On fat Hoffman graphs with smallest eigenvalue at least 3
|
|
- Nathan Norman
- 5 years ago
- Views:
Transcription
1 On fat Hoffman graphs with smallest eigenvalue at least 3 Qianqian Yang University of Science and Technology of China Ninth Shanghai Conference on Combinatorics Shanghai Jiaotong University May 24-28, 2017 (Joint work with Jack Koolen and Yan-Ran Li.) 1 / 24
2 1 Motivation 2 Main tool Hoffman graphs Representation of Hoffman graphs Special matrix 3 Result Special ( )-graphs Weighted graphs 2 / 24
3 In 1976, Cameron et al. showed that If G is a connected graph with smallest eigenvalue at least 2, then 1 G is a generalized line graph; or 2 G has at most 36 vertices. Denote A(G) the adjacency matrix of G. G is a generalized line graph, that is, there exists a matrix N such that A(G) + 2I = N T N, where N ij Z for all i, j. P.J. Cameron, J.-M. Goethals, J.J. Seidel and E.E. Shult, Line graphs, root systems, and elliptic geometry, J. Algebra 43 (1976), no. 1, / 24
4 Questions If G is a connected graph with smallest eigenvalue at least 3, then G is??? Suppose there exists a matrix N such that then G is??? A(G) + 3I = N T N, where N ij Z, Definition A graph G is called integrally representable of norm 3, if there exists a map ϕ : V (G) Z m for some positive integer m such that 3, if x = y; ϕ(x), ϕ(y) = 1, if x y; 0, otherwise. 4 / 24
5 Question What are the connected integrally representable graphs of norm 3? 5 / 24
6 Example Let G be the graph, then there are two integral representations ϕ 1 and ϕ 2 of G of norm 3, where ϕ 1 ( ) ϕ 1 ( ) ϕ 2 ( ) ϕ 2 ( ) f 1 f 1 f 2 f 3 f 4 f 5 f 1 f 1 f 2 f 2 f 1 f 1 f 2 f 2 f 3 f 3 f 4 f 4 f 5 f 5 g ϕ1 g ϕ2 6 / 24
7 Definition A Hoffman graph h is a pair (H, µ) of a graph H = (V, E) and a labeling map µ : V {f, s} satisfying the following conditions: 1 every vertex with label f is adjacent to at least one vertex with label s; 2 vertices with label f are pairwise non-adjacent. slim vertex f 1 f 2 fat vertex f 1 f 2 f 3 f 4 f 5 7 / 24
8 Definitions and notations f 1 f 2 h V f (h) = {f 1, f 2 }, V s (h) = {, }, N f h () = {f 1, f 2 }, Nh s() = { }, N f h () = {f 1, f 2 }, Nh s() = { }, h is 2-fat, slim graph of h is. V f (h) (V s (h)) the set of fat (slim) vertices of h. N f h (x) (N h s (x)) the set of fat (slim) neighbors of x in h; If every slim vertex has a fat neighbor, we call h fat; If every slim vertex has at least t fat neighbors, we call h t-fat. The slim graph of a Hoffman graph h = (H, µ) is the subgraph of H induced by V s (h). 8 / 24
9 Definition For a Hoffman graph h and a positive integer m, a mapping φ : V (h) Z m such that 3 if x = y and x, y V s (h); 1 if x = y and x, y V φ(x), φ(y) = f (h); 1 if x y; 0 otherwise, is called an integral representation of norm 3 of h. x1 x2 x1 ϕ 2 ( ) ϕ 2 ( ) f1 f2 f3 f4 f5 f1 x2 f2 φ( ) φ( ) φ(f 1 ) φ(f 2 ) G g ϕ2 Figure 1 9 / 24
10 Definition For a Hoffman graph h and a positive integer m, a mapping ψ : V s (h) Z m such that x1 x1 x2 ϕ 2 ( ) ϕ 2 ( ) x 3 N f f1 h (x) f2 if x = y; 1 1 ψ(x), 1 ψ(y) 1= 1 N f h (x, y) if x y; f1 f2 f3 f4 f5 x2 N f h (x, y) otherwise, G is called an integral reduced representation Figure 1 of norm 3 of h. g ϕ2 φ( ) φ( ) φ(f 1 ) φ(f 2 ) x1 x2 ϕ 2 ( ) ϕ 2 ( ) 1 1 f 1 f 2 f 3 1 f 4 1 f f1 f2 f3 f4 f5 f1 f 1 f 2 x1 x2 f2 ψ( ) ψ( ) G g ϕ2 Figure 2 f 1 f 2 10 / 24
11 Result Condition: Let G be a connected graph with an integral representation of norm 3. Conclusion: G is the slim graph of a fat Hoffman graph with an integral (reduced) representation of norm 3. Question: What are all fat Hoffman graphs with an integral (reduced) representation of norm 3? 11 / 24
12 Definition The special matrix of a Hoffman graph h is defined as follows: for any x, y V s (h), Sp(h) (x,y) = N f h (x), if x = y; 1 N f h (x, y), if x y; N f h (x, y), if x y. Eigenvalues of h are the eigenvalues of its special matrix. The smallest eigenvalue of h is denoted by λ min (h). f 3 f 4 f 5 f 1 f 2 ( 2 1 special matrix: 1 2 λ min (g ϕ2 ) = 3 ) g ϕ2 12 / 24
