Very few Moore Graphs
|
|
- Morgan Ward
- 6 years ago
- Views:
Transcription
1 Very few Moore Graphs Anurag Bishnoi June 7, 0 Abstract We prove here a well known result in graph theory, originally proved by Hoffman and Singleton, that any non-trivial Moore graph of diameter is regular of degree k =, 3, 7 or 57. The existence (and uniqueness) of these graphs is known for k =, 3, 7 while it is still an open problem if there is a moore graph of degree 57 or not. Basics We ll be talking about simple undirected connected graphs. There is a natural metric d on such graphs given by the shortest distance. Definition.. Diameter d of a graph is the maximum of all shortest distances between pair of vertices. Definition.. Girth g of a non-tree graph is the length of a shortest cycle. Theorem.. (Moore s Inequality) For a graph which is not a tree g d +. Proof. We ll prove this by contradiction. Say g > d +. Let C be a cycle of length g. Then there are two vertices u, v on C which are connected by a path of length d +. Since d is the diameter, there must be a path of length at most d between u, v. But these two paths, being distinct, will form a cycle of length at most d + which is less than g. This contradicts the fact that g is the girth. Graphs that achieve equality in this bound are known as Moore Graphs. Trivial examples are odd cycles and complete graphs. A non-trivial example is the Petersen Graph with d = and g = 5. Theorem.. Moore Graphs are regular. Proof. Note that in a Moore Graph any two vertices are joined by a unique shortest path and if they are joined by a path of length at most d then that is the unique shortest path. Now, let u, v be two vertices of a Moore Graph G at distance d from each other. Then we can deduce that deg(u) = deg(v) by exhibiting a bijection between N(u) and N(v) as follows. Let x N(u). Then x is joined to v by a shortest path. Let the image of x be the neighbor of v on this path. It is straightforward to check that this is a bijection. To show that all vertices have the same degree, take a cycle of length d +. Any two adjacent vertices vertices on this cycle are distance d apart from a single vertex on the cycle and therefore have the same degree. So, all vertices on this cycle have same degree, say k. Now let x be any vertex not on this cycle. Then there exists a point y on the cycle distance d apart from x. Hence deg(x) = k.
2 Theorem.3. If G is a n vertex Moore Graph of degree k and diameter d, then n = + k + k(k ) k(k ) d. Proof. Let u be a vertex of G. Then for all 0 i d and for each vertex x with d(u, x) = i there exists is a unique path from u to x. Hence, the number of vertices from u at distance i is k(k ) i where i, which can be proved by induction. So the total number of vertices in the graph is + k + k(k ) k(k ) d since d is the maximum distance a vertex can have from u. Distance Regular Graphs Definition.. Given a graph G with diameter d we define A i, for 0 i d to be the n n matrix with rows and columns indexed by V (G) such that A i (x, y) = if d(x, y) = i and 0 otherwise. Note that A 0 is the identity matrix and A the adjacency matrix of G. We also have d i=0 A i = J. Definition.. A graph G with diameter d is called distance regular graph if there exists constants a 0,..., a d, b 0,..., b d, c,..., c d such that the following holds : Given any two vertices x, y at distance i, the number of neighbors of x at distance i /i/i + from y is c i /a i /b i. Note that a distance regular graph is k-regular with k = b 0 and hence it is a stronger condition than regularity. Also by triangle inequality we see that if d(x, y) = i and z N(x) then i d(z, y) i +. Therefore we have a i + b i + c i = k for all i, taking c 0 = b d = 0. Theorem.. Moore graphs are distance regular. Proof. It is easy to see that for k regular Moore graphs with diameter d, c = c =... = c d =, a 0 =... = a d = 0, a d = k, b 0 = k, b =... = b d = k. Let A i s be the matrices as defined before for a graph G. For distance regular graphs we can see that each A i is a polynomial in A as follows : Lets calculare A A i. Then A A i (x, z) = #{y v(g) : d(x, y) =, d(y, z) = i}. Therefore we have A A i = b i A i + a i A i + c i+ A i+, 0 i d. Hence by finite induction on i we see that Theorem.. There are polynomials p i, 0 i d s.t. A i = p i (A ). They can be found by the following recurrence relation : p 0 (z) =, p (z) = z and c i+ p i+ (z) = p (z)p i (z) b i p i (z) a i p i (z). Let q(z) = p i (z). Then q(a ) = J. Therefore, Theorem.3. If A is the adjacency matrix of a distance regular graph G with diameter d and degree k then there is an explicitly computable polynomial q of degree d such that q(a) = J. And the eigenvalues of A are either k or some root of q(z). Proof. We see that since the graph is k regular, k is an eigenvalue of A with eigenvector. Any other eigenvalue λ corresponds to an eigenvector x such that x is orthogonal to. So Jx = 0. Now, q(λ)x = q(a)x = Jx = 0 and x 0. Therefore q(λ) = 0.
