Graph Theory Package for Giac/Xcas User Manual
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Drawing Graphs and Maps with Curves
Report from Dagstuhl Seminar 13151 Drawing Graphs and Maps with Curves Edited by Stephen Kobourov1, Martin Nöllenburg2, and Monique Teillaud3 1 University of Arizona – Tucson, US, [email protected] 2 KIT – Karlsruhe Institute of Technology, DE, [email protected] 3 INRIA Sophia Antipolis – Méditerranée, FR, [email protected] Abstract This report documents the program and the outcomes of Dagstuhl Seminar 13151 “Drawing Graphs and Maps with Curves”. The seminar brought together 34 researchers from different areas such as graph drawing, information visualization, computational geometry, and cartography. During the seminar we started with seven overview talks on the use of curves in the different communities represented in the seminar. Abstracts of these talks are collected in this report. Six working groups formed around open research problems related to the seminar topic and we report about their findings. Finally, the seminar was accompanied by the art exhibition Bending Reality: Where Arc and Science Meet with 40 exhibits contributed by the seminar participants. Seminar 07.–12. April, 2013 – www.dagstuhl.de/13151 1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling, G.2.2 Graph Theory, F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases graph drawing, information visualization, computational cartography, computational geometry Digital Object Identifier 10.4230/DagRep.3.4.34 Edited in cooperation with Benjamin Niedermann 1 Executive Summary Stephen Kobourov Martin Nöllenburg Monique Teillaud License Creative Commons BY 3.0 Unported license © Stephen Kobourov, Martin Nöllenburg, and Monique Teillaud Graphs and networks, maps and schematic map representations are frequently used in many fields of science, humanities and the arts. -
Maximizing the Order of a Regular Graph of Given Valency and Second Eigenvalue∗
SIAM J. DISCRETE MATH. c 2016 Society for Industrial and Applied Mathematics Vol. 30, No. 3, pp. 1509–1525 MAXIMIZING THE ORDER OF A REGULAR GRAPH OF GIVEN VALENCY AND SECOND EIGENVALUE∗ SEBASTIAN M. CIOABA˘ †,JACKH.KOOLEN‡, HIROSHI NOZAKI§, AND JASON R. VERMETTE¶ Abstract. From Alon√ and Boppana, and Serre, we know that for any given integer k ≥ 3 and real number λ<2 k − 1, there are only finitely many k-regular graphs whose second largest eigenvalue is at most λ. In this paper, we investigate the largest number of vertices of such graphs. Key words. second eigenvalue, regular graph, expander AMS subject classifications. 05C50, 05E99, 68R10, 90C05, 90C35 DOI. 10.1137/15M1030935 1. Introduction. For a k-regular graph G on n vertices, we denote by λ1(G)= k>λ2(G) ≥ ··· ≥ λn(G)=λmin(G) the eigenvalues of the adjacency matrix of G. For a general reference on the eigenvalues of graphs, see [8, 17]. The second eigenvalue of a regular graph is a parameter of interest in the study of graph connectivity and expanders (see [1, 8, 23], for example). In this paper, we investigate the maximum order v(k, λ) of a connected k-regular graph whose second largest eigenvalue is at most some given parameter λ. As a consequence of work of Alon and Boppana and of Serre√ [1, 11, 15, 23, 24, 27, 30, 34, 35, 40], we know that v(k, λ) is finite for λ<2 k − 1. The recent result of Marcus, Spielman, and Srivastava [28] showing the existence of infinite families of√ Ramanujan graphs of any degree at least 3 implies that v(k, λ) is infinite for λ ≥ 2 k − 1. -
The Intriguing Human Preference for a Ternary Patterned Reality
