebra II Lecture 13 Xi Chen 1 1 University of Alberta November 14, 2014 Outline 1 2 If v is an eigenvector of T : V V corresponding to λ, then v is an eigenvector of T m corresponding to λ m since

More information

The Number of Spanning Trees in a Graph

The Number of Spanning Trees in a Graph Course Trees The Ubiquitous Structure in Computer Science and Mathematics, JASS 08 The Number of Spanning Trees in a Graph Konstantin Pieper Fakultät für Mathematik TU München April 28, 2008 Konstantin

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

More information

Diagonalization. Hung-yi Lee

Diagonalization. Hung-yi Lee Diagonalization Hung-yi Lee Review If Av = λv (v is a vector, λ is a scalar) v is an eigenvector of A excluding zero vector λ is an eigenvalue of A that corresponds to v Eigenvectors corresponding to λ

More information

Statistical Mechanics and Combinatorics : Lecture II

Statistical Mechanics and Combinatorics : Lecture II Statistical Mechanics and Combinatorics : Lecture II Kircho s matrix-tree theorem Deinition (Spanning tree) A subgraph T o a graph G(V, E) is a spanning tree i it is a tree that contains every vertex in

More information

ORIE 6334 Spectral Graph Theory September 8, Lecture 6. In order to do the first proof, we need to use the following fact.

ORIE 6334 Spectral Graph Theory September 8, Lecture 6. In order to do the first proof, we need to use the following fact. ORIE 6334 Spectral Graph Theory September 8, 2016 Lecture 6 Lecturer: David P. Williamson Scribe: Faisal Alkaabneh 1 The Matrix-Tree Theorem In this lecture, we continue to see the usefulness of the graph

More information

Cayley-Hamilton Theorem

Cayley-Hamilton Theorem Cayley-Hamilton Theorem Massoud Malek In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n Let A be an n n matrix Although det (λ I n A

More information

Course : Algebraic Combinatorics

Course : Algebraic Combinatorics Course 18.312: Algebraic Combinatorics Lecture Notes #29-31 Addendum by Gregg Musiker April 24th - 29th, 2009 The following material can be found in several sources including Sections 14.9 14.13 of Algebraic

More information

Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI?

Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Section 5. The Characteristic Polynomial Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Property The eigenvalues

More information

Lecture 1 and 2: Random Spanning Trees

Lecture 1 and 2: Random Spanning Trees Recent Advances in Approximation Algorithms Spring 2015 Lecture 1 and 2: Random Spanning Trees Lecturer: Shayan Oveis Gharan March 31st Disclaimer: These notes have not been subjected to the usual scrutiny

More information

DOMINO TILING. Contents 1. Introduction 1 2. Rectangular Grids 2 Acknowledgments 10 References 10

DOMINO TILING. Contents 1. Introduction 1 2. Rectangular Grids 2 Acknowledgments 10 References 10 DOMINO TILING KASPER BORYS Abstract In this paper we explore the problem of domino tiling: tessellating a region with x2 rectangular dominoes First we address the question of existence for domino tilings

More information

Advanced Combinatorial Optimization Feb 13 and 25, Lectures 4 and 6

Advanced Combinatorial Optimization Feb 13 and 25, Lectures 4 and 6 18.438 Advanced Combinatorial Optimization Feb 13 and 25, 2014 Lectures 4 and 6 Lecturer: Michel X. Goemans Scribe: Zhenyu Liao and Michel X. Goemans Today, we will use an algebraic approach to solve the

More information

Jordan normal form notes (version date: 11/21/07)

Jordan normal form notes (version date: 11/21/07) Jordan normal form notes (version date: /2/7) If A has an eigenbasis {u,, u n }, ie a basis made up of eigenvectors, so that Au j = λ j u j, then A is diagonal with respect to that basis To see this, let

More information

Lecture 1: Graphs, Adjacency Matrices, Graph Laplacian

Lecture 1: Graphs, Adjacency Matrices, Graph Laplacian Lecture 1: Graphs, Adjacency Matrices, Graph Laplacian Radu Balan January 31, 2017 G = (V, E) An undirected graph G is given by two pieces of information: a set of vertices V and a set of edges E, G =

More information

A lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo

A lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo A lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo A. E. Brouwer & W. H. Haemers 2008-02-28 Abstract We show that if µ j is the j-th largest Laplacian eigenvalue, and d