3 Let A be the adjacency matrix of a graph G of n vertices. Then A C n n and A is self adjoint. Definition.3. The unital subring of C n n generated by A is called the adjacency algebra of G denoted by A. Clearly the adjacency algebra is a vector space over C. In fact it is a finite dimensional one since it is a subalgebra of C n n. For a distance regular graph we have the following result on this dimension. Theorem.4. Let G be a distance regular graph with diameter d and adjacency algebra A. Then dim(a) = d +. Proof. We first produce a linearly independent set of d + elements in A to show that dim(a) d + and then show that the powers of A upto d span A. Let A be the adjacency matrix of G. A 0, A,..., A d are polynomials in A by. and hence they belong to A. These matrices are 0 disjoint matrices and hence linearly independent. Now from.3 we know that A has at most d + eigenvalues. Therefore the minimal polynomial of A has degree at most d+. Hence A d+ is a polynomial in I, A, A,..., A d. Hence by induction on m, A m is a polynomial in I, A, A,..., A d for all m d +. Corollary.5. The polynomial q(z) defined in.3 has all distinct roots and all of them are eigenvalues of the adjacency matrix. So given an n vertex distance regular graph G with diameter d, it has d + distinct eigenvalues. Call them λ 0, λ,..., λ d and and let V 0, V,..., V d be the corresponding eigenspaces associated with them. Then we know that C n = V 0 V... V d. Let E is be the projection matrices corresponding to the eigenspaces (after choosing some basis). Then we have E i = E i and E i E j = 0 when i j. Theorem.6. We also have the following properties for these E is :. E 0, E,..., E d is another basis of A.. tr(e i ) = multiplicity of λ i. 3. A i = d j=0 λi je j. 4. A i = d j=0 p i(λ j )E j. The previous theorem gives us some idea of how to attack the problem of finding Moore graphs. Since we must have that the (d + ) (d + ) matrix M = (p i (λ j )) is invertible and tr(e i ) is an integer for all i. 3 Strongly Regular Graphs Definition 3.. A strongly regular graph (srg) is a distance regular graph of diameter at most. If diameter is one then it will be the complete graph so we generally ignore that case. Therefore the parameters for srg are c =, c, a 0 = 0, a, b 0 = k, b = k a. The free parameters a and c are generally denoted as λ, µ and we have an equivalent definition : 3
4 Definition 3.. An srg with parameters (v, k, λ, µ) is a regular graph of degree k on v vertices such that :. Any two adjacent vertices have exactly λ common neighbors.. Any two non adjacent vertices have exactly µ common neighbors. Theorem 3.. If G is an srg with parameters (v, k, λ, µ) then its complement Ḡ is also an srg. Proof. We would show explicitly the parameters that make Ḡ an srg. Firstly, it is a (v k ) regular graph. Let x, y be two adjacent vertices in Ḡ. Then x, y are non adjacent in G. A vertex z is a common neighbors to both x, y in Ḡ iff it is non adjacent to both x and y in G. But number of such vertices is a constant given by (v ) (k µ) as both x, y have k neighbors each and µ of them are common. Now let x, y to be two non adjacent vertices in Ḡ. Then by a similar argument we get that they have (v ) ((k ) λ) common neighbors. Therefore, Ḡ is also an srg with parameters (v, v k, v + µ k, v + λ k). Theorem 3.. For an srg the parameters v, k, λ, µ satisfy k(k λ ) = µ(v k ). Proof. Fix a vertex x and count the number of induced length two paths from x in two different ways. When talking of srg s we exclude complete graphs, disconnected graphs and all those connected graphs whose complements are disconnected. Which is equivalent to saying that µ must be well defined and positive. Such srg s