75 Kragujevac J. Sci. 27 (2005) 75-114. THE INTRIGUING HUMAN PREFERENCE FOR A TERNARY PATTERNED REALITY Lionello Pogliani,* Douglas J. Klein,‡ and Alexandru T. Balaban¥ *Dipartimento di Chimica Università della Calabria, 87030 Rende (CS), Italy; ‡Texas A&M University at Galveston, Galveston, Texas, 77553-1675, USA; ¥ Texas A&M University at Galveston, Galveston, Texas, 77551, USA E-mails: [email protected]; [email protected]; balabana @tamug.tamu.edu (Received February 10, 2005) Three paths through the high spring grass / One is the quicker / and I take it (Haikku by Yosa Buson, Spring wind on the river bank Kema) How did number theory degrade to numerology? How did numerology influence different aspects of human creation? How could some numbers assume such an extraordinary meaning? Why some numbers appear so often throughout the fabric of reality? Has reality really a preference for a ternary pattern? Present excursion through the ‘life’ and ‘deeds’ of this pattern and of its parent number in different fields of human culture try to offer an answer to some of these questions giving at the same time a glance of the strange preference of the human mind for a patterned reality based on number three, which, like the 'unary' and 'dualistic' patterned reality, but in a more emphatic way, ended up assuming an archetypical character in the intellectual sphere of humanity. Introduction Three is not only the sole number that is the sum of all the natural numbers less than itself, but it lies between arguably the two most important irrational and transcendental numbers of all mathematics, i.e., e < 3 < π. -
The Veldkamp Space of GQ(2,4) Metod Saniga, Richard Green, Peter Levay, Petr Pracna, Peter Vrana
The Veldkamp Space of GQ(2,4) Metod Saniga, Richard Green, Peter Levay, Petr Pracna, Peter Vrana To cite this version: Metod Saniga, Richard Green, Peter Levay, Petr Pracna, Peter Vrana. The Veldkamp Space of GQ(2,4). International Journal of Geometric Methods in Modern Physics, World Scientific Publishing, 2010, pp.1133-1145. 10.1142/S0219887810004762. hal-00365656v2 HAL Id: hal-00365656 https://hal.archives-ouvertes.fr/hal-00365656v2 Submitted on 6 Jul 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Veldkamp Space of GQ(2,4) M. Saniga,1 R. M. Green,2 P. L´evay,3 P. Pracna4 and P. Vrana3 1Astronomical Institute, Slovak Academy of Sciences SK-05960 Tatransk´aLomnica, Slovak Republic ([email protected]) 2Department of Mathematics, University of Colorado Campus Box 395, Boulder CO 80309-0395, U. S. A. ([email protected]) 3Department of Theoretical Physics, Institute of Physics Budapest University of Technology and Economics, H-1521 Budapest, Hungary ([email protected] and [email protected]) and 4J. Heyrovsk´yInstitute of Physical Chemistry, v.v.i., Academy of Sciences of the Czech Republic, Dolejˇskova 3, CZ-182 23 Prague 8, Czech Republic ([email protected]) (6 July 2009) Abstract It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2,4) is isomor- phic to PG(5,2). -
Some Topics Concerning Graphs, Signed Graphs and Matroids
SOME TOPICS CONCERNING GRAPHS, SIGNED GRAPHS AND MATROIDS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Vaidyanathan Sivaraman, M.S. Graduate Program in Mathematics The Ohio State University 2012 Dissertation Committee: Prof. Neil Robertson, Advisor Prof. Akos´ Seress Prof. Matthew Kahle ABSTRACT We discuss well-quasi-ordering in graphs and signed graphs, giving two short proofs of the bounded case of S. B. Rao's conjecture. We give a characterization of graphs whose bicircular matroids are signed-graphic, thus generalizing a theorem of Matthews from the 1970s. We prove a recent conjecture of Zaslavsky on the equality of frus- tration number and frustration index in a certain class of signed graphs. We prove that there are exactly seven signed Heawood graphs, up to switching isomorphism. We present a computational approach to an interesting conjecture of D. J. A. Welsh on the number of bases of matroids. We then move on to study the frame matroids of signed graphs, giving explicit signed-graphic representations of certain families of matroids. We also discuss the cycle, bicircular and even-cycle matroid of a graph and characterize matroids arising as two different such structures. We study graphs in which any two vertices have the same number of common neighbors, giving a quick proof of Shrikhande's theorem. We provide a solution to a problem of E. W. Dijkstra. Also, we discuss the flexibility of graphs on the projective plane. We conclude by men- tioning partial progress towards characterizing signed graphs whose frame matroids are transversal, and some miscellaneous results. -
