The path polynomial of a complete graph
|
|
- Jonah Townsend
- 5 years ago
- Views:
Transcription
1 Electroic Joural of Liear Algebra Volume 10 Article The path polyomial of a complete graph C M da Foseca cmf@matucpt Follow this ad additioal wors at: Recommeded Citatio da Foseca, C M 2003, "The path polyomial of a complete graph", Electroic Joural of Liear Algebra, Volume 10 DOI: This Article is brought to you for free ad ope access by Wyomig Scholars Repository It has bee accepted for iclusio i Electroic Joural of Liear Algebra by a authorized editor of Wyomig Scholars Repository For more iformatio, please cotact scholcom@uwyoedu
2 Volume 10, pp , Jue 2003 THE PATH POLYNOMIAL OF A COMPLETE GRAPH C M DA FONSECA Abstract Let P x deote the polyomial of the path o vertices A complete descriptio of the matrix that is the obtaied by evaluatig P x at the adjacecy matrix of the complete graph, alog with computig the effect of evaluatig P x with Laplacia matrices of a path ad of a circuit Key words Graph, Adjacecy matrix, Laplacia matrix, Characteristic polyomial AMS subject classificatios 05C38, 05C50 1 Itroductio ad prelimiaries For a fiite ad udirected graph G without loops or multiple edges, with vertices, let us defie the polyomial of G, P G,as the characteristic polyomial of its adjacecy matrix, AG, ie, P G x =detxi AG Whe the graph is a path with vertices, we simply call P G the path polyomial ad deote it by P Defie A as the adjacecy matrix of a path o vertices For several iterestig classes of graphs, AG i is a polyomial i AG, where G i is the ith distace graph of G [5]Actually, for distace-regular graphs, AG i isa polyomial i AG ofdegreei, ad this property characterizes these id of graphs [14] I [4], Beezer has ased whe a polyomial of a adjacecy matrix will be the adjacecy matrix of aother graphbeezer gave a solutio i the case that the origial graph is a path Theorem 11 [4] Suppose that px is a polyomial of degree less tha The pa is the adjacecy matrix of graph if ad oly if px =P 2i+1 x, forsome i, with0 i 2 1 I the same paper, Beezer gave a elegat formula for P A with =1,,, ad Bapat ad Lal, i [1], completely described the structure of P A, for all itegers This result was also reached by Foseca ad Petroilho [10] i a oiductive way Theorem 12 [1],[4],[10] For 0 1, beig a positive iteger, { 1 if i + j = +2r,with1 r mi {i, j, } P A ij = 0 otherwise I [12], Shi Roghua obtaied some geeralizatios of the oes achieved by Bapat ad LalLater, i [10], Foseca ad Petroilho determied the matrix P C, where C is the adjacecy matrix of a circuit o vertices Received by the editors o 15 March 2003 Accepted for publicatio o 02 May 2003 Hadlig Editor: Ravidar B Bapat Departameto de Matemática, Uiversidade de Coimbra, COIMBRA, PORTUGAL cmf@matucpt Supported by CMUC - Cetro de Matemática da Uiversidade de Coimbra 155
3 Volume 10, pp , Jue C M da Foseca Cosider the permutatio σ =12 Theorem 13 [10] For ay oegative iteger, P C = δ 2r,+2+j ṅ P σ j, j=0 r=0 where δ is the Kroecer fuctio, σ is the permutatio 12, P σ j is the permutatio matrix of σ j ad ṅ rus over the multiples of Accordig to Bapat ad Lal cf[1], a graph G is called path-positive of order m if P G 0, for =1, 2,,m,adG is simply called path-positive if it is pathpositive of ay orderi [3], Bapat ad Lal have characterized all graphs that are path-positivethe followig corollary is immediate from the theorem above Corollary 14 The circuit C is path-positive We defie the complete graph K, to be the graph with vertices i which each pair of vertices is adjacetthe adjacecy matrix of a complete graph, which we idetify also by K,isthe matrix 11 K = I this ote, we evaluate P K 2 The polyomial P Let us cosider the tridiagoal matrix A whose etries are give by { 1 if i j =1 A ij = 0 otherwise The expasio of the determiat det xi A =P x alog the first row or colum gives us the recurrece relatio 21 P x =xp 1 x P 2 x, for ay positive iteger, with the covetio P 1 x =0adP 0 x =1 It is well ow that x 22 P x =U, x C, =0, 1,, 2 where U x are the Chebyshev polyomials of the secod id