More information

A Generalized Algorithm for Computing Matching Polynomials using Determinants. Asa Scherer

A Generalized Algorithm for Computing Matching Polynomials using Determinants. Asa Scherer A Generalized Algorithm for Computing Matching Polynomials using Determinants Asa Scherer April 17, 2007 Abstract We discuss the connection between determinants of modified matching matrices and the matching

More information

Combinatorial and physical content of Kirchhoff polynomials

Combinatorial and physical content of Kirchhoff polynomials Combinatorial and physical content of Kirchhoff polynomials Karen Yeats May 19, 2009 Spanning trees Let G be a connected graph, potentially with multiple edges and loops in the sense of a graph theorist.

More information

Lecture 23: Trace and determinants! (1) (Final lecture)

Lecture 23: Trace and determinants! (1) (Final lecture) Lecture 23: Trace and determinants! (1) (Final lecture) Travis Schedler Thurs, Dec 9, 2010 (version: Monday, Dec 13, 3:52 PM) Goals (2) Recall χ T (x) = (x λ 1 ) (x λ n ) = x n tr(t )x n 1 + +( 1) n det(t

More information

Kernels of Directed Graph Laplacians. J. S. Caughman and J.J.P. Veerman

Kernels of Directed Graph Laplacians. J. S. Caughman and J.J.P. Veerman Kernels of Directed Graph Laplacians J. S. Caughman and J.J.P. Veerman Department of Mathematics and Statistics Portland State University PO Box 751, Portland, OR 97207. caughman@pdx.edu, veerman@pdx.edu

More information

Lecture 12 : Graph Laplacians and Cheeger s Inequality

Lecture 12 : Graph Laplacians and Cheeger s Inequality CPS290: Algorithmic Foundations of Data Science March 7, 2017 Lecture 12 : Graph Laplacians and Cheeger s Inequality Lecturer: Kamesh Munagala Scribe: Kamesh Munagala Graph Laplacian Maybe the most beautiful

More information

Smith Normal Form and Combinatorics

Smith Normal Form and Combinatorics Smith Normal Form and Combinatorics Richard P. Stanley June 8, 2017 Smith normal form A: n n matrix over commutative ring R (with 1) Suppose there exist P,Q GL(n,R) such that PAQ := B = diag(d 1,d 1 d

More information

arxiv: v1 [math.co] 15 Jun 2009

arxiv: v1 [math.co] 15 Jun 2009 SANDPILE GROUPS AND SPANNING TREES OF DIRECTED LINE GRAPHS arxiv:0906.2809v1 [math.co] 15 Jun 2009 LIONEL LEVINE Abstract. We generalize a theorem of Knuth relating the oriented spanning trees of a directed

More information

A New Spectral Technique Using Normalized Adjacency Matrices for Graph Matching 1

A New Spectral Technique Using Normalized Adjacency Matrices for Graph Matching 1 CHAPTER-3 A New Spectral Technique Using Normalized Adjacency Matrices for Graph Matching Graph matching problem has found many applications in areas as diverse as chemical structure analysis, pattern

More information

Announcements Monday, November 06

Announcements Monday, November 06 Announcements Monday, November 06 This week s quiz: covers Sections 5 and 52 Midterm 3, on November 7th (next Friday) Exam covers: Sections 3,32,5,52,53 and 55 Section 53 Diagonalization Motivation: Difference

More information

Matching Polynomials of Graphs

Matching Polynomials of Graphs Spectral Graph Theory Lecture 25 Matching Polynomials of Graphs Daniel A Spielman December 7, 2015 Disclaimer These notes are not necessarily an accurate representation of what happened in class The notes

More information

Lecture 22: Counting

Lecture 22: Counting CS 710: Complexity Theory 4/8/2010 Lecture 22: Counting Instructor: Dieter van Melkebeek Scribe: Phil Rydzewski & Chi Man Liu Last time we introduced extractors and discussed two methods to construct them.

More information

City Suburbs. : population distribution after m years

City Suburbs. : population distribution after m years Section 5.3 Diagonalization of Matrices Definition Example: stochastic matrix To City Suburbs From City Suburbs.85.03 = A.15.97 City.15.85 Suburbs.97.03 probability matrix of a sample person s residence

More information

This section is an introduction to the basic themes of the course.