13 Let h be a Hoffman graph with an integral reduced representation ψ of norm 3. For any x, y V s (h), Sp(h) (x,y) = ψ(x), ψ(y) = N f h (x), if x = y; 1 N f h (x, y), if x y; N f h (x, y), if x y. 3 N f h (x) if x = y; 1 N f h (x, y) if x y; N f h (x, y) otherwise. A Hoffman graph h has an integral reduced representation of norm 3. Equivalently, there exists a matrix N, such that Sp(h) + 3I = N T N, where N ij Z for all i, j. It also implies that all Hoffman graphs with an integral reduced representation of norm 3 has smallest eigenvalue at least / 24
14 Definition Let µ be a real number with µ 1. A Hoffman graph h is called µ-saturated, if 1 λ min (h) µ; 2 no fat vertex can be attached to h in such a way that the resulting Hoffman graph has smallest eigenvalue at least µ. ( 3)-saturated ( 3)-saturated 14 / 24
15 Definition A Hoffman graph h is called decomposable, if there exists a partition {Vs 1 (h), Vs 2 (h)} of V s (h), such that ( V s 1 (h) Vs 2 (h) ) Vs 1 (h) 0 Sp(h) = Otherwise, h is indecomposable. V 2 s (h) 0. ( 3 0 Sp(g ϕ1 ) = 0 3 f 1 f 2 ) f 1 f 2 f 3 f 4 f 5 g ϕ1, decomposable 15 / 24
16 What are fat Hoffman graphs, which have an integral (reduced) representation of norm 3? What is the family H of Hoffman graphs? H is the family of fat, ( 3)-saturated, indecomposable integrally (reduced) representable Hoffman graphs of norm / 24
17 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 Definition The special graph of a Hoffman graph h is the edge-signed graph S(h) := (V (S(h)), E + (S(h)), E (S(h))), where V (S(h)) = V s (h) and E + (S(h)) = { {x, y} x, y V s (h), x y, sgn(sp(h) (x,y) ) = + }, x 3 E (S(h)) = { {x, y} x, y V s (h), x y, sgn(sp(h) (x,y) ) = }. The special ε-graph of h is the graph S ɛ (h) = (V s (h), E ɛ (S(h))) for ɛ {+, }. x 3 + x 3 x1 x2 x3 x 1 x Sp(h) = x x 3 x 3 h + x1 x2 x3 Figure 1 S(h) x 3 S (h) 17 / 24
18 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 Jang et al. showed that x 3 xn 2 x n 1 x n x 3 xn 2 x n 1 x n If a Hoffman graph h H has n slim vertices, then its special ( )-graph S (h) isx n one of the following: x n x 3 xn 2 x n 1 x n A n : 1 x 3 xn 2 x n 1 x n, Ã n 1 : D n : x 3 xn 3 x n 2 x n 1 x 3 x 4 xn 2 x n 1 x n x n 2 3 n 1 xn 3 n 2 x n 1 1 x 3 xn 3 x n 2 x n, Dn 1 : x 3 xn 3 x n 2 x n 1 x n 3 x 34 xn 3 xn 2 x n 2 n 1 x n 1 x n x 3 x 4 xn 2 x n 1 x n 1 n x 3 x 4 xn 3 x n 2. x n, x 3 x 4 xn 2 x n 1 x x n 1 x n 1 x 3 x 4 xn 3 x n 2 x n x 3 xn 3 x n 2 x n 1 H.J. Jang, J. Koolen, A. Munemasa and T. Taniguchi, On fat Hoffman x n graphs with smallest eigenvalue at least 3, Ars Math. Contemp. 7 (2014), no. 1, x n 1 x n 1 x 3 xn 3 x n 2 x 3 x 4 xn 3 x n 2 x n x n 18 / 24
19 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 Definition A weighted graph is a pair (G, w) of a graph G and a weight function w : V (G) Z 0. The weighted special ( )-graph of a Hoffman graph h is the weighted graph (S (h), w), where S (h) is the special ( )-graph of h and w(x) := N f h (x). Question What are the necessary and sufficient conditions for a weighted graph to be the weighted special ( )-graph of a Hoffman graph in H? 19 / 24
20 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 A leaf of a graph G is a vertex of valency 1 in G. A t-leaf is a leaf with its unique neighbor having valency t. Necessary conditions Let (G, w) be a weighted special ( )-graph of a Hoffman graph in H, which has n slim vertices where n 6. 1 G is isomorphic to A n, Ã n 1, D n or D n 1 and w(x) {1, 2} for any x V (G). 2 For, V (G), if and w() = w() = 2, then one of them is a leaf of G. 3 If is a 3-leaf of G, then w() = 1. 4 If is a 2-leaf of G with w() = 1, then there exists a unique 4-path x 3x 4 containing in G. Moreover, w() = 2, w(x 3) = w(x 4) = 1. 5 If G is isomorphic to Ãn 1 and w(x) = 2 for some x V (G), then there exist at least 3 vertices having weight 2 in (G, w). 20 / 24