are called primitive. Theorem 3.3. For any primitive srg v k + with equality holding iff λ = 0 and µ =. Proof. This follows from the previous theorem, since we have k(k ) k(k λ ) = µ(v k ) v k. From the above theorem we have another characterisation of Moore Graphs of diameter two. They are precisely the (k +, k, 0, ) strongly regular graphs! Computing the polynomials for a srg(v, k, λ, µ) we see that p 0 (z) =, p (z) = z and p (z) = (z λz k)/µ Therefore the eigenvalues except for k are roots of the polynomial z + (µ λ)z + µ k. So, if r, s are the its roots then we have µ = k + rs and λ = k + r + s + rs. Theorem 3.4. A primitive srg(v, k, λ, µ) has exactly three eigen values, k, [λ µ + D] and [λ µ D] with corresponding multiplicities, k+(v )(λ µ) [v D ] and k+(v )(λ µ) [v + D ] where D = (λ µ) + 4(k µ). Proof. We know that the eigenvalues are k, r, s where r, s are roots of z +(µ λ)z +µ k. So we just need to prove the multiplicities.. We show that the dimension of eigenspace corresponding to eigenvalue k is. Let A be the adjacency matrix and x = [x, x,..., x v ] t a non-zero vector such that Ax = kx. Suppose x j is the entry of x having latgest absolute value. Then we have kx j = x i where summation is over all those k i s where v i is adjacent to v j. Therefore by maximality of x j we have x j = x i for all such i. Since the graph is connected we can continue in this manner to show that all coordinates of x are equal and hence x = t. 4
5 . Let f, g be the multiplicities of r, s respectively. Then from Theorem.6 we see that + f + g = v and k + rf + sg = 0, since tr(e 0 ) =, tr(e ) = f, tr(e ) = g, tr(i) = v, tr(a) = 0. To solve this pair of equations, let f = (v )/ + x and g = (v )/ + y. Then we have x = y. Hence we get the result. Putting v = k +, λ = 0, µ = we get the following corollary. Corollary 3.5. If G is a Moore graph of degree k, then k k 4k 3 is an integer. Now we prove our main theorem, Theorem 3.6. A k regular graph G of diameter is a Moore graph only if k =, 3, 7 or 57. Proof. We know from previous corollary that k k 4k 3 must be an integer. One possibility is that k k = 0, this gives us k =. Now let 4k 3 = n. Then we get that k k 0 (mod n). Multiplying both sides by 6 we see that (n + 3) 8(n + 3) 0 (mod n). So n 5. Only non-trivial possibilities are n = 3, 5, 5 giving us k = 3, 7, 57. It has been proved that any non trivial Moore graph must have diameter at most two, but we don t discuss that here. And it can be proved that srg(5,, 0, ) (pentagon), srg(0, 3, 0, ) (Peterson graph) and srg(50, 7, 0, ) (Hoffman Singleton graph) are unique upto isomorphism. Therefore, the list of moore graphs is Odd cycles, Complete Graphs, Peterson Graph, Hoffman Singleton Graph and possibly an srg(350, 57, 0, ). 5
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 informationReal symmetric matrices/1. 1 Eigenvalues and eigenvectors
Real symmetric matrices 1 Eigenvalues and eigenvectors We use the convention that vectors are row vectors and matrices act on the right. Let A be a square matrix with entries in a field F; suppose that
More information1.10 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 informationEcon Slides from Lecture 7
Econ 205 Sobel Econ 205 - Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for
More informationStrongly Regular Decompositions of the Complete Graph
Journal of Algebraic Combinatorics, 17, 181 201, 2003 c 2003 Kluwer Academic Publishers. Manufactured in The Netherlands. Strongly Regular Decompositions of the Complete Graph EDWIN R. VAN DAM Edwin.vanDam@uvt.nl
More informationHOMEWORK #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 informationA short course on matching theory, ECNU Shanghai, July 2011.