4 Volume 10, pp , Jue 2003 The Path Polyomial of a Complete Graph 157 From 22, it is straightforward to prove that 23 P x P y x y 1 = P l x P 1 l y The, from 21 ad 23, we may coclude the followig lemma Lemma 21 For ay positive iteger ad square matrices A ad B, 1 P A P B = P l AA B P 1 l B As i Bapat ad Lal [1], ote that a coected graph is path-positive if it has a spaig subgraph which is path-positivethus we have this immediate corollary from Corollary 14 Corollary 22 The complete graph K is path-positive 3 Evaluatig P of a complete graph If a matrix A =a ij satisfiesthe relatio a ij = a 1σ 1 i j we say that A is a circulat matrixtherefore, to defie a circulat matrix A is equivalet to presetig a tuple, say a 1,,a The ad its eigevalues are give by A = a i P σ i, i=0 31 λ h = ζ hl a l, where ζ =exp i 2π Give a polyomial px, the image of A is pa =p The, a i P σ i = 1 ζ hj p i=0 j=0 h=0 ζ hl a l P σ j P a i P σ i = 1 ζ hj P λ h P σ j, where λ h is defied as i 31 i=0 j=0 h=0
5 Volume 10, pp , Jue C M da Foseca The matrix K, defied i 11, is a circulat matrix ad it ca be writte K = P σ i i=1 By 31, K has the eigevalues λ 0 = 1adλ l = 1, for l =1,, 1 Therefore, P K =P P σ i i=1 = 1 ζ hj P λ h P σ j j=0 h=0 = 1 P 1 + P 1 ζ hj P σ j j=0 h=1 = P 1 P σ P 1 P 1 P σ j Note that P σ 0 is the idetity matrix We have thus proved the mai result of this sectio: Theorem 31 For ay oegative iteger, the diagoal etries of P K are the weighted average 1 P 1 + P 1 ad the off-diagoal etries are 1 P 1 1 P 1 We ca easily evaluate the differet values of each term of the sum P K Accordigto22, 1 if 1mod3 P 1 = 0 if 2mod3 1 if 0mod3 Aother relatio already ow [11, p72] for P x is j=0 P x = /2 1 l l l x 2l, where z deotes the greatest iteger less or equal to ztherefore we have also P 1 P 1 = = /2 /2 1 l l l 2l 1 j+l j=1 1 2l 1 2l l! l!j! 2l j! j 1
6 Volume 10, pp , Jue 2003 The Path Polyomial of a Complete Graph Evaluatig P of some Laplacia matrices Let G be a graphdeote DG the diagoal matrix of its vertex degrees ad by AG its adjacecy matrix The LG =DG AG is the Laplacia matrix of G I this sectio, expressios for P LA ad P LC, the path polyomials of the Laplacia matrices of a path ad a circuit, respectively, with vertices, are determied Let us cosider the followig recurrece relatio: P 0 x =1, P1 x =x +1, ad P x =x +2 P 1 x P 2 x, for 2 1, P x =x +1 P x P 2 x Therefore x x P x =U 2 +1 U , for 2 1, ad P x =xu x 2 +1 where U x are the Chebyshev polyomials of the secod id The the zeroes of P x are λ j =2cos jπ 2, j =0,, 1 The recurrece relatio above ca be writte i the followig matricial way: P 0 x P 0 x 0 P 1 x P 1 x x = + P 0 x P 2 x P 2 x 0 P x P x 1 Thus, for j =0,, 1, the vector P 0 λ j P 1 λ j 41 = cos jπ 2 P 2 λ j P λ j 1 cos jπ 2 cos 3 jπ 2 cos2 3 jπ 2 cos2 1 jπ 2
7 Volume 10, pp , Jue C M da Foseca is a eigevector associated to the eigevalue λ j of LA Therefore the matrix LA is diagoalizable ad, for 0, thei, jth etry of P LA is give by P LA ij = 1 P i 1 λ l P λ l P j 1 λ l 2 s=1 Ps 1 λ l which is equal to 1 cos π cos2i 1 lπ 2 U cos lπ 1 cos2j 1 lπ 2 If we defie l=1 α p m = cos m lπ lπ cosp, l=1 the α p m = 1 2 α p 1 m 1 + αp 1 m+1 ad 42 with α p m = 1 2 p p p l α 0 m+2l p, α 0 m = δ m,2ṅ m, where ṅ represets a multiple of Usig the trigoometric trasformatio formula ad the Taylor formula U cos lπ 1 = p=0 U p 1 cos p lπ p!, we ca state the followig propositio Theorem 41 For 0, beig a positive iteger, P LA ij = 1 cos π where α p m is defied as i 42 p=0 U p 1 α p i j p! + i+j 1 αp,