21 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 Given a weighted graph (G, w) which satisfies the above necessary conditions and has at least 6 vertices, we will construct a Hoffman graph h (G,w) H, h (G,w) has (G, w) as its weighted special ( )-graph for each fat vertex f of h (G,w), the induced subgraph of S (h (G,w) ) on N s h (G,w) (f) is connected. We call the Hoffman graphs satisfying the second condition generic. 21 / 24
22 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 Construction For each pair of vertices x and y of V (G), they share exactly one common fat neighbor in h (G,w) if one of the following conditions is satisfied: There exists a path P = x... x py in G such that w(x i) = 1 for i = 1,..., p. Note that the value of p can be zero as well. Both x and y are 3-leaves of G and they have one common neighbor in G. For each pair of vertices x and y of V (G), they are adjacent in h (G,w) if one of the following conditions is satisfied: N f h (G,w) (x, y) = 1 and x y in G; N f h (G,w) (x, y) = 0 and there exists a vertex z with w(z) = 2 in (G, w) such that x and y are adjacent to z in G, except the case where x or y is a 2-leaf of G with weight 1; N f h (G,w) (x, y) = 0, one of them is a 2-leaf with weight 1 in (G, w) and the distance of x and y is 3 in G. 22 / 24
23 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 How to obtain the family H Weighted graph (G, w), which satisfies the necessary conditions Generic Hoffman graph h (G,w), which has (G, w) as weighted special ( )-graph identify fat vertices identify fat vertices Hoffman graph h H Note that the for a Hoffman graph h 1 H, two fat vertices f 1 and f 2 of h 1 can be identified, if and only if N s h 1 (f 1, f 2) = and for any fat vertex f V f (h 1) {f 1, f 2}, N s h 1 (f) (N s h 1 (f 1) N s h 1 (f 2)) / 24
24 Thank you for your attention! 24 / 24
arxiv: v1 [math.co] 21 Dec 2016
The integrall representable trees of norm 3 Jack H. Koolen, Masood Ur Rehman, Qianqian Yang Februar 6, 08 Abstract In this paper, we determine the integrall representable trees of norm 3. arxiv:6.0697v
More informationOn graphs with smallest eigenvalue at least 3 and their lattices
On graphs with smallest eigenvalue at least 3 and their lattices arxiv:1804.00369v1 [math.co] 2 Apr 2018 e-mail: Jack H. Koolen, Jae Young Yang, Qianqian Yang School of Mathematical Sciences, University
More informationFat Hoffman graphs with smallest eigenvalue greater than 3
Fat Hoffman graphs with smallest eigenvalue greater than 3 arxiv:1211.3929v4 [math.co] 24 Dec 2013 Akihiro MUNEMASA Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan E-mail
More informationLine Graphs Eigenvalues and Root Systems
Line Graphs Eigenvalues and Root Systems Thomas Zaslavsky Binghamton University (State University of New York) C R Rao Advanced Institute of Mathematics, Statistics and Computer Science 2 August 2010 Outline
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 informationApplying block intersection polynomials to study graphs and designs
Applying block intersection polynomials to study graphs and designs Leonard Soicher Queen Mary University of London CoCoA15, Colorado State University, Fort Collins, July 2015 Leonard Soicher (QMUL) Block
More informationButson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4
Butson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4 Akihiro Munemasa Tohoku University (joint work with Takuya Ikuta) November 17, 2017 Combinatorics Seminar
More informationButson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4
Butson-Hadamard matrices in association schemes of class 6 on Galois rings of characteristic 4 Akihiro Munemasa Tohoku University (joint work with Takuya Ikuta) November 17, 2017 Combinatorics Seminar
More informationGraphs With Many Valencies and Few Eigenvalues
Graphs With Many Valencies and Few Eigenvalues By Edwin van Dam, Jack H. Koolen, Zheng-Jiang Xia ELA Vol 28. 2015 Vicente Valle Martinez Math 595 Spectral Graph Theory Apr 14, 2017 V. Valle Martinez (Math
More informationMATH 829: Introduction to Data Mining and Analysis Graphical Models I
MATH 829: Introduction to Data Mining and Analysis Graphical Models I Dominique Guillot Departments of Mathematical Sciences University of Delaware May 2, 2016 1/12 Independence and conditional independence:
