# On the spectral radius, k-degree and the upper bound of energy in a graph

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 57 (007) ISSN On the spectral radius, k-degree and the upper bound of energy in a graph Yaoping Hou a, Zhen Tang a and Chingwah Woo b a Department of Mathematics, Hunan Normal University Changsha, Hunan , China b Department of Mathematics, City University of Hong Kong Hong Kong, China (Received November 8, 006) Abstract: Let G be a simple graph. For v V (G), k-degree d k of v is the number of walks of length k of G starting at v. In this paper a lower bounds of the spectral radius of G in terms of the k-degree of vertices is presented and the upper bounds of energy of a connected graph is obtained. 1 Introduction Let G = (V, E) be a finite graph. For v V (G), the degree of v written by d, is the number of edges incident with v. The number of walks of length k of G starting at v is denoted by d k, is also called k-degree of the vertex v and d k is called average k-degree d

2 of the vertex v. Clearly, one has d 0 = 1, d 1 = d, and d k+1 = where N is the set of all neighbors of the vertex v. w N d k (w), In view of the well-known fact that with j denote the all one vector defined on the set V (G), the vector D k = (d k ) coincides with A k j, where A = A(G) is the adjacency matrix of the graph G. In [3], Dress and Gutman showed an interesting inequality on the number of walks in a graph, and the discussion of the equality holding resulted classification graphs in exactly five classes. A graph G is called regular graph if there exists a constant r such that d = r holds for every v V (G), in which case G is also called r-regular. Obviously, this is equivalent with the assertion that Aj = rj holds. Further, a graph G is called a, b-semiregular if {d, d(w)} = {a, b} holds for all edges vw E(G). Clearly, this implies A j = abj. A semiregular graph that is not regular will henceforth be called strictly semiregular. Clearly, a connected strictly semiregular graph must be bipartite. A graph G is called harmonic [3](pseudoregular [13]) if there exists a constants µ such that d = µd holds for every v V (G) in which case G is also called µ-harmonic, clearly, a graph G is µ-harmonic if and only if A j = µaj holds.( In other words, A j and Aj are linear dependent.) A graph G is called semiharmonic [3] if there exists a constants µ such that d 3 = µd holds for every v V (G) in which case G is also called µ-semiharmonic, clearly, a graph G is µ-semiharmonic if and only if A 3 j = µaj holds. (In other words, A 3 j and Aj are linear dependent.) Thus every µ-harmonic graph is µ -semiharmonic. Also every a, b-semiregular graph is ab-semiharmonic. A semiharmonic graph that is not harmonic will henceforth be called strictly semiharmonic. Finally, a graph G is called (a, b)-pseudosemiregular if { d d, d (w) } = {a, b} holds d(w) for all edges vw E(G). A pseudosemiregular graph that is not pseudoregular will henceforth be called strictly pseudosemiregular. Clearly, a connected strictly pseudosemiregular graph must be bipartite. In this paper, we present a lower bound for the spectral radius of a connected graph and give a upper bound for the energy of a graph by using this new lower bound of spectral radius.

3 The spectral radius a graph and k-degree Let G be a connected graph. The spectral radius (largest eigenvalue of A(G)) of G is denoted by ρ(g). In [4], the following theorem was given: Theorem 1 Let G be a connected graph with degree sequence d 1, d,..., d n. Then ρ(g) d, (1) n with equality if and only if G is regular or a semiregular. In [13], the following theorem was given: Theorem Let G be a connected graph. Then ρ(g) d d, () with equality if and only if G is a pseudo-regular graph or a strictly pseudo-semiregular graph. In this section we prove a more general result which generalizes and unifies the above two results. Lemma 3 Let G be a connected graph and k 0. Then ρ(g) d k+1, (3) d k with equality if and only if A k+ (G)j = ρ (G)A k (G)j. Proof. Since Dk T A D k = A k+1 j, A k+1 j = D k+1, D k+1 = d k+1, thus, by Rayleigh-Ritz Theorem, ρ(g) = ρ(a ) DT k A D k D k, D k d k+1, and equality d k holds if and only if D k is a eigenvector of A (G) corresponding to the eigenvalue ρ(g), that is, if and only if A k+ (G)j = ρ (G)A k (G)j. For k = 0, the above lemma becomes Theorem 1, this is because that A (G)j = µj if and only if G is regular or semiregular (see [3]). By Theorem 1 of [3], every graph G for