This section is an introduction to the basic themes of the course. Chapter 1 Matrices and Graphs 1.1 The Adjacency Matrix This section is an introduction to the basic themes of the course. Definition 1.1.1. A simple undirected graph G = (V, E) consists of a non-empty

More information

Laplacians of Graphs, Spectra and Laplacian polynomials

Laplacians of Graphs, Spectra and Laplacian polynomials Laplacians of Graphs, Spectra and Laplacian polynomials Lector: Alexander Mednykh Sobolev Institute of Mathematics Novosibirsk State University Winter School in Harmonic Functions on Graphs and Combinatorial

More information

COUNTING AND ENUMERATING SPANNING TREES IN (di-) GRAPHS

COUNTING AND ENUMERATING SPANNING TREES IN (di-) GRAPHS COUNTING AND ENUMERATING SPANNING TREES IN (di-) GRAPHS Xuerong Yong Version of Feb 5, 9 pm, 06: New York Time 1 1 Introduction A directed graph (digraph) D is a pair (V, E): V is the vertex set of D,

More information

SANDPILE GROUPS AND SPANNING TREES OF DIRECTED LINE GRAPHS

SANDPILE GROUPS AND SPANNING TREES OF DIRECTED LINE GRAPHS SANDPILE GROUPS AND SPANNING TREES OF DIRECTED LINE GRAPHS LIONEL LEVINE arxiv:0906.2809v2 [math.co] 6 Apr 2010 Abstract. We generalize a theorem of Knuth relating the oriented spanning trees of a directed

More information

Energy of Graphs. Sivaram K. Narayan Central Michigan University. Presented at CMU on October 10, 2015

Energy of Graphs. Sivaram K. Narayan Central Michigan University. Presented at CMU on October 10, 2015 Energy of Graphs Sivaram K. Narayan Central Michigan University Presented at CMU on October 10, 2015 1 / 32 Graphs We will consider simple graphs (no loops, no multiple edges). Let V = {v 1, v 2,..., v

More information

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,

More information

HW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name.

HW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name. HW2 - Due 0/30 Each answer must be mathematically justified. Don t forget your name. Problem. Use the row reduction algorithm to find the inverse of the matrix 0 0, 2 3 5 if it exists. Double check your

More information

Math 3191 Applied Linear Algebra

Math 3191 Applied Linear Algebra Math 9 Applied Linear Algebra Lecture 9: Diagonalization Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./9 Section. Diagonalization The goal here is to develop a useful

More information

33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM

33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM 33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM (UPDATED MARCH 17, 2018) The final exam will be cumulative, with a bit more weight on more recent material. This outline covers the what we ve done since the

More information

Fiedler s Theorems on Nodal Domains

Fiedler s Theorems on Nodal Domains Spectral Graph Theory Lecture 7 Fiedler s Theorems on Nodal Domains Daniel A. Spielman September 19, 2018 7.1 Overview In today s lecture we will justify some of the behavior we observed when using eigenvectors

More information

On the inverse matrix of the Laplacian and all ones matrix

On the inverse matrix of the Laplacian and all ones matrix On the inverse matrix of the Laplacian and all ones matrix Sho Suda (Joint work with Michio Seto and Tetsuji Taniguchi) International Christian University JSPS Research Fellow PD November 21, 2012 Sho

More information

Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam

Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system

More information

Dimension. Eigenvalue and eigenvector

Dimension. Eigenvalue and eigenvector Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,

More information

DIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix

DIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that

More information

Spanning Trees of Shifted Simplicial Complexes

Spanning Trees of Shifted Simplicial Complexes Art Duval (University of Texas at El Paso) Caroline Klivans (Brown University) Jeremy Martin (University of Kansas) Special Session on Extremal and Probabilistic Combinatorics University of Nebraska, Lincoln

More information

MAT 1302B Mathematical Methods II

MAT 1302B Mathematical Methods II MAT 1302B Mathematical Methods II Alistair Savage Mathematics and Statistics University of Ottawa Winter 2015 Lecture 19 Alistair Savage (uottawa) MAT 1302B Mathematical Methods II Winter 2015 Lecture