More informationOriented covers of the triangular graphs
Oriented covers of the triangular graphs Akihiro Munemasa Tohoku University (joint work with Keiji Ito) March 30, 2018 Beijing Normal University A. Munemasa Triangular graphs March 30, 2018 1 / 23 Contents
More informationOn non-bipartite distance-regular graphs with valency k and smallest eigenvalue not larger than k/2
On non-bipartite distance-regular graphs with valency k and smallest eigenvalue not larger than k/2 J. Koolen School of Mathematical Sciences USTC (Based on joint work with Zhi Qiao) CoCoA, July, 2015
More informationThe Terwilliger algebra of a Q-polynomial distance-regular graph with respect to a set of vertices
The Terwilliger algebra of a Q-polynomial distance-regular graph with respect to a set of vertices Hajime Tanaka (joint work with Rie Tanaka and Yuta Watanabe) Research Center for Pure and Applied Mathematics
More informationStrongly regular graphs and Borsuk s conjecture
Seminar talk Department of Mathematics, Shanghai Jiao Tong University Strongly regular graphs and Borsuk s conjecture Andriy Bondarenko Norwegian University of Science and Technology and National Taras
More informationRelation between Graphs
Max Planck Intitute for Math. in the Sciences, Leipzig, Germany Joint work with Jan Hubička, Jürgen Jost, Peter F. Stadler and Ling Yang SCAC2012, SJTU, Shanghai Outline Motivation and Background 1 Motivation
More informationEigenvalues and edge-connectivity of regular graphs
Eigenvalues and edge-connectivity of regular graphs Sebastian M. Cioabă University of Delaware Department of Mathematical Sciences Newark DE 19716, USA cioaba@math.udel.edu August 3, 009 Abstract In this
More informationDiscrete Mathematics
Discrete Mathematics 3 (0) 333 343 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: wwwelseviercom/locate/disc The Randić index and the diameter of graphs Yiting Yang a,
More informationExtension of Strongly Regular Graphs
Extension of Strongly Regular Graphs Ralucca Gera Department of Applied Mathematics Naval Postgraduate School, Monterey, CA 93943 email: rgera@nps.edu, phone (831) 656-2206, fax (831) 656-2355 and Jian
More informationApplicable Analysis and Discrete Mathematics available online at GRAPHS WITH TWO MAIN AND TWO PLAIN EIGENVALUES
Applicable Analysis and Discrete Mathematics available online at http://pefmath.etf.rs Appl. Anal. Discrete Math. 11 (2017), 244 257. https://doi.org/10.2298/aadm1702244h GRAPHS WITH TWO MAIN AND TWO PLAIN
More informationMAP Examples. Sargur Srihari
MAP Examples Sargur srihari@cedar.buffalo.edu 1 Potts Model CRF for OCR Topics Image segmentation based on energy minimization 2 Examples of MAP Many interesting examples of MAP inference are instances
More informationSpectral Characterization of Generalized Cocktail-Party Graphs
Journal of Mathematical Research with Applications Nov., 01, Vol. 3, No. 6, pp. 666 67 DOI:10.3770/j.issn:095-651.01.06.005 Http://jmre.dlut.edu.cn Spectral Characterization of Generalized Cocktail-Party
More informationSKEW-SPECTRA AND SKEW ENERGY OF VARIOUS PRODUCTS OF GRAPHS. Communicated by Ivan Gutman
Transactions on Combinatorics ISSN (print: 2251-8657, ISSN (on-line: 2251-8665 Vol. 4 No. 2 (2015, pp. 13-21. c 2015 University of Isfahan www.combinatorics.ir www.ui.ac.ir SKEW-SPECTRA AND SKEW ENERGY
More informationUniform Star-factors of Graphs with Girth Three
Uniform Star-factors of Graphs with Girth Three Yunjian Wu 1 and Qinglin Yu 1,2 1 Center for Combinatorics, LPMC Nankai University, Tianjin, 300071, China 2 Department of Mathematics and Statistics Thompson
More informationRoot systems and optimal block designs
Root systems and optimal block designs Peter J. Cameron School of Mathematical Sciences Queen Mary, University of London Mile End Road London E1 4NS, UK p.j.cameron@qmul.ac.uk Abstract Motivated by a question
More informationComplex Hadamard matrices and 3-class association schemes