A short course on matching theory, ECNU Shanghai, July 2011. Sergey Norin LECTURE 3 Tight cuts, bricks and braces. 3.1. Outline of Lecture Ear decomposition of bipartite graphs. Tight cut decomposition.
More informationDimension. 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 informationDistance-regular graphs where the distance-d graph has fewer distinct eigenvalues
NOTICE: this is the author s version of a work that was accepted for publication in . Changes resulting from the publishing process, such as peer review, editing, corrections,
More informationA characterization of graphs by codes from their incidence matrices
A characterization of graphs by codes from their incidence matrices Peter Dankelmann Department of Mathematics University of Johannesburg P.O. Box 54 Auckland Park 006, South Africa Jennifer D. Key pdankelmann@uj.ac.za
More informationUsing 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 informationDiagonalization of Matrix
of Matrix King Saud University August 29, 2018 of Matrix Table of contents 1 2 of Matrix Definition If A M n (R) and λ R. We say that λ is an eigenvalue of the matrix A if there is X R n \ {0} such that
More informationLaplacian 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 informationThen x 1,..., x n is a basis as desired. Indeed, it suffices to verify that it spans V, since n = dim(v ). We may write any v V as r
Practice final solutions. I did not include definitions which you can find in Axler or in the course notes. These solutions are on the terse side, but would be acceptable in the final. However, if you
More informationSome Notes on Distance-Transitive and Distance-Regular Graphs
Some Notes on Distance-Transitive and Distance-Regular Graphs Robert Bailey February 2004 These are notes from lectures given in the Queen Mary Combinatorics Study Group on 13th and 20th February 2004,
More informationLinear Algebra and its Applications
Linear Algebra and its Applications xxx (2008) xxx xxx Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: wwwelseviercom/locate/laa Graphs with three distinct
More informationGraph G = (V, E). V ={vertices}, E={edges}. V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)}
Graph Theory Graph G = (V, E). V ={vertices}, E={edges}. a b c h k d g f e V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)} E =16. Digraph D = (V, A). V ={vertices}, E={edges}.
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW JC Stuff you should know for the exam. 1. Basics on vector spaces (1) F n is the set of all n-tuples (a 1,... a n ) with a i F. It forms a VS with the operations of + and scalar multiplication
More informationLINEAR ALGEBRA BOOT CAMP WEEK 2: LINEAR OPERATORS
LINEAR ALGEBRA BOOT CAMP WEEK 2: LINEAR OPERATORS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts
More information11 Block Designs. Linear Spaces. Designs. By convention, we shall
11 Block Designs Linear Spaces In this section we consider incidence structures I = (V, B, ). always let v = V and b = B. By convention, we shall Linear Space: We say that an incidence structure (V, B,
More informationSpectral radius, symmetric and positive matrices
Spectral radius, symmetric and positive matrices Zdeněk Dvořák April 28, 2016 1 Spectral radius Definition 1. The spectral radius of a square matrix A is ρ(a) = max{ λ : λ is an eigenvalue of A}. For an
More informationStrongly Regular Graphs, part 1
Spectral Graph Theory Lecture 23 Strongly Regular Graphs, part 1 Daniel A. Spielman November 18, 2009 23.1 Introduction In this and the next lecture, I will discuss strongly regular graphs. Strongly regular
More informationPainting Squares in 2 1 Shades
Painting Squares in 1 Shades Daniel W. Cranston Landon Rabern May 1, 014 Abstract Cranston and Kim conjectured that if G is a connected graph with maximum degree and G is not a Moore Graph, then χ l (G
More informationPartial cubes: structures, characterizations, and constructions
Partial cubes: structures, characterizations, and constructions Sergei Ovchinnikov San Francisco State University, Mathematics Department, 1600 Holloway Ave., San Francisco, CA 94132 Abstract Partial cubes