8 Volume 10, pp , Jue 2003 The Path Polyomial of a Complete Graph 161 Note that U p 1 ca be easily evaluated, sice ad the U p U x = /2 1 l l l 2x 2l, /2 1 = 1 l p 2 2l l! l! 2l p! Now, we ca fid the matrix P LC usig the same techiques of the last sectio LC is the circulat matrix Hece The eigevalues of LC are for l =0,, 1 ad thus LC =2P σ 0 P σ P σ 2 2cos 2lπ, P LC =P 2P σ 0 P σ P σ = 1 j=0 = 1 j=0 p=0 2ljπ i e U 1 cos 2lπ p P σ j U p 1 l!p l!2 p δ j+2l p,ṅp σ j REFERENCES [1] RB Bapat ad AK Lal Path-positive Graphs Liear Algebra ad Its Applicatios, 149: , 1991 [2] RB Bapat ad VS Suder O hypergroups of matrices Liear ad Multiliear Algebra, 29: , 1991 [3] RB Bapat ad AK Lal Path positivity ad ifiite Coxeter groups Liear Algebra ad Its Applicatios, 196:19 35, 1994 [4] Robert A Beezer O the polyomial of a path Liear Algebra ad Its Applicatios, 63: , 1984
9 Volume 10, pp , Jue C M da Foseca [5] NL Biggs Algebraic Graph Theory Cambridge Uiversity Press, Cambridge, 1974 [6] TS Chihara A itroductio to orthogoal polyomials Gordo ad Breach, New Yor, 1978 [7] DMCvetović, M Doobad H Sachs Spectra of Graphs, Theory ad Applicatios Academic Press, New Yor, 1979 [8] PJ Davis Circulat Matrices Joh Wiley & Sos, New Yor, 1979 [9] A Erdélyi, W Magus, F Oberhettiger ad FG Tricomi Higher Trascedetal Fuctios Vol II Robert E Krieger Publishig Co, Melboure, FL, 1981 [10] CM da Foseca ad J Petroilho Path polyomials of a circuit: a costructive approach Liear ad Multiliear Algebra, 44: , 1998 [11] László Lovász Combiatorial Problems ad Exercises North-Hollad, Amsterdam, 1979 [12] Shi Roghua Path polyomials of a graph Liear Algebra ad Its Applicatios, 31: , 1996 [13] Ala C Wilde Differetial equatios ivolvig circulat matrices Rocy Moutai Joural of Mathematics, 13:1 13, 1983 [14] PM Weichsel O distace-regularity i graphs Joural of Combiatorial Theory Series B, 32: , 1982
Disjoint unions of complete graphs characterized by their Laplacian spectrum
Electroic Joural of Liear Algebra Volume 18 Volume 18 (009) Article 56 009 Disjoit uios of complete graphs characterized by their Laplacia spectrum Romai Boulet boulet@uiv-tlse.fr Follow this ad additioal
More informationSome Trigonometric Identities Involving Fibonacci and Lucas Numbers
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 (2009), Article 09.8.4 Some Trigoometric Idetities Ivolvig Fiboacci ad Lucas Numbers Kh. Bibak ad M. H. Shirdareh Haghighi Departmet of Mathematics Shiraz
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationA FIBONACCI MATRIX AND THE PERMANENT FUNCTION
A FIBONACCI MATRIX AND THE PERMANENT FUNCTION BRUCE W. KING Burt Hiils-Ballsto Lake High School, Ballsto Lake, New York ad FRANCIS D. PARKER The St. Lawrece Uiversity, Cato, New York The permaet of a -square
More informationNumber of Spanning Trees of Circulant Graphs C 6n and their Applications
Joural of Mathematics ad Statistics 8 (): 4-3, 0 ISSN 549-3644 0 Sciece Publicatios Number of Spaig Trees of Circulat Graphs C ad their Applicatios Daoud, S.N. Departmet of Mathematics, Faculty of Sciece,
More informationEXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX
EXPLICIT INVERSE OF A TRIDIAGONAL k TOEPLITZ MATRIX C M DA FONSECA AND J PETRONILHO Abstract We obtai explicit formulas for the etries of the iverse of a osigular ad irreducible tridiagoal k Toeplitz matrix
More informationA new error bound for linear complementarity problems for B-matrices
Electroic Joural of Liear Algebra Volume 3 Volume 3: (206) Article 33 206 A ew error boud for liear complemetarity problems for B-matrices Chaoqia Li Yua Uiversity, lichaoqia@yueduc Megtig Ga Shaorog Yag
More informationEigenvalue localization for complex matrices
Electroic Joural of Liear Algebra Volume 7 Article 1070 014 Eigevalue localizatio for complex matrices Ibrahim Halil Gumus Adıyama Uiversity, igumus@adiyama.edu.tr Omar Hirzallah Hashemite Uiversity, o.hirzal@hu.edu.jo
More informationA Note On The Exponential Of A Matrix Whose Elements Are All 1
Applied Mathematics E-Notes, 8(208), 92-99 c ISSN 607-250 Available free at mirror sites of http://wwwmaththuedutw/ ame/ A Note O The Expoetial Of A Matrix Whose Elemets Are All Reza Farhadia Received
More informationFormulas for the Number of Spanning Trees in a Maximal Planar Map