Complex Hadamard matrices and 3-class association schemes Akihiro Munemasa 1 (joint work with Takuya Ikuta) 1 Graduate School of Information Sciences Tohoku University June 27, 2014 Algebraic Combinatorics:
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 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 informationTilburg University. Strongly Regular Graphs with Maximal Energy Haemers, W. H. Publication date: Link to publication
Tilburg University Strongly Regular Graphs with Maximal Energy Haemers, W. H. Publication date: 2007 Link to publication Citation for published version (APA): Haemers, W. H. (2007). Strongly Regular Graphs
More informationRoot systems. S. Viswanath
Root systems S. Viswanath 1. (05/07/011) 1.1. Root systems. Let V be a finite dimensional R-vector space. A reflection is a linear map s α,h on V satisfying s α,h (x) = x for all x H and s α,h (α) = α,
More informationEquiangular lines in Euclidean spaces
Equiangular lines in Euclidean spaces Gary Greaves 東北大学 Tohoku University 14th August 215 joint work with J. Koolen, A. Munemasa, and F. Szöllősi. Gary Greaves Equiangular lines in Euclidean spaces 1/23
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 informationThe Signless Laplacian Spectral Radius of Graphs with Given Degree Sequences. Dedicated to professor Tian Feng on the occasion of his 70 birthday
The Signless Laplacian Spectral Radius of Graphs with Given Degree Sequences Xiao-Dong ZHANG Ü À Shanghai Jiao Tong University xiaodong@sjtu.edu.cn Dedicated to professor Tian Feng on the occasion of his
More informationStrongly regular graphs and Borsuk s conjecture
Optimal Point Configurations and Orthogonal Polynomials 2017 Strongly regular graphs and Borsuk s conjecture Andriy Bondarenko Norwegian University of Science and Technology 19 April 2017 Andriy Bondarenko
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 informationProof of a Conjecture on Monomial Graphs
Proof of a Conjecture on Monomial Graphs Xiang-dong Hou Department of Mathematics and Statistics University of South Florida Joint work with Stephen D. Lappano and Felix Lazebnik New Directions in Combinatorics
More informationLOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP W a (Ãn 1) Jian-yi Shi
LOWER BOUNDARY HYPERPLANES OF THE CANONICAL LEFT CELLS IN THE AFFINE WEYL GROUP W a (Ãn 1) Jian-yi Shi Department of Mathematics, East China Normal University, Shanghai, 200062, China and Center for Combinatorics,
More informationThe query register and working memory together form the accessible memory, denoted H A. Thus the state of the algorithm is described by a vector
1 Query model In the quantum query model we wish to compute some function f and we access the input through queries. The complexity of f is the number of queries needed to compute f on a worst-case input
More informationMa/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 informationCS281A/Stat241A Lecture 19
CS281A/Stat241A Lecture 19 p. 1/4 CS281A/Stat241A Lecture 19 Junction Tree Algorithm Peter Bartlett CS281A/Stat241A Lecture 19 p. 2/4 Announcements My office hours: Tuesday Nov 3 (today), 1-2pm, in 723
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 informationIntroduction to Association Schemes
Introduction to Association Schemes Akihiro Munemasa Tohoku University June 5 6, 24 Algebraic Combinatorics Summer School, Sendai Assumed results (i) Vandermonde determinant: a a m =. a m a m m i
More informationRecall: Matchings. Examples. K n,m, K n, Petersen graph, Q k ; graphs without perfect matching
Recall: Matchings A matching is a set of (non-loop) edges with no shared endpoints. The vertices incident to an edge of a matching M are saturated by M, the others are unsaturated. A perfect matching of
More informationarxiv: v2 [math.co] 24 Mar 2011
The non-bipartite integral graphs with spectral radius three TAEYOUNG CHUNG E-mail address: tae7837@postech.ac.kr JACK KOOLEN arxiv:1011.6133v2 [math.co] 24 Mar 2011 Department of Mathematics, POSTECH,
More informationQUIVERS AND LATTICES.