Vertex-Transitive Graphs That Have No Hamilton Decomposition
Vertex-transitive graphs that have no Hamilton decomposition Darryn Bryant ∗ Matthew Dean † Abstract It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency. 1 Introduction A famous question of Lov´asz concerns the existence of Hamilton paths in vertex-transitive graphs [28], and no example of a connected vertex-transitive graph with no Hamilton path is known. The related question concerning the existence of Hamilton cycles in vertex-transitive graphs is another interesting and well-studied problem in graph theory, see the survey [23]. A Hamiltonian graph is a graph containing a Hamilton cycle. Thomassen (see [10, 23]) has conjectured that there are only finitely many non-Hamiltonian connected vertex-transitive graphs. On the other hand, Babai [8, 9] has conjectured that there are infinitely many such graphs. To date only five are known. These arXiv:1408.5211v3 [math.CO] 12 Nov 2014 are the complete graph of order 2, the Petersen graph, the Coxeter graph, and the two graphs obtained from the Petersen and Coxeter graphs by replacing each vertex with a triangle. For a regular graph of valency at least 4, a stronger property than the existence of a Hamilton cycle is the existence of a Hamilton decomposition. If X is a k-valent graph, then a Hamilton k decomposition of X is a set of ⌊ 2 ⌋ pairwise edge-disjoint Hamilton cycles in X. Given the small number of non-Hamiltonian connected vertex-transitive graphs, and the uncertainty concerning the existence of others, it is natural to ask how many connected vertex-transitive graphs have no Hamilton decomposition. -
Lattices from Group Frames and Vertex Transitive Graphs
Frames Lattices Graphs Examples Lattices from group frames and vertex transitive graphs Lenny Fukshansky Claremont McKenna College (joint work with Deanna Needell, Josiah Park and Jessie Xin) Tight frame F is rational if there exists a real number α so that αpf i ; f j q P Q @ 1 ¤ i; j ¤ n: Frames Lattices Graphs Examples Tight frames k A spanning set tf 1;:::; f nu Ă R , n ¥ k, is called a tight frame k if there exists a real constant γ such that for every x P R , n 2 2 }x} “ γ px; f j q ; j“1 ¸ where p ; q stands for the usual dot-product. Frames Lattices Graphs Examples Tight frames k A spanning set tf 1;:::; f nu Ă R , n ¥ k, is called a tight frame k if there exists a real constant γ such that for every x P R , n 2 2 }x} “ γ px; f j q ; j“1 ¸ where p ; q stands for the usual dot-product. Tight frame F is rational if there exists a real number α so that αpf i ; f j q P Q @ 1 ¤ i; j ¤ n: k Let f P R be a nonzero vector, then Gf “ tUf : U P Gu k is called a group frame (or G-frame). If G acts irreducibly on R , Gf is called an irreducible group frame. Irreducible group frames are always tight. Frames Lattices Graphs Examples Group frames Let G be a finite subgroup of Ok pRq, the k-dimensional real orthog- k onal group, then G acts on R by left matrix multiplication. -
Lombardi Drawings of Graphs 1 Introduction