4 which some integers k, l with 0 l < k and some constant µ such that d k = µd l for all v V (G) (equivalently, A k j = µa l j ) is semiharmonic, and even harmonic in case k l is odd.) Therefore, for k 1, ρ(g) d k+1 d k and the equality holds if and only if G is a semiharmonic graph. In the next lemma we prove that strictly semiharmonic connected graphs and strictly pseudosemiregular connected graphs are the same. Lemma 4 Let G be a connected graph. Then following are equivalent: (1). G is strictly semiharmonic. (). G is strictly pseudosemiregular. Proof. Let G be a strictly semiharmonic graph. Then G must have a vertex v 0 of the smallest average -degree, say a = d (v 0 ) d(v 0 ). Let w 0 be the vertex that is adjacent to v 0 and has the largest average -degree among the neighbors of v 0, say b = d (w 0 ) d(w 0 ). Then d 3 (w 0 ) = u N(v 0) d (u) d (v 0 ) d(v 0 ) d (w 0 ) = abd(w 0 ), with equality holding if and only if all neighbors of w 0 have average -degree a. d 3 (v 0 ) = u N(w 0) d (u) d (w 0 ) d(w 0 ) d (v 0 ) = abd(v 0 ), with equality holding if and only if all neighbors of v 0 have average -degree b. Since G is strictly semiharmonic, a b and d 3 = µd holds for all v V (G), all neighbors of v 0 must have average -degree b and all neighbors of w 0 must have average -degree a. Since G is connected, it follows immediately that G is (bipartite) strictly pseudosemiregular. If G is strictly pseudosemiregular and let V = V 1 V be a bipartition of V (G). Set a = d (u) d(u) (u V 1) and b = d d (v V ) and a b. Hence G is not harmonic. For all u V 1, we have d 3 (u) = v N(u) d = v N(u) d d d = bd (u) = abd(u) and similarly, d 3 = abd for all v V. Thus d 3 = abd for all v V and hence G is a strictly ab-semiharmonic graph.

5 Combining lemmas 3 and 4 we have the main result which generalizes Theorem. Theorem 5 Let G be a connected graph and k 1. Then ρ(g) d k+1 d k, with equality if and only if G is pseudoregular or strictly pseudosemiregular. Theorem 6 Let G be a connected graph and f(k) = k+1, k 0. (4) Then f(k) is an increasing sequence and d k lim f(k) = ρ(g). n Proof. We prove the monotone increasing of f(k) by showing that f(k + 1) f(k). Note that the inner product A k j, A k j = (A k j) T A k j = d k. By Cauchy-Schwarz inequality, we have A k j, A k j A k+ j, A k+ j A k j, A k+ j = A k+1 j, A k+1 j. That is, d k+ d k+1 Hence, f(k + 1) f(k) for k 0. d k+1 d k. Since the sequence f(k) is an monotonically increasing sequence and has a upper bound ρ(g), the limit lim k f(k) must exist. In order to prove the limit it suffices to prove lim k f(k) = ρ(g). Let ρ(g) = λ 1 > λ λ n be all eigenvalues of A(G) and X 1, X,, X n be unit eigenvectors corresponding these eigenvalues of A(G). Then X 1, X,, X n consist of a orthonormal basis of R n. Assume that j = n i=1 θ ix i, then θ i = j T X i, i = 1,,..., n. D In order to prove lim f(k) = ρ(g) it suffices to prove that k k Dk, D k approaches a unit eigenvector corresponding the eigenvalue ρ (G) of A (G) when k. Note that D k Dk, D k = A k j A k j, A k j = A k j and A k j = A k d k n θ i X i = i=1 n i=1 θ i λ k i X i,