Applied Mathematical Scieces Vol. 5 011 o. 64 3147-3159 Formulas for the Number of Spaig Trees i a Maximal Plaar Map A. Modabish D. Lotfi ad M. El Marraki Departmet of Computer Scieces Faculty of Scieces
More informationResistance matrix and q-laplacian of a unicyclic graph
Resistace matrix ad q-laplacia of a uicyclic graph R. B. Bapat Idia Statistical Istitute New Delhi, 110016, Idia e-mail: rbb@isid.ac.i Abstract: The resistace distace betwee two vertices of a graph ca
More informationSOME RELATIONS ON HERMITE MATRIX POLYNOMIALS. Levent Kargin and Veli Kurt
Mathematical ad Computatioal Applicatios, Vol. 18, No. 3, pp. 33-39, 013 SOME RELATIONS ON HERMITE MATRIX POLYNOMIALS Levet Kargi ad Veli Kurt Departmet of Mathematics, Faculty Sciece, Uiversity of Adeiz
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More information1 Last time: similar and diagonalizable matrices
Last time: similar ad diagoalizable matrices Let be a positive iteger Suppose A is a matrix, v R, ad λ R Recall that v a eigevector for A with eigevalue λ if v ad Av λv, or equivaletly if v is a ozero
More informationMath 4707 Spring 2018 (Darij Grinberg): homework set 4 page 1
Math 4707 Sprig 2018 Darij Griberg): homewor set 4 page 1 Math 4707 Sprig 2018 Darij Griberg): homewor set 4 due date: Wedesday 11 April 2018 at the begiig of class, or before that by email or moodle Please
More informationA class of spectral bounds for Max k-cut
A class of spectral bouds for Max k-cut Miguel F. Ajos, José Neto December 07 Abstract Let G be a udirected ad edge-weighted simple graph. I this paper we itroduce a class of bouds for the maximum k-cut
More informationCourse : Algebraic Combinatorics
Course 18.312: Algebraic Combiatorics Lecture Notes # 18-19 Addedum by Gregg Musier March 18th - 20th, 2009 The followig material ca be foud i a umber of sources, icludig Sectios 7.3 7.5, 7.7, 7.10 7.11,
More informationOn Net-Regular Signed Graphs
Iteratioal J.Math. Combi. Vol.1(2016), 57-64 O Net-Regular Siged Graphs Nuta G.Nayak Departmet of Mathematics ad Statistics S. S. Dempo College of Commerce ad Ecoomics, Goa, Idia E-mail: ayakuta@yahoo.com
More informationOn the Determinants and Inverses of Skew Circulant and Skew Left Circulant Matrices with Fibonacci and Lucas Numbers
WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog O the Determiats ad Iverses of Skew Circulat ad Skew Left Circulat Matrices with Fiboacci ad Lucas Numbers YUN GAO Liyi Uiversity Departmet
More informationGENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION
J Korea Math Soc 44 (2007), No 2, pp 487 498 GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION Gi-Sag Cheo ad Moawwad E A El-Miawy Reprited from the Joural of the Korea Mathematical
More informationSPECTRA OF GRAPH OPERATIONS BASED ON CORONA AND NEIGHBORHOOD CORONA OF GRAPH G AND K 1
JOURNAL OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4866, ISSN (o) 2303-4947 www.imvibl.org / JOURNALS / JOURNAL Vol. 5(2015), 55-69 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS
More informationA GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca
Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity
More informationEigenvalues and Eigenvectors
5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math E-2b Lecture #8 Notes This week is all about determiats. We ll discuss how to defie them, how to calculate them, lear the allimportat property kow as multiliearity, ad show that a square matrix A
More informationTHE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES
Joural of Mathematical Aalysis ISSN: 17-341, URL: http://iliriascom/ma Volume 7 Issue 4(16, Pages 13-19 THE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES ABEDALLAH RABABAH, AYMAN AL
More informationNon-singular circulant graphs and digraphs
Electroic Joural of Liear Algebra Volume 26 Volume 26 2013) Article 19 2013 No-sigular circulat graphs ad digraphs A. K. Lal arlal@iitk.ac.i A. Satyaarayaa Reddy Follow this ad additioal works at: http://repository.uwyo.edu/ela