More information

Smith Normal Form and Combinatorics

Smith Normal Form and Combinatorics Smith Normal Form and Combinatorics p. 1 Smith Normal Form and Combinatorics Richard P. Stanley Smith Normal Form and Combinatorics p. 2 Smith normal form A: n n matrix over commutative ring R (with 1)

More information

PROBLEMS ON LINEAR ALGEBRA

PROBLEMS ON LINEAR ALGEBRA 1 Basic Linear Algebra PROBLEMS ON LINEAR ALGEBRA 1. Let M n be the (2n + 1) (2n + 1) for which 0, i = j (M n ) ij = 1, i j 1,..., n (mod 2n + 1) 1, i j n + 1,..., 2n (mod 2n + 1). Find the rank of M n.

More information

MATH 235. Final ANSWERS May 5, 2015

MATH 235. Final ANSWERS May 5, 2015 MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your

More information

Networks and Matrices

Networks and Matrices Bowen-Lanford Zeta function, Math table talk Oliver Knill, 10/18/2016 Networks and Matrices Assume you have three friends who do not know each other. This defines a network containing 4 nodes in which

More information

Spectral Generative Models for Graphs

Spectral Generative Models for Graphs Spectral Generative Models for Graphs David White and Richard C. Wilson Department of Computer Science University of York Heslington, York, UK wilson@cs.york.ac.uk Abstract Generative models are well known

More information

and let s calculate the image of some vectors under the transformation T.

and let s calculate the image of some vectors under the transformation T. Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

More information

Random Walks on Graphs. One Concrete Example of a random walk Motivation applications

Random Walks on Graphs. One Concrete Example of a random walk Motivation applications Random Walks on Graphs Outline One Concrete Example of a random walk Motivation applications shuffling cards universal traverse sequence self stabilizing token management scheme random sampling enumeration

More information

Eigenvectors. Prop-Defn

Eigenvectors. Prop-Defn Eigenvectors Aim lecture: The simplest T -invariant subspaces are 1-dim & these give rise to the theory of eigenvectors. To compute these we introduce the similarity invariant, the characteristic polynomial.

More information

MTH 5102 Linear Algebra Practice Final Exam April 26, 2016

MTH 5102 Linear Algebra Practice Final Exam April 26, 2016 Name (Last name, First name): MTH 5 Linear Algebra Practice Final Exam April 6, 6 Exam Instructions: You have hours to complete the exam. There are a total of 9 problems. You must show your work and write

More information

1 T 1 = where 1 is the all-ones vector. For the upper bound, let v 1 be the eigenvector corresponding. u:(u,v) E v 1(u)

1 T 1 = where 1 is the all-ones vector. For the upper bound, let v 1 be the eigenvector corresponding. u:(u,v) E v 1(u) CME 305: Discrete Mathematics and Algorithms Instructor: Reza Zadeh (rezab@stanford.edu) Final Review Session 03/20/17 1. Let G = (V, E) be an unweighted, undirected graph. Let λ 1 be the maximum eigenvalue

More information

MATH 829: Introduction to Data Mining and Analysis Clustering II

MATH 829: Introduction to Data Mining and Analysis Clustering II his lecture is based on U. von Luxburg, A Tutorial on Spectral Clustering, Statistics and Computing, 17 (4), 2007. MATH 829: Introduction to Data Mining and Analysis Clustering II Dominique Guillot Departments

More information

Spring 2019 Exam 2 3/27/19 Time Limit: / Problem Points Score. Total: 280

Spring 2019 Exam 2 3/27/19 Time Limit: / Problem Points Score. Total: 280 Math 307 Spring 2019 Exam 2 3/27/19 Time Limit: / Name (Print): Problem Points Score 1 15 2 20 3 35 4 30 5 10 6 20 7 20 8 20 9 20 10 20 11 10 12 10 13 10 14 10 15 10 16 10 17 10 Total: 280 Math 307 Exam

More information

Recitation 8: Graphs and Adjacency Matrices

Recitation 8: Graphs and Adjacency Matrices Math 1b TA: Padraic Bartlett Recitation 8: Graphs and Adjacency Matrices Week 8 Caltech 2011 1 Random Question Suppose you take a large triangle XY Z, and divide it up with straight line segments into