QUIVERS AND LATTICES. KEVIN MCGERTY We will discuss two classification results in quite different areas which turn out to have the same answer. This note is an slightly expanded version of the talk given
More informationOn the Dynamic Chromatic Number of Graphs
On the Dynamic Chromatic Number of Graphs Maryam Ghanbari Joint Work with S. Akbari and S. Jahanbekam Sharif University of Technology m_phonix@math.sharif.ir 1. Introduction Let G be a graph. A vertex
More informationIntriguing sets of vertices of regular graphs
Intriguing sets of vertices of regular graphs Bart De Bruyn and Hiroshi Suzuki February 18, 2010 Abstract Intriguing and tight sets of vertices of point-line geometries have recently been studied in the
More informationEvery SOMA(n 2, n) is Trojan
Every SOMA(n 2, n) is Trojan John Arhin 1 Marlboro College, PO Box A, 2582 South Road, Marlboro, Vermont, 05344, USA. Abstract A SOMA(k, n) is an n n array A each of whose entries is a k-subset of a knset
More informationOn star forest ascending subgraph decomposition
On star forest ascending subgraph decomposition Josep M. Aroca and Anna Lladó Department of Mathematics, Univ. Politècnica de Catalunya Barcelona, Spain josep.m.aroca@upc.edu,aina.llado@upc.edu Submitted:
More informationA lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo
A lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo A. E. Brouwer & W. H. Haemers 2008-02-28 Abstract We show that if µ j is the j-th largest Laplacian eigenvalue, and d
More informationCombinatorics 3: Finite geometry and strongly regular graphs
Combinatorics 3: Finite geometry and strongly regular graphs 1 Preface This is the third of a three-part set of lecture notes on Advanced Combinatorics, for the module MT5821 of that title at the University
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 informationarxiv: v2 [math.co] 4 Sep 2009
Bounds for the Hückel energy of a graph Ebrahim Ghorbani a,b,, Jack H. Koolen c,d,, Jae Young Yang c a Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran,
More informationOn a lower bound on the Laplacian eigenvalues of a graph
On a lower bound on the Laplacian eigenvalues of a graph Akihiro Munemasa (joint work with Gary Greaves and Anni Peng) Graduate School of Information Sciences Tohoku University May 22, 2016 JCCA 2016,
More informationHomomorphism Testing
Homomorphism Shpilka & Wigderson, 2006 Daniel Shahaf Tel-Aviv University January 2009 Table of Contents 1 2 3 1 / 26 Algebra Definitions 1 Algebra Definitions 2 / 26 Groups Algebra Definitions Definition
More informationMarkov chains, graph spectra, and some static/dynamic scaling limits
Markov chains, graph spectra, and some static/dynamic scaling limits Akihito Hora Hokkaido University I will talk about how I began to get interested in spectra of graphs and then was led to beautiful
More information3-Class Association Schemes and Hadamard Matrices of a Certain Block Form
Europ J Combinatorics (1998) 19, 943 951 Article No ej980251 3-Class Association Schemes and Hadamard Matrices of a Certain Block Form R W GOLDBACH AND H L CLAASEN We describe 3-class association schemes
More informationNotes on Freiman s Theorem
Notes on Freiman s Theorem Jacques Verstraëte Department of Mathematics University of California, San Diego La Jolla, CA, 92093. E-mail: jacques@ucsd.edu. 1 Introduction Freiman s Theorem describes the
More informationNew feasibility conditions for directed strongly regular graphs
New feasibility conditions for directed strongly regular graphs Sylvia A. Hobart Jason Williford Department of Mathematics University of Wyoming Laramie, Wyoming, U.S.A sahobart@uwyo.edu, jwillif1@uwyo.edu
More informationDominating a family of graphs with small connected subgraphs
Dominating a family of graphs with small connected subgraphs Yair Caro Raphael Yuster Abstract Let F = {G 1,..., G t } be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive
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 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 informationZero sum partition of Abelian groups into sets of the same order and its applications
Zero sum partition of Abelian groups into sets of the same order and its applications Sylwia Cichacz Faculty of Applied Mathematics AGH University of Science and Technology Al. Mickiewicza 30, 30-059 Kraków,
More informationA note on the Isomorphism Problem for Monomial Digraphs
A note on the Isomorphism Problem for Monomial Digraphs Aleksandr Kodess Department of Mathematics University of Rhode Island kodess@uri.edu Felix Lazebnik Department of Mathematical Sciences University
More informationA manifestation of the Grothendieck-Teichmueller group in geometry
A manifestation of the Grothendieck-Teichmueller group in geometry Vasily Dolgushev Temple University Vasily Dolgushev (Temple University) GRT in geometry 1 / 14 Kontsevich s conjecture on the action of
More informationON THE NUMBERS OF CUT-VERTICES AND END-BLOCKS IN 4-REGULAR GRAPHS
Discussiones Mathematicae Graph Theory 34 (2014) 127 136 doi:10.7151/dmgt.1724 ON THE NUMBERS OF CUT-VERTICES AND END-BLOCKS IN 4-REGULAR GRAPHS Dingguo Wang 2,3 and Erfang Shan 1,2 1 School of Management,
More informationarxiv: v4 [math.gr] 17 Jun 2015
On finite groups all of whose cubic Cayley graphs are integral arxiv:1409.4939v4 [math.gr] 17 Jun 2015 Xuanlong Ma and Kaishun Wang Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, 100875,
More informationExtremal Graphs Having No Stable Cutsets
Extremal Graphs Having No Stable Cutsets Van Bang Le Institut für Informatik Universität Rostock Rostock, Germany le@informatik.uni-rostock.de Florian Pfender Department of Mathematics and Statistics University
More informationThe Hierarchical Product of Graphs
The Hierarchical Product of Graphs Lali Barrière Francesc Comellas Cristina Dalfó Miquel Àngel Fiol Universitat Politècnica de Catalunya - DMA4 March 22, 2007 Outline 1 Introduction 2 The hierarchical
More informationVertex subsets with minimal width and dual width
Vertex subsets with minimal width and dual width in Q-polynomial distance-regular graphs University of Wisconsin & Tohoku University February 2, 2011 Every face (or facet) of a hypercube is a hypercube...