More informationGraphs with Large Variance
Graphs with Large Variance Yair Caro Raphael Yuster Abstract For a graph G, let V ar(g) denote the variance of the degree sequence of G, let sq(g) denote the sum of the squares of the degrees of G, and
More informationTHE RADIO NUMBERS OF ALL GRAPHS OF ORDER n AND DIAMETER n 2
LE MATEMATICHE Vol LXVIII (2013) Fasc II, pp 167 190 doi: 104418/201368213 THE RADIO NUMBERS OF ALL GRAPHS OF ORDER n AND DIAMETER n 2 K F BENSON - M PORTER - M TOMOVA A radio labeling of a simple connected
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationThe converse is clear, since
14. The minimal polynomial For an example of a matrix which cannot be diagonalised, consider the matrix ( ) 0 1 A =. 0 0 The characteristic polynomial is λ 2 = 0 so that the only eigenvalue is λ = 0. The
More informationLecture 12: Diagonalization
Lecture : Diagonalization A square matrix D is called diagonal if all but diagonal entries are zero: a a D a n 5 n n. () Diagonal matrices are the simplest matrices that are basically equivalent to vectors
More information2 Eigenvectors and Eigenvalues in abstract spaces.
MA322 Sathaye Notes on Eigenvalues Spring 27 Introduction In these notes, we start with the definition of eigenvectors in abstract vector spaces and follow with the more common definition of eigenvectors
More informationON DOMINATING THE CARTESIAN PRODUCT OF A GRAPH AND K 2. Bert L. Hartnell
Discussiones Mathematicae Graph Theory 24 (2004 ) 389 402 ON DOMINATING THE CARTESIAN PRODUCT OF A GRAPH AND K 2 Bert L. Hartnell Saint Mary s University Halifax, Nova Scotia, Canada B3H 3C3 e-mail: bert.hartnell@smu.ca
More informationOn Hadamard Diagonalizable Graphs
On Hadamard Diagonalizable Graphs S. Barik, S. Fallat and S. Kirkland Department of Mathematics and Statistics University of Regina Regina, Saskatchewan, Canada S4S 0A2 Abstract Of interest here is a characterization
More informationGraph Theory. Thomas Bloom. February 6, 2015
Graph Theory Thomas Bloom February 6, 2015 1 Lecture 1 Introduction A graph (for the purposes of these lectures) is a finite set of vertices, some of which are connected by a single edge. Most importantly,
More informationLectures 2 3 : Wigner s semicircle law
Fall 009 MATH 833 Random Matrices B. Való Lectures 3 : Wigner s semicircle law Notes prepared by: M. Koyama As we set up last wee, let M n = [X ij ] n i,j= be a symmetric n n matrix with Random entries
More informationChapter 5. Eigenvalues and Eigenvectors
Chapter 5 Eigenvalues and Eigenvectors Section 5. Eigenvectors and Eigenvalues Motivation: Difference equations A Biology Question How to predict a population of rabbits with given dynamics:. half of the
More informationAlgebraically defined graphs and generalized quadrangles
Department of Mathematics Kutztown University of Pennsylvania Combinatorics and Computer Algebra 2015 July 22, 2015 Cages and the Moore bound For given positive integers k and g, find the minimum number
More informationAn Algorithmist s Toolkit September 15, Lecture 2
18.409 An Algorithmist s Toolkit September 15, 007 Lecture Lecturer: Jonathan Kelner Scribe: Mergen Nachin 009 1 Administrative Details Signup online for scribing. Review of Lecture 1 All of the following
More informationLectures 2 3 : Wigner s semicircle law
Fall 009 MATH 833 Random Matrices B. Való Lectures 3 : Wigner s semicircle law Notes prepared by: M. Koyama As we set up last wee, let M n = [X ij ] n i,j=1 be a symmetric n n matrix with Random entries
More informationCographs; chordal graphs and tree decompositions
Cographs; chordal graphs and tree decompositions Zdeněk Dvořák September 14, 2015 Let us now proceed with some more interesting graph classes closed on induced subgraphs. 1 Cographs The class of cographs
More informationFurther Mathematical Methods (Linear Algebra) 2002
Further Mathematical Methods (Linear Algebra) 2002 Solutions For Problem Sheet 4 In this Problem Sheet, we revised how to find the eigenvalues and eigenvectors of a matrix and the circumstances under which
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationMATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2.
MATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2. Diagonalization Let L be a linear operator on a finite-dimensional vector space V. Then the following conditions are equivalent:
More informationMATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.
MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of
More informationThis 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 informationSPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS
SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly
More informationBipartite graphs with at most six non-zero eigenvalues
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 11 (016) 315 35 Bipartite graphs with at most six non-zero eigenvalues
More informationNotes 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 informationMATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003
MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003 1. True or False (28 points, 2 each) T or F If V is a vector space
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors week -2 Fall 26 Eigenvalues and eigenvectors The most simple linear transformation from R n to R n may be the transformation of the form: T (x,,, x n ) (λ x, λ 2,, λ n x n
More informationLINEAR 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 informationLecture 13: Spectral Graph Theory
CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 13: Spectral Graph Theory Lecturer: Shayan Oveis Gharan 11/14/18 Disclaimer: These notes have not been subjected to the usual scrutiny reserved
More informationKernels 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 information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More information235 Final exam review questions
5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an
More informationACO Comprehensive Exam March 17 and 18, Computability, Complexity and Algorithms
1. Computability, Complexity and Algorithms (a) Let G(V, E) be an undirected unweighted graph. Let C V be a vertex cover of G. Argue that V \ C is an independent set of G. (b) Minimum cardinality vertex
More informationarxiv: v2 [math.co] 7 Jan 2016
Global Cycle Properties in Locally Isometric Graphs arxiv:1506.03310v2 [math.co] 7 Jan 2016 Adam Borchert, Skylar Nicol, Ortrud R. Oellermann Department of Mathematics and Statistics University of Winnipeg,
More informationLecture 13 Spectral Graph Algorithms
COMS 995-3: Advanced Algorithms March 6, 7 Lecture 3 Spectral Graph Algorithms Instructor: Alex Andoni Scribe: Srikar Varadaraj Introduction Today s topics: Finish proof from last lecture Example of random
More informationAutomorphisms of strongly regular graphs and PDS in Abelian groups
Automorphisms of strongly regular graphs and PDS in Abelian groups Zeying Wang Department of Mathematical Sciences Michigan Technological University Joint work with Stefaan De Winter and Ellen Kamischke
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationThe least eigenvalue of the signless Laplacian of non-bipartite unicyclic graphs with k pendant vertices
Electronic Journal of Linear Algebra Volume 26 Volume 26 (2013) Article 22 2013 The least eigenvalue of the signless Laplacian of non-bipartite unicyclic graphs with k pendant vertices Ruifang Liu rfliu@zzu.edu.cn
More informationTopics 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 informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
More informationGraph 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 informationA Questionable Distance-Regular Graph
A Questionable Distance-Regular Graph Rebecca Ross Abstract In this paper, we introduce distance-regular graphs and develop the intersection algebra for these graphs which is based upon its intersection
More informationHomework 11 Solutions. Math 110, Fall 2013.