6 we have d k = n i=1 θ i λ 4k i. If G is nonbipartite, since θ 1 > 0 and λ 1 > λ i for all i =, 3,..., n, then the vector θ i λ k i X i D k approaches X 1 if i = 1, 0 if i > 1. Thus the eigenvector Dk, D k n i=1 θ i λ4k i approaches X 1 and the result follows. If G is bipartite, since θ 1 > 0 and λ 1 > λ i for all i =, 3,..., n 1, λ n = λ 1, then the θ i λ k i X i θ 1 X 1 θ vector approaches n X if i = 1, 0 if 1 < i < n, n if i = n. θ 1 + θn θ 1 + θn n i=1 θ i λ4k i D k Thus the eigenvector Dk, D k approaches θ 1X 1 + θ n X n, which is a unit eigenvector θ 1 + θn corresponding the eigenvalue ρ (G) of A (G) and the result follows. 3 Upper bounds for the energy of a graph The energy of a graph G, denoted by E(G), is defined as E(G) = n i=1 λ i, where λ 1 λ λ n are the eigenvalues of the adjacency matrix of G. This concept was introduced by I. Gutman and is studied intensively in mathematical chemistry, since it can be used to approximate the total π-electron energy of a molecule(see [5] ). In 1971 McClelland discovered the first upper bound for E(G) as follows: E(G) mn. (5) Since then, numerous other bounds for E(G) were found (see [1,,6 10,1,14 16]). Recently, Yu et. al [14] proved the following result: Theorem 7 Let G be a nonempty graph with n vertices and m edges. Then ) (G) E(G) d (G) d + (G) (n 1) (m d. (6) (G) d Equality holds if and only if one of the following statements holds: (1) G = n K ; () G = K n ; (3) G is a non-bipartite connected µ-pseudoregular graph with three distinct eigenvalues m µ µ,, m µ, where µ > m. n 1 n 1 n

7 Remark In fact, the graph appears in (3) of above Theorem must be non-completed m ( strongly regular graph with two nontrivial eigenvalues both with absolute value m n ), n 1 for details see the proof of the next Theorem. The following is the main results of this section which generalize the above result. Since general case is not substantially different, we concentrate on the particular case of connected graph for the sake of readability. Theorem 8 Let G be a connected graph with n(n ) vertices and m edges. Then (G) E(G) d k+1 (G) d k + (G) (n 1) (m d k+1 ) (G) d k. (7) Equality holds if and only if G is the complete graph K n or G is a strongly regular graph m ( with two nontrivial eigenvalues both with absolute value m n ). n 1 Proof. Let λ 1 λ λ n be the eigenvalues of G. By the Cauchy-Scharz inequality we have E(G) = n n λ i = λ 1 + λ i i=1 i= n (n 1) λ i (n = 1)(m λ 1). i= Now since the function F (x) = x+ (n 1)(m x ) decreases on the interval m/n x m. By Theorem 6 we have (G) λ 1 f(k) f(0) = d n Hence F (λ 1 ) F ( (G) d k+1 /d k ) and m n. (G) E(G) d k+1 (G) d k + (G) (n 1) (m d k+1 ) (G) d k. (8) If the connected graph G is a complete graph K n or a strong regular graph with two m ( non-trivial eigenvalues both with absolute values m n ) then it is easy to check that n 1 the equality holds.

8 Conversely, if the equality holds, according the above proof, we have (G) λ 1 = d k+1, (G) d k m λ which implies that G is pseudo-semiregular and λ i = 1 for i =, 3,..., n. Note n 1 that a graph has only one distinct eigenvalue if and only if it has no edges and a graph has two distinct eigenvalues if and only if it is a complete graph. Since G is connected, we reduced to the following two cases: (1) G has two distinct eigenvalues. In this case G = K n. (G) () G has three distinct eigenvalues. In this case, λ 1 = d k+1 and λ i = (G) d k m λ 1 for i =, 3,..., n. Since G is connected, λ 1 > λ i, λ i 0 for i =,..., n. Thus n 1 G must be regular (else G has 0 as an eigenvalue) and non-bipartite ( else G has least four distinct eigenvalues). Hence G is λ 1 -regular (λ 1 = m ) and has exact three distinct n eigenvalues. Thus G is strong regular graph with two non-trivial eigenvalues both with (m ( mn ) ) absolute value. n 1 Similarly, (similar to [7, 14, 15]) we may prove Theorem 9 Let G be a connected bipartite graph with n(n ) vertices and m edges. Then (G) E(G) d k+1 (G) d k + (G) (n ) (m d k+1 ) (G) d k. (9) Equality holds if and only if G is the complete bipartite graph or G is the incidence graph of a symmetric -(ν, k, λ)-design with k = m k(k 1), n = ν and λ = n ν 1. Acknowledgements: This work was supported by NSF ( ) of China and by Scientific Research Fund of Hunan Provinical Education Department(06A037 ).