More informationA new upper bound on the clique number of a strongly regular graph
A new upper bound on the clique number of a strongly regular graph Leonard Soicher Queen Mary University of London LSBU Maths Study Group 2017 New Year Lecture Leonard Soicher (QMUL) Clique number of an
More informationFunctional Analysis HW #1
Functional Analysis HW #1 Sangchul Lee October 9, 2015 1 Solutions Solution of #1.1. Suppose that X
More informationTHE DISTINGUISHING NUMBER OF THE ITERATED LINE GRAPH. 1. Introduction
THE DISTINGUISHING NUMBER OF THE ITERATED LINE GRAPH IAN SHIPMAN Abstract. We show that for all simple graphs G other than the cycles C 3, C 4, C 5, and the claw K 1,3 there exists a K > such that whenever
More informationOn zero sum-partition of Abelian groups into three sets and group distance magic labeling
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 417 425 On zero sum-partition of Abelian groups into three
More informationA lower bound for the spectral radius of graphs with fixed diameter
A lower bound for the spectral radius of graphs with fixed diameter Sebastian M. Cioabă Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA e-mail: cioaba@math.udel.edu Edwin
More informationDistinguishing infinite graphs
Distinguishing infinite graphs Wilfried Imrich Montanuniversität Leoben, A-8700 Leoben, Austria wilfried.imrich@uni-leoben.at Sandi Klavžar Department of Mathematics and Computer Science FNM, University
More informationLearning from Sensor Data: Set II. Behnaam Aazhang J.S. Abercombie Professor Electrical and Computer Engineering Rice University
Learning from Sensor Data: Set II Behnaam Aazhang J.S. Abercombie Professor Electrical and Computer Engineering Rice University 1 6. Data Representation The approach for learning from data Probabilistic
More informationNew Constructions of Antimagic Graph Labeling
New Constructions of Antimagic Graph Labeling Tao-Ming Wang and Cheng-Chih Hsiao Department of Mathematics Tunghai University, Taichung, Taiwan wang@thu.edu.tw Abstract An anti-magic labeling of a finite
More informationLocating-Total Dominating Sets in Twin-Free Graphs: a Conjecture
Locating-Total Dominating Sets in Twin-Free Graphs: a Conjecture Florent Foucaud Michael A. Henning Department of Pure and Applied Mathematics University of Johannesburg Auckland Park, 2006, South Africa
More informationDistinguishing infinite graphs
Distinguishing infinite graphs Wilfried Imrich Montanuniversität Leoben, A-8700 Leoben, Austria wilfried.imrich@mu-leoben.at Sandi Klavžar Department of Mathematics and Computer Science FNM, University
More informationExplicit Construction of Small Folkman Graphs
Michael Spectral Graph Theory Final Presentation April 17, 2017 Notation Rado s Arrow Consider two graphs G and H. Then G (H) p is the statement that if the edges of G are p-colored, then there exists
More informationPairs of a tree and a nontree graph with the same status sequence
arxiv:1901.09547v1 [math.co] 28 Jan 2019 Pairs of a tree and a nontree graph with the same status sequence Pu Qiao, Xingzhi Zhan Department of Mathematics, East China Normal University, Shanghai 200241,
More informationMath 5707: Graph Theory, Spring 2017 Midterm 3
University of Minnesota Math 5707: Graph Theory, Spring 2017 Midterm 3 Nicholas Rancourt (edited by Darij Grinberg) December 25, 2017 1 Exercise 1 1.1 Problem Let G be a connected multigraph. Let x, y,
More informationCombinatorial semigroups and induced/deduced operators
Combinatorial semigroups and induced/deduced operators G. Stacey Staples Department of Mathematics and Statistics Southern Illinois University Edwardsville Modified Hypercubes Particular groups & semigroups
More informationGraceful Tree Conjecture for Infinite Trees
Graceful Tree Conjecture for Infinite Trees Tsz Lung Chan Department of Mathematics The University of Hong Kong, Pokfulam, Hong Kong h0592107@graduate.hku.hk Wai Shun Cheung Department of Mathematics The
More informationOn spectral radius and energy of complete multipartite graphs
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn., ISSN 1855-3974 (electronic edn. ARS MATHEMATICA CONTEMPORANEA 9 (2015 109 113 On spectral radius and energy of complete multipartite