Homework 11 Solutions Math 110, Fall 2013 1 a) Suppose that T were self-adjoint Then, the Spectral Theorem tells us that there would exist an orthonormal basis of P 2 (R), (p 1, p 2, p 3 ), consisting
More informationScribes: Po-Hsuan Wei, William Kuzmaul Editor: Kevin Wu Date: October 18, 2016
CS 267 Lecture 7 Graph Spanners Scribes: Po-Hsuan Wei, William Kuzmaul Editor: Kevin Wu Date: October 18, 2016 1 Graph Spanners Our goal is to compress information about distances in a graph by looking
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationA = 3 1. We conclude that the algebraic multiplicity of the eigenvalues are both one, that is,
65 Diagonalizable Matrices It is useful to introduce few more concepts, that are common in the literature Definition 65 The characteristic polynomial of an n n matrix A is the function p(λ) det(a λi) Example
More informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
More informationQuick Tour of Linear Algebra and Graph Theory
Quick Tour of Linear Algebra and Graph Theory CS224w: Social and Information Network Analysis Fall 2012 Yu Wayne Wu Based on Borja Pelato s version in Fall 2011 Matrices and Vectors Matrix: A rectangular
More informationLecture 7. Econ August 18
Lecture 7 Econ 2001 2015 August 18 Lecture 7 Outline First, the theorem of the maximum, an amazing result about continuity in optimization problems. Then, we start linear algebra, mostly looking at familiar
More informationReview of Some Concepts from Linear Algebra: Part 2
Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, 2019 1 / 22 Vector spaces A set
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
More informationJORDAN NORMAL FORM. Contents Introduction 1 Jordan Normal Form 1 Conclusion 5 References 5
JORDAN NORMAL FORM KATAYUN KAMDIN Abstract. This paper outlines a proof of the Jordan Normal Form Theorem. First we show that a complex, finite dimensional vector space can be decomposed into a direct
More informationChapter 6: Orthogonality
Chapter 6: Orthogonality (Last Updated: November 7, 7) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved around.. Inner products
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationLinear Algebra Final Exam Solutions, December 13, 2008
Linear Algebra Final Exam Solutions, December 13, 2008 Write clearly, with complete sentences, explaining your work. You will be graded on clarity, style, and brevity. If you add false statements to a
More informationLINEAR ALGEBRA SUMMARY SHEET.
LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized
More informationA Characterization of Distance-Regular Graphs with Diameter Three
Journal of Algebraic Combinatorics 6 (1997), 299 303 c 1997 Kluwer Academic Publishers. Manufactured in The Netherlands. A Characterization of Distance-Regular Graphs with Diameter Three EDWIN R. VAN DAM
More informationThe non-bipartite graphs with all but two eigenvalues in
The non-bipartite graphs with all but two eigenvalues in [ 1, 1] L.S. de Lima 1, A. Mohammadian 1, C.S. Oliveira 2 1 Departamento de Engenharia de Produção, Centro Federal de Educação Tecnológica Celso
More informationOn Moore Graphs with Diameters 2 and 3
.., A. J. Hoffman* R. R. Singleton On Moore Graphs with Diameters 2 and 3 Abstract: This note treats the existence of connected, undirected graphs homogeneous of degree d and of diameter k, having a number
More information3 (Maths) Linear Algebra
3 (Maths) Linear Algebra References: Simon and Blume, chapters 6 to 11, 16 and 23; Pemberton and Rau, chapters 11 to 13 and 25; Sundaram, sections 1.3 and 1.5. The methods and concepts of linear algebra
More informationHamiltonian Graphs Graphs
COMP2121 Discrete Mathematics Hamiltonian Graphs Graphs Hubert Chan (Chapter 9.5) [O1 Abstract Concepts] [O2 Proof Techniques] [O3 Basic Analysis Techniques] 1 Hamiltonian Paths and Circuits [O1] A Hamiltonian
More informationx = π m (a 0 + a 1 π + a 2 π ) where a i R, a 0 = 0, m Z.