9 References [1] R. Balakrishnan, The energy of a graph, Lin. Algebra Appl., 387 (004) [] A. Chen, A. Chang, W.C. Shiu, Energy ordering of unicyclic graphs, MATCH Commun. Math. Comput. Chem., 55 (006) [3] A. Dress and I. Gutman, On the number of walks in a graph, Appl. Math. Lett., 16 (003) [4] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs: Theory and Application, Academic Press, New York, [5] I. Guman, The energy of a graph: old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wassermann(Eds.), Algebraic Combinatorics and Applications, Springer- Verlag, Berlin, 001, pp [6] J. H. Koolem, V. Moulton, Maximal energy graphs, Adv. Appl. Math., 6 (001) [7] J. H. Koolem, V. Moulton, Maximal energy bipartite graphs, Graphs Combin., 19 (003) [8] J. H. Koolem, V. Moulton, I. Gutman, Improving the McCelland inequality for total π-electron energy, Chem. Phys. Lett., 30(000) [9] F. Li, B. Zhou, Minimal energy of bipartite unicyclic graphs of a given bipartion, MATCH Commun. Math. Comput. Chem., 54 (005) [10] W. Lin, X. Guo, H. Li, On the extermal energies of trees with a given maximum degree,match Commun. Math. Comput. Chem., 54 (005) [11] B. J. McClelland, Properties of the latent roots of a matrix: The estimation of π-electron energies, J. Chem. Phys., 54 (1971) [1] W. Yan, L. Ye, On the maximal energy and Hosoya index of a type of trees with many pendant vertices, MATCH Commun. Math. Comput. Chem., 53 (005)

10 [13] A. Yu, M. Lu, F. Tian, On the spectral radius of graphs, Lin. Algbera Appl., 387 (004) [14] A. Yu, M. Lu, F. Tian, New upper bounds for the energy of graphs, MATCH Commun. Math. Comput. Chem., 53 (005) [15] B. Zhou, Energy of a graph, MATCH Commun. Math. Comput. Chem., 51 (004) [16] B. Zhou, Lower bounds for the energy of quadrangle-free graphs, MATCH Commun. Math. Comput. Chem., 55 (006)

### Tricyclic biregular graphs whose energy exceeds the number of vertices

MATHEMATICAL COMMUNICATIONS 213 Math. Commun., Vol. 15, No. 1, pp. 213-222 (2010) Tricyclic biregular graphs whose energy exceeds the number of vertices Snježana Majstorović 1,, Ivan Gutman 2 and Antoaneta

### On Some Vertex Degree Based Graph Invariants

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (20) 723-730 ISSN 0340-6253 On Some Vertex Degree Based Graph Invariants Batmend Horoldagva a and Ivan

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### YILUN SHANG. e λi. i=1

LOWER BOUNDS FOR THE ESTRADA INDEX OF GRAPHS YILUN SHANG Abstract. Let G be a graph with n vertices and λ 1,λ,...,λ n be its eigenvalues. The Estrada index of G is defined as EE(G = n eλ i. In this paper,

### Extremal Wiener Index of Trees with All Degrees Odd

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong

### Linear Algebra and its Applications

Linear Algebra and its Applications 438 2013) 1393 1397 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Note on the

### Connectivity and cuts

Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every

### Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

### An inequality for the group chromatic number of a graph

Discrete Mathematics 307 (2007) 3076 3080 www.elsevier.com/locate/disc Note An inequality for the group chromatic number of a graph Hong-Jian Lai a, Xiangwen Li b,,1, Gexin Yu c a Department of Mathematics,

### On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

### Product irregularity strength of certain graphs

Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

### The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

### BOUNDARY EDGE DOMINATION IN GRAPHS

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

### Conductance, the Normalized Laplacian, and Cheeger s Inequality

Spectral Graph Theory Lecture 6 Conductance, the Normalized Laplacian, and Cheeger s Inequality Daniel A. Spielman September 21, 2015 Disclaimer These notes are not necessarily an accurate representation

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### Continuity of the Perron Root

Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North

### 1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### School of Industrial and System Engineering Georgia Institute of Technology Atlanta, Georgia 30332

NORI~H- ~ Bounds on Eigenvalues and Chromatic Numbers Dasong Cao School of Industrial and System Engineering Georgia Institute of Technology Atlanta, Georgia 30332 Submitted by Richard A. Brualdi ABSTRACT

### A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

### A characterization of trace zero symmetric nonnegative 5x5 matrices

A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the

### Special Classes of Divisor Cordial Graphs

International Mathematical Forum, Vol. 7,, no. 35, 737-749 Special Classes of Divisor Cordial Graphs R. Varatharajan Department of Mathematics, Sri S.R.N.M. College Sattur - 66 3, Tamil Nadu, India varatharajansrnm@gmail.com

### A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### Frans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages

Frans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages Article (Published version) (Refereed) Original citation: de Ruiter, Frans and Biggs, Norman (2015) Applications