More information2-bondage in graphs. Marcin Krzywkowski*
International Journal of Computer Mathematics Vol. 00, No. 00, January 2012, 1 8 2-bondage in graphs Marcin Krzywkowski* e-mail: marcin.krzywkowski@gmail.com Department of Algorithms and System Modelling
More informationThe tiling by the minimal separators of a junction tree and applications to graphical models Durham, 2008
The tiling by the minimal separators of a junction tree and applications to graphical models Durham, 2008 G. Letac, Université Paul Sabatier, Toulouse. Joint work with H. Massam 1 Motivation : the Wishart
More informationCommutative Banach algebras 79
8. Commutative Banach algebras In this chapter, we analyze commutative Banach algebras in greater detail. So we always assume that xy = yx for all x, y A here. Definition 8.1. Let A be a (commutative)
More informationOn S-packing edge-colorings of cubic graphs
On S-packing edge-colorings of cubic graphs arxiv:1711.10906v1 [cs.dm] 29 Nov 2017 Nicolas Gastineau 1,2 and Olivier Togni 1 1 LE2I FRE2005, CNRS, Arts et Métiers, Université Bourgogne Franche-Comté, F-21000
More informationThe maximum forcing number of a polyomino
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 69(3) (2017), Pages 306 314 The maximum forcing number of a polyomino Yuqing Lin Mujiangshan Wang School of Electrical Engineering and Computer Science The
More informationThe anti-ramsey number of perfect matching.
The anti-ramsey number of perfect matching. Ruth Haas and Michael Young June 1, 011 Abstract An r-edge coloring of a graph G is a mapping h : E(G [r], where h(e is the color assigned to edge e E(G. An
More informationProperties of θ-super positive graphs
Properties of θ-super positive graphs Cheng Yeaw Ku Department of Mathematics, National University of Singapore, Singapore 117543 matkcy@nus.edu.sg Kok Bin Wong Institute of Mathematical Sciences, University
More informationMATH 829: Introduction to Data Mining and Analysis Clustering II
his lecture is based on U. von Luxburg, A Tutorial on Spectral Clustering, Statistics and Computing, 17 (4), 2007. MATH 829: Introduction to Data Mining and Analysis Clustering II Dominique Guillot Departments
More informationCharacterization of quasi-symmetric designs with eigenvalues of their block graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 68(1) (2017), Pages 62 70 Characterization of quasi-symmetric designs with eigenvalues of their block graphs Shubhada M. Nyayate Department of Mathematics,
More informationThe Mixed Chinese Postman Problem Parameterized by Pathwidth and Treedepth
The Mixed Chinese Postman Problem Parameterized by Pathwidth and Treedepth Gregory Gutin, Mark Jones, and Magnus Wahlström Royal Holloway, University of London Egham, Surrey TW20 0EX, UK Abstract In the
More informationThe maximum size of a partial spread in H(4n + 1,q 2 ) is q 2n+1 + 1
The maximum size of a partial spread in H(4n +, 2 ) is 2n+ + Frédéric Vanhove Dept. of Pure Mathematics and Computer Algebra, Ghent University Krijgslaan 28 S22 B-9000 Ghent, Belgium fvanhove@cage.ugent.be
More informationComplex Hadamard matrices and 3-class association schemes
Complex Hadamard matrices and 3-class association schemes Akihiro Munemasa 1 1 Graduate School of Information Sciences Tohoku University (joint work with Takuya Ikuta) June 26, 2013 The 30th Algebraic
More informationDECOMPOSITIONS OF MULTIGRAPHS INTO PARTS WITH THE SAME SIZE
Discussiones Mathematicae Graph Theory 30 (2010 ) 335 347 DECOMPOSITIONS OF MULTIGRAPHS INTO PARTS WITH THE SAME SIZE Jaroslav Ivančo Institute of Mathematics P.J. Šafári University, Jesenná 5 SK-041 54
More informationRandom Walks and Electric Resistance on Distance-Regular Graphs
Random Walks and Electric Resistance on Distance-Regular Graphs Greg Markowsky March 16, 2011 Graphs A graph is a set of vertices V (can be taken to be {1, 2,..., n}) and edges E, where each edge is an
More informationAn algebraic proof of the Erdős-Ko-Rado theorem for intersecting families of perfect matchings
Also available at http://amc-journal.eu ISSN 855-3966 (printed edn.), ISSN 855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 2 (207) 205 27 An algebraic proof of the Erdős-Ko-Rado theorem for intersecting
More information