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationSome remarks for codes and lattices over imaginary quadratic
Some remarks for codes ad lattices over imagiary quadratic fields Toy Shaska Oaklad Uiversity, Rochester, MI, USA. Caleb Shor Wester New Eglad Uiversity, Sprigfield, MA, USA. shaska@oaklad.edu Abstract
More informationHarmonic Number Identities Via Euler s Transform
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810
More informationSOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS
Folia Mathematica Vol. 5, No., pp. 4 6 Acta Uiversitatis Lodziesis c 008 for Uiversity of Lódź Press SOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS ROMAN WITU LA, DAMIAN S
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math S-b Lecture # Notes This wee is all about determiats We ll discuss how to defie them, how to calculate them, lear the allimportat property ow as multiliearity, ad show that a square matrix A is ivertible
More informationSymmetric Matrices and Quadratic Forms
7 Symmetric Matrices ad Quadratic Forms 7.1 DIAGONALIZAION OF SYMMERIC MARICES SYMMERIC MARIX A symmetric matrix is a matrix A such that. A = A Such a matrix is ecessarily square. Its mai diagoal etries
More informationTRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES. 1. Introduction
Math Appl 6 2017, 143 150 DOI: 1013164/ma201709 TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES PANKAJ KUMAR DAS ad LALIT K VASHISHT Abstract We preset some iequality/equality for traces of Hadamard
More informationGraphs with few distinct distance eigenvalues irrespective of the diameters
Electroic Joural of Liear Algebra Volume 29 Special volume for Proceedigs of the Iteratioal Coferece o Liear Algebra ad its Applicatios dedicated to Professor Ravidra B. Bapat Article 13 2015 Graphs with
More informationREGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS
REGULARIZATION OF CERTAIN DIVERGENT SERIES OF POLYNOMIALS LIVIU I. NICOLAESCU ABSTRACT. We ivestigate the geeralized covergece ad sums of series of the form P at P (x, where P R[x], a R,, ad T : R[x] R[x]
More informationFibonacci numbers and orthogonal polynomials
Fiboacci umbers ad orthogoal polyomials Christia Berg April 10, 2006 Abstract We prove that the sequece (1/F +2 0 of reciprocals of the Fiboacci umbers is a momet sequece of a certai discrete probability,
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
More informationCOMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q n } Sang Pyo Jun
Korea J. Math. 23 2015) No. 3 pp. 371 377 http://dx.doi.org/10.11568/kjm.2015.23.3.371 COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q } Sag Pyo Ju Abstract. I this ote we cosider a geeralized
More informationOn Energy and Laplacian Energy of Graphs
Electroic Joural of Liear Algebra Volume 31 Volume 31: 016 Article 15 016 O Eergy ad Laplacia Eergy of Graphs Kikar Ch Das Sugkyukwa Uiversity, kikardas003@googlemailcom Seyed Ahmad Mojalal Sugkyukwa Uiversity,
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
More informationMATH10212 Linear Algebra B Proof Problems
MATH22 Liear Algebra Proof Problems 5 Jue 26 Each problem requests a proof of a simple statemet Problems placed lower i the list may use the results of previous oes Matrices ermiats If a b R the matrix
More informationBounds for the Extreme Eigenvalues Using the Trace and Determinant
ISSN 746-7659, Eglad, UK Joural of Iformatio ad Computig Sciece Vol 4, No, 9, pp 49-55 Bouds for the Etreme Eigevalues Usig the Trace ad Determiat Qi Zhog, +, Tig-Zhu Huag School of pplied Mathematics,
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationEnergy of a Hypercube and its Complement
Iteratioal Joural of Algebra, Vol. 6, 01, o. 16, 799-805 Eergy of a Hypercube ad its Complemet Xiaoge Che School of Iformatio Sciece ad Techology, Zhajiag Normal Uiversity Zhajiag Guagdog, 54048 P.R. Chia