ALGEBRAIC NUMBER THEORY LECTURE 7 NOTES Material covered: Local fields, Hensel s lemma. Remark. The non-archimedean topology: Recall that if K is a field with a valuation, then it also is a metric space
More informationMATH 31BH Homework 1 Solutions
MATH 3BH Homework Solutions January 0, 04 Problem.5. (a) (x, y)-plane in R 3 is closed and not open. To see that this plane is not open, notice that any ball around the origin (0, 0, 0) will contain points
More informationOn the mean connected induced subgraph order of cographs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 71(1) (018), Pages 161 183 On the mean connected induced subgraph order of cographs Matthew E Kroeker Lucas Mol Ortrud R Oellermann University of Winnipeg Winnipeg,
More informationLECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT FALL 2006 PRINCETON UNIVERSITY
LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT 204 - FALL 2006 PRINCETON UNIVERSITY ALFONSO SORRENTINO 1 Adjoint of a linear operator Note: In these notes, V will denote a n-dimensional euclidean vector
More informationInduced Saturation of Graphs
Induced Saturation of Graphs Maria Axenovich a and Mónika Csikós a a Institute of Algebra and Geometry, Karlsruhe Institute of Technology, Englerstraße 2, 76128 Karlsruhe, Germany Abstract A graph G is
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationSEMI-STRONG SPLIT DOMINATION IN GRAPHS. Communicated by Mehdi Alaeiyan. 1. Introduction
Transactions on Combinatorics ISSN (print): 2251-8657, ISSN (on-line): 2251-8665 Vol. 3 No. 2 (2014), pp. 51-63. c 2014 University of Isfahan www.combinatorics.ir www.ui.ac.ir SEMI-STRONG SPLIT DOMINATION
More informationQuick Tour of Linear Algebra and Graph Theory
Quick Tour of Linear Algebra and Graph Theory CS224W: Social and Information Network Analysis Fall 2014 David Hallac Based on Peter Lofgren, Yu Wayne Wu, and Borja Pelato s previous versions Matrices and
More informationLaplacians of Graphs, Spectra and Laplacian polynomials
Mednykh A. D. (Sobolev Institute of Math) Laplacian for Graphs 27 June - 03 July 2015 1 / 30 Laplacians of Graphs, Spectra and Laplacian polynomials Alexander Mednykh Sobolev Institute of Mathematics Novosibirsk
More informationOn a problem of Bermond and Bollobás
On a problem of Bermond and Bollobás arxiv:1803.07501v1 [math.co] 20 Mar 2018 Slobodan Filipovski University of Primorska, Koper, Slovenia slobodan.filipovski@famnit.upr.si Robert Jajcay Comenius University,
More informationOn the difference between the revised Szeged index and the Wiener index
On the difference between the revised Szeged index and the Wiener index Sandi Klavžar a,b,c M J Nadjafi-Arani d June 3, 01 a Faculty of Mathematics and Physics, University of Ljubljana, Slovenia sandiklavzar@fmfuni-ljsi
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More information18.312: Algebraic Combinatorics Lionel Levine. Lecture 19
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,
More informationGeneralized Eigenvectors and Jordan Form
Generalized Eigenvectors and Jordan Form We have seen that an n n matrix A is diagonalizable precisely when the dimensions of its eigenspaces sum to n. So if A is not diagonalizable, there is at least
More informationMATH 221, Spring Homework 10 Solutions
MATH 22, Spring 28 - Homework Solutions Due Tuesday, May Section 52 Page 279, Problem 2: 4 λ A λi = and the characteristic polynomial is det(a λi) = ( 4 λ)( λ) ( )(6) = λ 6 λ 2 +λ+2 The solutions to the
More information= 10 such triples. If it is 5, there is = 1 such triple. Therefore, there are a total of = 46 such triples.
. Two externally tangent unit circles are constructed inside square ABCD, one tangent to AB and AD, the other to BC and CD. Compute the length of AB. Answer: + Solution: Observe that the diagonal of the
More information