More information

EIGENVECTOR NORMS MATTER IN SPECTRAL GRAPH THEORY

EIGENVECTOR NORMS MATTER IN SPECTRAL GRAPH THEORY EIGENVECTOR NORMS MATTER IN SPECTRAL GRAPH THEORY Franklin H. J. Kenter Rice University ( United States Naval Academy 206) orange@rice.edu Connections in Discrete Mathematics, A Celebration of Ron Graham

More information

Ir O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )

Ir O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v ) Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O

More information

Math 315: Linear Algebra Solutions to Assignment 7

Math 315: Linear Algebra Solutions to Assignment 7 Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are

More information

Math Camp Notes: Linear Algebra II

Math Camp Notes: Linear Algebra II Math Camp Notes: Linear Algebra II Eigenvalues Let A be a square matrix. An eigenvalue is a number λ which when subtracted from the diagonal elements of the matrix A creates a singular matrix. In other

More information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

Using Laplacian Eigenvalues and Eigenvectors in the Analysis of Frequency Assignment Problems

Using Laplacian Eigenvalues and Eigenvectors in the Analysis of Frequency Assignment Problems Using Laplacian Eigenvalues and Eigenvectors in the Analysis of Frequency Assignment Problems Jan van den Heuvel and Snežana Pejić Department of Mathematics London School of Economics Houghton Street,

More information

ELA

ELA THE DISTANCE MATRIX OF A BIDIRECTED TREE R. B. BAPAT, A. K. LAL, AND SUKANTA PATI Abstract. A bidirected tree is a tree in which each edge is replaced by two arcs in either direction. Formulas are obtained

More information

= main diagonal, in the order in which their corresponding eigenvectors appear as columns of E.

= main diagonal, in the order in which their corresponding eigenvectors appear as columns of E. 3.3 Diagonalization Let A = 4. Then and are eigenvectors of A, with corresponding eigenvalues 2 and 6 respectively (check). This means 4 = 2, 4 = 6. 2 2 2 2 Thus 4 = 2 2 6 2 = 2 6 4 2 We have 4 = 2 0 0

More information

Topics in Graph Theory

Topics in Graph Theory Topics in Graph Theory September 4, 2018 1 Preliminaries A graph is a system G = (V, E) consisting of a set V of vertices and a set E (disjoint from V ) of edges, together with an incidence function End

More information

Determinants. Beifang Chen

Determinants. Beifang Chen Determinants Beifang Chen 1 Motivation Determinant is a function that each square real matrix A is assigned a real number, denoted det A, satisfying certain properties If A is a 3 3 matrix, writing A [u,

More information

Math 110 Linear Algebra Midterm 2 Review October 28, 2017

Math 110 Linear Algebra Midterm 2 Review October 28, 2017 Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections

More information

Online Exercises for Linear Algebra XM511

Online Exercises for Linear Algebra XM511 This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2

More information

Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur

Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. #07 Jordan Canonical Form Cayley Hamilton Theorem (Refer Slide Time:

More information

vibrations, light transmission, tuning guitar, design buildings and bridges, washing machine, Partial differential problems, water flow,...

vibrations, light transmission, tuning guitar, design buildings and bridges, washing machine, Partial differential problems, water flow,... 6 Eigenvalues Eigenvalues are a common part of our life: vibrations, light transmission, tuning guitar, design buildings and bridges, washing machine, Partial differential problems, water flow, The simplest

More information

On the adjacency matrix of a block graph

On the adjacency matrix of a block graph On the adjacency matrix of a block graph R. B. Bapat Stat-Math Unit Indian Statistical Institute, Delhi 7-SJSS Marg, New Delhi 110 016, India. email: rbb@isid.ac.in Souvik Roy Economics and Planning Unit

More information

Damped random walks and the characteristic polynomial of the weighted Laplacian on a graph

Damped random walks and the characteristic polynomial of the weighted Laplacian on a graph Damped random walks and the characteristic polynomial of the weighted Laplacian on a graph arxiv:mathpr/0506460 v1 Jun 005 MADHAV P DESAI and HARIHARAN NARAYANAN September 11, 006 Abstract For λ > 0, we

More information

DETERMINANTS. , x 2 = a 11b 2 a 21 b 1

DETERMINANTS. , x 2 = a 11b 2 a 21 b 1 DETERMINANTS 1 Solving linear equations The simplest type of equations are linear The equation (1) ax = b is a linear equation, in the sense that the function f(x) = ax is linear 1 and it is equated to