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationA Hadamard-type lower bound for symmetric diagonally dominant positive matrices
A Hadamard-type lower boud for symmetric diagoally domiat positive matrices Christopher J. Hillar, Adre Wibisoo Uiversity of Califoria, Berkeley Jauary 7, 205 Abstract We prove a ew lower-boud form of
More informationThe inverse eigenvalue problem for symmetric doubly stochastic matrices
Liear Algebra ad its Applicatios 379 (004) 77 83 www.elsevier.com/locate/laa The iverse eigevalue problem for symmetric doubly stochastic matrices Suk-Geu Hwag a,,, Sug-Soo Pyo b, a Departmet of Mathematics
More informationA NOTE ON PASCAL S MATRIX. Gi-Sang Cheon, Jin-Soo Kim and Haeng-Won Yoon
J Korea Soc Math Educ Ser B: Pure Appl Math 6(1999), o 2 121 127 A NOTE ON PASCAL S MATRIX Gi-Sag Cheo, Ji-Soo Kim ad Haeg-Wo Yoo Abstract We ca get the Pascal s matrix of order by takig the first rows
More informationLAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH
Kragujevac Joural of Mathematics Volume 4 018, Pages 99 315 LAPLACIAN ENERGY OF GENERALIZED COMPLEMENTS OF A GRAPH H J GOWTHAM 1, SABITHA D SOUZA 1, AND PRADEEP G BHAT 1 Abstract Let P = {V 1, V, V 3,,
More informationBenaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco
EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco e-mail: Beaissa@fstt.ac.ma Mustapha
More informationc 2006 Society for Industrial and Applied Mathematics
SIAM J. MATRIX ANAL. APPL. Vol. 7, No. 3, pp. 851 860 c 006 Society for Idustrial ad Applied Mathematics EXTREMAL EIGENVALUES OF REAL SYMMETRIC MATRICES WITH ENTRIES IN AN INTERVAL XINGZHI ZHAN Abstract.
More informationMath 778S Spectral Graph Theory Handout #3: Eigenvalues of Adjacency Matrix
Math 778S Spectral Graph Theory Hadout #3: Eigevalues of Adjacecy Matrix The Cartesia product (deoted by G H) of two simple graphs G ad H has the vertex-set V (G) V (H). For ay u, v V (G) ad x, y V (H),
More informationLECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK)
LECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK) Everythig marked by is ot required by the course syllabus I this lecture, all vector spaces is over the real umber R. All vectors i R is viewed as a colum
More informationON SOLVING A FORMAL HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION IN THE COMPLEX FIELD
A. Şt. Uiv. Ovidius Costaţa Vol. (), 003, 69 78 ON SOLVING A FORMAL HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION IN THE COMPLEX FIELD Nicolae Popoviciu To Professor Silviu Sburla, at his 60 s aiversary Abstract
More informationSeries with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers
3 47 6 3 Joural of Iteger Sequeces, Vol. 5 (0), Article..7 Series with Cetral Biomial Coefficiets, Catala Numbers, ad Harmoic Numbers Khristo N. Boyadzhiev Departmet of Mathematics ad Statistics Ohio Norther
More informationAn Asymptotic Expansion for the Number of Permutations with a Certain Number of Inversions
A Asymptotic Expasio for the Number of Permutatios with a Certai Number of Iversios Lae Clark Departmet of Mathematics Souther Illiois Uiversity Carbodale Carbodale, IL 691-448 USA lclark@math.siu.edu
More informationBinomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationOn Some Properties of Digital Roots
Advaces i Pure Mathematics, 04, 4, 95-30 Published Olie Jue 04 i SciRes. http://www.scirp.org/joural/apm http://dx.doi.org/0.436/apm.04.46039 O Some Properties of Digital Roots Ilha M. Izmirli Departmet
More informationOn Some Inverse Singular Value Problems with Toeplitz-Related Structure
O Some Iverse Sigular Value Problems with Toeplitz-Related Structure Zheg-Jia Bai Xiao-Qig Ji Seak-Weg Vog Abstract I this paper, we cosider some iverse sigular value problems for Toeplitz-related matrices
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationPROBLEM SET I (Suggested Solutions)