Lombardi Drawings of Graphs Christian A. Duncan1, David Eppstein2, Michael T. Goodrich2, Stephen G. Kobourov3, and Martin Nollenburg¨ 2 1Department of Computer Science, Louisiana Tech. Univ., Ruston, Louisiana, USA 2Department of Computer Science, University of California, Irvine, California, USA 3Department of Computer Science, University of Arizona, Tucson, Arizona, USA Abstract. We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs. 1 Introduction The American artist Mark Lombardi [24] was famous for his drawings of social net- works representing conspiracy theories. Lombardi used curved arcs to represent edges, leading to a strong aesthetic quality and high readability. Inspired by this work, we intro- duce the notion of a Lombardi drawing of a graph, in which edges are drawn as circular arcs with perfect angular resolution: consecutive edges are evenly spaced around each vertex. While not all vertices have perfect angular resolution in Lombardi’s work, the even spacing of edges around vertices is clearly one of his aesthetic criteria; see Fig. 1. Traditional graph drawing methods rarely guarantee perfect angular resolution, but poor edge distribution can nevertheless lead to unreadable drawings. Additionally, while some tools provide options to draw edges as curves, most rely on straight-line edges, and it is known that maintaining good angular resolution can result in exponential draw- ing area for straight-line drawings of planar graphs [17,25]. -
The Extendability of Matchings in Strongly Regular Graphs
The extendability of matchings in strongly regular graphs Sebastian M. Cioab˘a∗ Weiqiang Li Department of Mathematical Sciences Department of Mathematical Sciences University of Delaware University of Delaware Newark, DE 19707-2553, U.S.A. Newark, DE 19707-2553, U.S.A. [email protected] [email protected] Submitted: Feb 25, 2014; Accepted: Apr 29, 2014; Published: May 13, 2014 Mathematics Subject Classifications: 05E30, 05C50, 05C70 Abstract A graph G of even order v is called t-extendable if it contains a perfect matching, t < v=2 and any matching of t edges is contained in some perfect matching. The extendability of G is the maximum t such that G is t-extendable. In this paper, we study the extendability properties of strongly regular graphs. We improve previous results and classify all strongly regular graphs that are not 3-extendable. We also show that strongly regular graphs of valency k > 3 with λ > 1 are bk=3c-extendable k+1 (when µ 6 k=2) and d 4 e-extendable (when µ > k=2), where λ is the number of common neighbors of any two adjacent vertices and µ is the number of common neighbors of any two non-adjacent vertices. Our results are close to being best pos- sible as there are strongly regular graphs of valency k that are not dk=2e-extendable. We show that the extendability of many strongly regular graphs of valency k is at least dk=2e − 1 and we conjecture that this is true for all primitive strongly regular graphs. We obtain similar results for strongly regular graphs of odd order. -
Use the K-Neighborhood Subgraphs to Compute Canonical Labelings of Graphs
mathematics Article Use the K-Neighborhood Subgraphs to Compute Canonical Labelings of Graphs Jianqiang Hao 1,*, Yunzhan Gong 2, Jianzhi Sun 1 and Li Tan 1 1 Beijing Key Laboratory of Big Data Technology for Food Safety, Beijing Technology and Business University, No. 11, Fu Cheng Road, Beijing 100048, China 2 State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, No 10, Xitucheng Road, Haidian District, Beijing 100876, China * Correspondence: [email protected]; Tel.: +86-10-6898-5704 Received: 5 July 2019; Accepted: 27 July 2019; Published: 31 July 2019 Abstract: This paper puts forward an innovative theory and method to calculate the canonical labelings of graphs that are distinct to Nauty’s. It shows the correlation between the canonical labeling of a graph and the canonical labeling of its complement graph. It regularly examines the link between computing the canonical labeling of a graph and the canonical labeling of its open k-neighborhood subgraph . It defines dif fusion degree sequences and entire dif fusion degree sequence . For each node of a graph G, it designs a characteristic m_NearestNode to improve the precision for calculating canonical labeling. Two theorems established here display how to compute the first nodes of MaxQ(G). Another theorem presents how to determine the second nodes of MaxQ(G). When computing Cmax(G), if MaxQ(G) already holds the first i nodes u1, u2, ··· , ui, Diffusion and Nearest Node theorems provide skill on how to pick the succeeding node of MaxQ(G). Further, it also establishes two theorems to determine the Cmax(G) of disconnected graphs. -