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices

MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 47-58 (2010) Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang

### Split Nonthreshold Laplacian Integral Graphs

Split Nonthreshold Laplacian Integral Graphs Stephen Kirkland University of Regina, Canada kirkland@math.uregina.ca Maria Aguieiras Alvarez de Freitas Federal University of Rio de Janeiro, Brazil maguieiras@im.ufrj.br

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### PETER G. CASAZZA AND GITTA KUTYNIOK

A GENERALIZATION OF GRAM SCHMIDT ORTHOGONALIZATION GENERATING ALL PARSEVAL FRAMES PETER G. CASAZZA AND GITTA KUTYNIOK Abstract. Given an arbitrary finite sequence of vectors in a finite dimensional Hilbert

### ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER

Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group

### USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS

USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu ABSTRACT This

### Analysis of Algorithms, I

Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

### GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

### Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

### (January 14, 2009) End k (V ) End k (V/W )

(January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.

### Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

### Perron Frobenius theory and some extensions

and some extensions Department of Mathematics University of Ioannina GREECE Como, Italy, May 2008 History Theorem (1) The dominant eigenvalue of a matrix with positive entries is positive and the corresponding

### High degree graphs contain large-star factors

High degree graphs contain large-star factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### 10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES

58 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 you saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c nn c nn...

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### SCORE SETS IN ORIENTED GRAPHS

Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### The chromatic spectrum of mixed hypergraphs

The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### On three zero-sum Ramsey-type problems

On three zero-sum Ramsey-type problems Noga Alon Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and Yair Caro Department of Mathematics

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### PROBLEM SET 7: PIGEON HOLE PRINCIPLE

PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Essential spectrum of the Cariñena operator

Theoretical Mathematics & Applications, vol.2, no.3, 2012, 39-45 ISSN: 1792-9687 (print), 1792-9709 (online) Scienpress Ltd, 2012 Essential spectrum of the Cariñena operator Y. Mensah 1 and M.N. Hounkonnou

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### 3. Eulerian and Hamiltonian Graphs

3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### Class One: Degree Sequences

Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

### The Open University s repository of research publications and other research outputs

Open Research Online The Open University s repository of research publications and other research outputs The degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

### Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### arxiv: v2 [math.co] 30 Nov 2015

PLANAR GRAPH IS ON FIRE PRZEMYSŁAW GORDINOWICZ arxiv:1311.1158v [math.co] 30 Nov 015 Abstract. Let G be any connected graph on n vertices, n. Let k be any positive integer. Suppose that a fire breaks out

### The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

### Network (Tree) Topology Inference Based on Prüfer Sequence

Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### Odd induced subgraphs in graphs of maximum degree three

Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### The minimum number of distinct areas of triangles determined by a set of n points in the plane

The minimum number of distinct areas of triangles determined by a set of n points in the plane Rom Pinchasi Israel Institute of Technology, Technion 1 August 6, 007 Abstract We prove a conjecture of Erdős,

### Uniform Multicommodity Flow through the Complete Graph with Random Edge-capacities

Combinatorics, Probability and Computing (2005) 00, 000 000. c 2005 Cambridge University Press DOI: 10.1017/S0000000000000000 Printed in the United Kingdom Uniform Multicommodity Flow through the Complete

### ENERGY OF COMPLEMENT OF STARS

J Indones Math Soc Vol 19, No 1 (2013), pp 15 21 ENERGY OF COMPLEMENT OF STARS P R Hampiholi 1, H B Walikar 2, and B S Durgi 3 1 Department of Master of Computer Applications, Gogte Institute of Technology,

### A note on companion matrices

Linear Algebra and its Applications 372 (2003) 325 33 www.elsevier.com/locate/laa A note on companion matrices Miroslav Fiedler Academy of Sciences of the Czech Republic Institute of Computer Science Pod

### DISTANCE MAGIC CARTESIAN PRODUCTS OF GRAPHS

1 2 Discussiones Mathematicae Graph Theory xx (xxxx) 1 9 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 DISTANCE MAGIC CARTESIAN PRODUCTS OF GRAPHS Christian Barrientos Department of Mathematics,

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### ON HAMMER'S X-RAY PROBLEM

ON HAMMER'S X-RAY PROBLEM R. J. GARDNER AND P. McMULLEN ABSTRACT Suppose that two distinct convex bodies in E" have the same Steiner symmetrals in a number of hyperplanes whose normals lie in a plane.

### Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

### Sample Induction Proofs

Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

### CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010

CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected

### The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can