More information

Determinants - Uniqueness and Properties

Determinants - Uniqueness and Properties Determinants - Uniqueness and Properties 2-2-2008 In order to show that there s only one determinant function on M(n, R), I m going to derive another formula for the determinant It involves permutations

More information

The Cayley-Hamilton Theorem and the Jordan Decomposition

The Cayley-Hamilton Theorem and the Jordan Decomposition LECTURE 19 The Cayley-Hamilton Theorem and the Jordan Decomposition Let me begin by summarizing the main results of the last lecture Suppose T is a endomorphism of a vector space V Then T has a minimal

More information

Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures

More information

v = 1(1 t) + 1(1 + t). We may consider these components as vectors in R n and R m : w 1. R n, w m

v = 1(1 t) + 1(1 + t). We may consider these components as vectors in R n and R m : w 1. R n, w m 20 Diagonalization Let V and W be vector spaces, with bases S = {e 1,, e n } and T = {f 1,, f m } respectively Since these are bases, there exist constants v i and w such that any vectors v V and w W can

More information

Chapter 2 Spectra of Finite Graphs

Chapter 2 Spectra of Finite Graphs Chapter 2 Spectra of Finite Graphs 2.1 Characteristic Polynomials Let G = (V, E) be a finite graph on n = V vertices. Numbering the vertices, we write down its adjacency matrix in an explicit form of n

More information

HOMEWORK #2 - MATH 3260

HOMEWORK #2 - MATH 3260 HOMEWORK # - MATH 36 ASSIGNED: JANUARAY 3, 3 DUE: FEBRUARY 1, AT :3PM 1) a) Give by listing the sequence of vertices 4 Hamiltonian cycles in K 9 no two of which have an edge in common. Solution: Here is

More information

Math 443/543 Graph Theory Notes 5: Graphs as matrices, spectral graph theory, and PageRank

Math 443/543 Graph Theory Notes 5: Graphs as matrices, spectral graph theory, and PageRank Math 443/543 Graph Theory Notes 5: Graphs as matrices, spectral graph theory, and PageRank David Glickenstein November 3, 4 Representing graphs as matrices It will sometimes be useful to represent graphs

More information

Ma/CS 6b Class 23: Eigenvalues in Regular Graphs

Ma/CS 6b Class 23: Eigenvalues in Regular Graphs Ma/CS 6b Class 3: Eigenvalues in Regular Graphs By Adam Sheffer Recall: The Spectrum of a Graph Consider a graph G = V, E and let A be the adjacency matrix of G. The eigenvalues of G are the eigenvalues

More information

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them. Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence

More information

The Structure of the Jacobian Group of a Graph. A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College

The Structure of the Jacobian Group of a Graph. A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College The Structure of the Jacobian Group of a Graph A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College In Partial Fulfillment of the Requirements for the Degree Bachelor of

More information

Energies of Graphs and Matrices

Energies of Graphs and Matrices Energies of Graphs and Matrices Duy Nguyen T Parabola Talk October 6, 2010 Summary 1 Definitions Energy of Graph 2 Laplacian Energy Laplacian Matrices Edge Deletion 3 Maximum energy 4 The Integral Formula

More information

MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006

MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006 MATH 511 ADVANCED LINEAR ALGEBRA SPRING 2006 Sherod Eubanks HOMEWORK 2 2.1 : 2, 5, 9, 12 2.3 : 3, 6 2.4 : 2, 4, 5, 9, 11 Section 2.1: Unitary Matrices Problem 2 If λ σ(u) and U M n is unitary, show that

More information

AMS10 HW7 Solutions. All credit is given for effort. (-5 pts for any missing sections) Problem 1 (20 pts) Consider the following matrix 2 A =

AMS10 HW7 Solutions. All credit is given for effort. (-5 pts for any missing sections) Problem 1 (20 pts) Consider the following matrix 2 A = AMS1 HW Solutions All credit is given for effort. (- pts for any missing sections) Problem 1 ( pts) Consider the following matrix 1 1 9 a. Calculate the eigenvalues of A. Eigenvalues are 1 1.1, 9.81,.1

More information

22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices

22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix

More information