Math.RA] 25 Sep 2013 Previous Paper [3], Also Relying in Conceptually Separated Tools from Them, Such As Graphs and Digraphs
Certain particular families of graphicable algebras Juan Núñez, María Luisa Rodríguez-Arévalo and María Trinidad Villar Dpto. Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla. Apdo. 1160. 41080-Sevilla, Spain. [email protected] [email protected] [email protected] Abstract In this paper, we introduce some particular families of graphicable algebras obtained by following a relatively new line of research, ini- tiated previously by some of the authors. It consists of the use of certain objects of Discrete Mathematics, mainly graphs and digraphs, to facilitate the study of graphicable algebras, which are a subset of evolution algebras. 2010 Mathematics Subject Classification: 17D99; 05C20; 05C50. Keywords: Graphicable algebras; evolution algebras; graphs. Introduction The main goal of this paper is to advance in the research of a novel mathematical topic emerged not long ago, the evolution algebras in general, and the graphicable algebras (a subset of them) in particular, in order to obtain new results starting from those by Tian (see [4, 5]) and others already obtained by some of us in a arXiv:1309.6469v1 [math.RA] 25 Sep 2013 previous paper [3], also relying in conceptually separated tools from them, such as graphs and digraphs. Concretely, our goal is to find some particular types of graphicable algebras associated with well-known types of graphs. The motivation to deal with evolution algebras in general and graphicable al- gebras in particular is due to the fact that at present, the study of these algebras is very booming, due to the numerous connections between them and many other branches of Mathematics, such as Graph Theory, Group Theory, Markov pro- cesses, dynamic systems and the Theory of Knots, among others. -
A Godsil E Thomason I Solomon M Abiad B Van Dam F Balbuena J
8.30–9.15 am A Godsil E Thomason I Solomon M Abiad 9.20–9.40 am 1 Zhan 10 Bukh 19 Srinivasan 24 Reichard 9.45–10.05 am 2 Ye 11 Martin 20 Sumalroj 25 Xu 10.05–10.35 am break break break break 10.35–10.55 am 3 Dalfó 12 Kamat 21 Bencs 26 Peng 11.00–11.20 am 4 McGinnis 13 Timmons 22 Guo 27 Kravitz 11.25–12.10 pm B Van Dam F Balbuena J Muzychuk N Xiang 12.10–2.00 pm lunch lunch lunch lunch 2.00–2.45 pm C Kantor G Füredi K Williford 2.50–3.10 pm 5 Gu 14 Tait 23 Ducey 3.15–3.35 pm 6 Coutinho 15 Kodess L Woldar break 3.40–4.00 pm 7 Greaves 16 Y. Wang 4.00–4.30 pm break break 4.30–4.50 pm 8 Fiol 17 Kronenthal 4:55–5.15 pm 9 W. Wang 18 Moorhouse 5.20–6.10 pm D Haemers H Lazebnik 2 8.30–9.15 am A Chris Godsil Spectral invariants from embeddings 9.20–9.40 am 1 Harmony Zhan Quantum walks and mixing 9.45–10.05 am 2 Dong Ye Median eigenvalues and graph inverse 10.35–10.55 am 3 Cristina Dalfó Characterizing identifying codes from the spectrum of a graph or digraph 11.00–11.20 am 4 Matt McGinnis The smallest eigenvalues of the Hamming graphs 11.25–12:10 pm B Edwin van Dam Partially metric association schemes with a small multiplicity 2.00–2.45 pm C Bill Kantor MUBs 2.50–3.10 pm 5 Xiaofeng Gu Toughness, connectivity and the spectrum of regular graphs 3.15–3.35 pm 6 Gabriel Coutinho Average mixing matrix 3.40–4.00 pm 7 Gary Greaves Edge-regular graphs and regular cliques 4.30–4.50 pm 8 Miguel Angel Fiol An algebraic approach to lifts of digraphs 4.55–5.15 pm 9 Wei Wang A positive proportion of multigraphs are determined by their generalized spectra 5.20–6.10