Eco3-Fall3 PROBLE SET I (Suggested Solutios). a) Cosider the followig: x x = x The quadratic form = T x x is the required oe i matrix form. Similarly, for the followig parts: x 5 b) x = = x c) x x x x
More informationDeterminants of order 2 and 3 were defined in Chapter 2 by the formulae (5.1)
5. Determiats 5.. Itroductio 5.2. Motivatio for the Choice of Axioms for a Determiat Fuctios 5.3. A Set of Axioms for a Determiat Fuctio 5.4. The Determiat of a Diagoal Matrix 5.5. The Determiat of a Upper
More informationChapter 2. Periodic points of toral. automorphisms. 2.1 General introduction
Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationThe Discrete Fourier Transform
The Discrete Fourier Trasform Complex Fourier Series Represetatio Recall that a Fourier series has the form a 0 + a k cos(kt) + k=1 b k si(kt) This represetatio seems a bit awkward, sice it ivolves two
More informationHoggatt and King [lo] defined a complete sequence of natural numbers
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
More informationBounds of Balanced Laplacian Energy of a Complete Bipartite Graph
Iteratioal Joural of Computatioal Itelligece Research ISSN 0973-1873 Volume 13, Number 5 (2017), pp. 1157-1165 Research Idia Publicatios http://www.ripublicatio.com Bouds of Balaced Laplacia Eergy of a
More informationON GRAPHS WITH THREE DISTINCT LAPLACIAN EIGENVALUES. 1 Introduction
Appl. Math. J. Chiese Uiv. Ser. B 2007, 22(4): 478-484 ON GRAPHS WITH THREE DISTINCT LAPLACIAN EIGENVALUES Wag Yi 1 Fa Yizheg 1 Ta Yigyig 1,2 Abstract. I this paper, a equivalet coditio of a graph G with
More informationOn the distribution of coefficients of powers of positive polynomials
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 49 (2011), Pages 239 243 O the distributio of coefficiets of powers of positive polyomials László Major Istitute of Mathematics Tampere Uiversity of Techology
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationRoger Apéry's proof that zeta(3) is irrational
Cliff Bott cliffbott@hotmail.com 11 October 2011 Roger Apéry's proof that zeta(3) is irratioal Roger Apéry developed a method for searchig for cotiued fractio represetatios of umbers that have a form such
More informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationLinear combinations of graph eigenvalues
Electroic Joural of Liear Algebra Volume 5 Volume 5 2006 Article 27 2006 Liear combiatios of graph eigevalues Vladimir ikiforov vikifrv@memphis.edu Follow this ad additioal works at: http://repository.uwyo.edu/ela
More informationOn the Number of 1-factors of Bipartite Graphs
Math Sci Lett 2 No 3 181-187 (2013) 181 Mathematical Scieces Letters A Iteratioal Joural http://dxdoiorg/1012785/msl/020306 O the Number of 1-factors of Bipartite Graphs Mehmet Akbulak 1 ad Ahmet Öteleş
More informationAnalytical solutions for multi-wave transfer matrices in layered structures
Joural of Physics: Coferece Series PAPER OPEN ACCESS Aalytical solutios for multi-wave trasfer matrices i layered structures To cite this article: Yu N Belyayev 018 J Phys: Cof Ser 109 01008 View the article
More informationOn Nonsingularity of Saddle Point Matrices. with Vectors of Ones
Iteratioal Joural of Algebra, Vol. 2, 2008, o. 4, 197-204 O Nosigularity of Saddle Poit Matrices with Vectors of Oes Tadeusz Ostrowski Istitute of Maagemet The State Vocatioal Uiversity -400 Gorzów, Polad
More informationTopics in Eigen-analysis
Topics i Eige-aalysis Li Zajiag 28 July 2014 Cotets 1 Termiology... 2 2 Some Basic Properties ad Results... 2 3 Eige-properties of Hermitia Matrices... 5 3.1 Basic Theorems... 5 3.2 Quadratic Forms & Noegative
More informationarxiv: v3 [math.nt] 24 Dec 2017
DOUGALL S 5 F SUM AND THE WZ-ALGORITHM Abstract. We show how to prove the examples of a paper by Chu ad Zhag usig the WZ-algorithm. arxiv:6.085v [math.nt] Dec 07 Keywords. Geeralized hypergeometric series;
More informationAssignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.
PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio
More informationBinary codes from graphs on triples and permutation decoding
Biary codes from graphs o triples ad permutatio decodig J. D. Key Departmet of Mathematical Scieces Clemso Uiversity Clemso SC 29634 U.S.A. J. Moori ad B. G. Rodrigues School of Mathematics Statistics
More informationANOTHER GENERALIZED FIBONACCI SEQUENCE 1. INTRODUCTION
ANOTHER GENERALIZED FIBONACCI SEQUENCE MARCELLUS E. WADDILL A N D LOUIS SACKS Wake Forest College, Wisto Salem, N. C., ad Uiversity of ittsburgh, ittsburgh, a. 1. INTRODUCTION Recet issues of umerous periodicals
More information1/(1 -x n ) = \YJ k=0
ADVANCED PROBLEMS AND SOLUTIONS Edited by RAYMOND E.WHITNEY Lock Have State College, Lock Have, Pesylvaia Sed all commuicatios cocerig Advaced Problems ad Solutios to Eaymod E. Whitey, Mathematics Departmet,
More informationA brief introduction to linear algebra
CHAPTER 6 A brief itroductio to liear algebra 1. Vector spaces ad liear maps I what follows, fix K 2{Q, R, C}. More geerally, K ca be ay field. 1.1. Vector spaces. Motivated by our ituitio of addig ad
More informationDecoupling Zeros of Positive Discrete-Time Linear Systems*
Circuits ad Systems,,, 4-48 doi:.436/cs..7 Published Olie October (http://www.scirp.org/oural/cs) Decouplig Zeros of Positive Discrete-Time Liear Systems* bstract Tadeusz Kaczorek Faculty of Electrical
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationLinear Algebra and its Applications
Liear Algebra ad its Applicatios 433 (2010) 1148 1153 Cotets lists available at ScieceDirect Liear Algebra ad its Applicatios joural homepage: www.elsevier.com/locate/laa The algebraic coectivity of graphs
More informationGAMALIEL CERDA-MORALES 1. Blanco Viel 596, Valparaíso, Chile. s: /
THE GELIN-CESÀRO IDENTITY IN SOME THIRD-ORDER JACOBSTHAL SEQUENCES arxiv:1810.08863v1 [math.co] 20 Oct 2018 GAMALIEL CERDA-MORALES 1 1 Istituto de Matemáticas Potificia Uiversidad Católica de Valparaíso
More informationThe Binet formula, sums and representations of generalized Fibonacci p-numbers
Europea Joural of Combiatorics 9 (008) 70 7 wwwelseviercom/locate/ec The Biet formula, sums ad represetatios of geeralized Fiboacci p-umbers Emrah Kilic TOBB ETU Uiversity of Ecoomics ad Techology, Mathematics
More informationk-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
More informationOn the Jacobsthal-Lucas Numbers by Matrix Method 1
It J Cotemp Math Scieces, Vol 3, 2008, o 33, 1629-1633 O the Jacobsthal-Lucas Numbers by Matrix Method 1 Fikri Köke ad Durmuş Bozkurt Selçuk Uiversity, Faculty of Art ad Sciece Departmet of Mathematics,
More informationMT5821 Advanced Combinatorics
MT5821 Advaced Combiatorics 9 Set partitios ad permutatios It could be said that the mai objects of iterest i combiatorics are subsets, partitios ad permutatios of a fiite set. We have spet some time coutig
More informationTheorem: Let A n n. In this case that A does reduce to I, we search for A 1 as the solution matrix X to the matrix equation A X = I i.e.
Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse
More informationMatrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.
2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a
More informationRandom Matrices with Blocks of Intermediate Scale Strongly Correlated Band Matrices
Radom Matrices with Blocks of Itermediate Scale Strogly Correlated Bad Matrices Jiayi Tog Advisor: Dr. Todd Kemp May 30, 07 Departmet of Mathematics Uiversity of Califoria, Sa Diego Cotets Itroductio Notatio
More informationMath 104: Homework 2 solutions
Math 04: Homework solutios. A (0, ): Sice this is a ope iterval, the miimum is udefied, ad sice the set is ot bouded above, the maximum is also udefied. if A 0 ad sup A. B { m + : m, N}: This set does
More informationAN INTRODUCTION TO SPECTRAL GRAPH THEORY
AN INTRODUCTION TO SPECTRAL GRAPH THEORY JIAQI JIANG Abstract. Spectral graph theory is the study of properties of the Laplacia matrix or adjacecy matrix associated with a graph. I this paper, we focus
More informationPAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 9-20 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v94i.2 PAijpam.eu
More informationReview Problems 1. ICME and MS&E Refresher Course September 19, 2011 B = C = AB = A = A 2 = A 3... C 2 = C 3 = =
Review Problems ICME ad MS&E Refresher Course September 9, 0 Warm-up problems. For the followig matrices A = 0 B = C = AB = 0 fid all powers A,A 3,(which is A times A),... ad B,B 3,... ad C,C 3,... Solutio:
More informationSome remarks on a generalized vector product
Revista Itegració Escuela de Matemáticas Uiversidad Idustrial de Satader Vol. 9, No., 011, pág. 151 16 Some remarks o a geeralized vector product Primitivo Acosta-Humáez a, Moisés Arada b, Reialdo Núñez
More information