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Symmetries of Graphs and Networks 2nd International Workshop on Symmetries of Graphs and Networks August 1 – August 6, 2010, Rogla, Slovenia BOOKOF ABSTRACTS University of Primorska, 2010 The 2nd International Workshop on Symmetries of Graphs and Networks August 1 - August 6, 2010, Rogla, Slovenia BOOKOF ABSTRACTS Edited by D. Marušicˇ et al. University of Primorska, 2010 The 2nd International Workshop on Symmetries of Graphs and Networks, August 1 - August 6, 2010, Rogla, Slovenia BOOKOF ABSTRACTS Edited by: Dragan Marušicˇ (Editor-in-Chief) Klavdija Kutnar Boštjan Kuzman Primož Šparl Designed by: Boštjan Kuzman Printed by: Published by: Koper, August 2010. Supported by ARRS (Javna agencija za raziskovalno dejavnost Republike Slovenije). CIP – Kataložni zapis o publikaciji Narodna in univerzitetna knjižnica, Ljubljana ISBN WELCOME TO SYGN 2010 Dear Colleagues. On behalf of the organizers, I am pleased to personally welcome you to the 2nd Workshop on Symmetries of Graphs and Networks. A number of leading international researchers in the field have gathered here in order to pursue the line of research that was started by the 1st SYGN Workshop in Banff, Canada, 2008. Besides the conference talks, much of our time on Rogla will be devoted to discussion and work on open problems. You are kindly invited to submit any papers related to this workshop to the special issue of the international journal Ars Mathematica Contemporanea. Hopefully, your work shall inspire a new generation of young researchers to get involved into this fascinating branch of mathematics. As the Chair of the Scientific Committee, I also wish to thank our funders. The organiza- tion of this meeting would not have been possible without financial support from Primorska Institute for Natural Science and Technology (UP PINT) and Faculty of Mathematics, Natural Sciences and Information Technologies (UP FAMNIT) at University of Primorska, Institute of Mathematics, Physics and Mechanics, Ljubljana (IMFM), and Slovenian Research Agency (ARRS). I thank you all for your attendance and wish you a pleasant and mathematically fruitful meeting in Rogla. We are looking forward to meeting you again at the 7th Slovenian Interna- tional Conference on Graph Theory, which will be held in Bled, 2011. Dragan Marušic,ˇ Chair of the Scientific Committee SYGN 2010 6 SYGN 2010 CONTENTS Welcome to SYGN 2010 . .5 Basic Information . .8 Keynote Talks . .9 Conference Logo . .9 Academic Program . 10 Abstracts . 12 List of Participants . 23 Call for papers: Ars Mathematica Contemporanea . 25 First announcement: Bled 2011 . 26 7 SYGN 2010 BASIC INFORMATION 2nd International Workshop on Symmetries of Graphs and Networks AUGUST 1 - AUGUST 6, 2010, ROGLA,SLOVENIA ORGANIZED BY University of Primorska IN COLLABORATION WITH Primorska Institute for Natural Science and Technology (UP PINT) Faculty of Mathematics, Natural Sciences and Information Technologies (UP FAMNIT) Institute of Mathematics, Physics and Mechanics (IMFM). SPONSOREDBY Javna agencija za raziskovalno dejavnost Republike Slovenije (ARRS) SCIENTIFIC COMMITTEE Aleksander Malnic,ˇ Dragan Marušic,ˇ Štefko Miklavic,ˇ Joy Morris, Cheryl E. Praeger ORGANIZING COMMITTEE Klavdija Kutnar, Boštjan Kuzman, Primož Šparl CONFERENCE VENUE Hotel Planja, Rogla, Slovenia CONFERENCE WEBSITE www.famnit.upr.si/sl/konference/sygn/ CONFERENCE INFORMATION E-mail to [email protected] 8 SYGN 2010 KEYNOTE TALKS Regular (flag-transitive) polytopes with few flags Marston Conder, University of Auckland, New Zealand. Normal Cayley graphs Edward Dobson, Mississippi State University, USA. Majorana Theory Alexander A. Ivanov, Imperial College London, United Kingdom. Strongly regular graphs with no triangles Mikhail Klin, Ben-Gurion University of the Negev, Israel. Finite edge-transitive graphs Caiheng Li, University of Western Australia, Australia. The classification of tridiagonal pairs Paul Terwilliger, University of Wisconsin – Madison, USA. CONFERENCE LOGO The letters of the logo are inspired by the question of Lovász from 1969 on the existence of Hamiltonian paths in vertex-transitive graphs. For a certain planar representation of the Cayley graph Cay(S4, (123),(132),(1234),(1432) ) each letter of the logo represents a tree of faces whose boudaryf yields a Hamiltonian cycle.g 9 SYGN 2010 ACADEMIC PROGRAM All talks will be held in Hotel Planja, Rogla. Sunday, August 1 15.00 – 18.00 Registration 18.00 – 20.30 DINNER Monday, August 2 07.30 – 09.20 BREAKFAST 09.20 – 09.30 Opening remarks 09.30 – 10.10 E. Dobson: Normal Cayley graphs 10.15 – 10.35 M. Muzychuk: On isomorphism problem for cyclic codes 10.40 – 11.00 J. Morris: Structure of strongly regular vertex- and edge-transitive graphs 11.00 – 11.30 COFFEEBREAK 11.30 – 11.50 Z. P.Lu On edge transitive Cayley graphs of square-free order 11.55 – 12.15 X. Wang: The primitive permutation groups with an orbital of length 5 12.20 – 12.40 C. Amarra: Symmetric graphs of diameter 2 with complete normal quotients 12.45 – 14.00 LUNCH 15.00 – 18.00 Discussion / Afternoon activities 18.00 – 20.30 DINNER Tuesday, August 3 07.30 – 09.30 BREAKFAST 09.30 – 10.10 A. A. Ivanov: Majorana Theory 10.15 – 11.00 M. Klin: Strongly regular graphs with no triangles 11.00 – 11.30 COFFEEBREAK 11.30 – 11.50 A. Woldar: Dale Mesner, Higman & Sims, and the SRG(100,22,0,6) 11.55 – 12.15 G. Wang: Edge-transitive graphs of square-free order 12.20 – 12.40 B. Kuzman: Arc-transitive elementary abelian covers of graphs 12.45 – 14.00 LUNCH 15.00 – 18.00 Discussion / Afternoon activities 18.00 – 20.30 DINNER 10 SYGN 2010 Wednesday, August 4 07.30 – 09.30 BREAKFAST 09.30 – 10.10 P.Terwilliger: The classification of tridiagonal pairs 10.15 – 10.35 I. Ponomarenko: The schurity problem for quasi-thin association schemes 10.40 – 11.00 J. Lauri: Two-fold orbital graphs: I 11.00 – 11.30 COFFEEBREAK 11.30 – 11.50 R. Scapellato: Two-fold orbital graphs: II 11.55 – 12.15 G. Pretel: A-like elements for a tridiagonal pair 12.20 – 12.40 H. Zhang: On finite 2-path-transitive graphs 12.45 – 14.00 LUNCH 15.00 – 18.00 Discussion / Afternoon activities 18.00 – 20.30 DINNER Thursday, August 5 07.30 – 09.30 BREAKFAST 09.30 – 10.10 M. Conder: Regular (flag-transitive) polytopes with few flags 10.15 – 10.35 Q. Xiang: TBA 10.40 – 11.00 L. M. Fernández: Semiregular actions in combinatorial structures 11.00 – 11.30 COFFEEBREAK 11.30 – 11.50 R. Nedela: Map enumeration 11.55 – 12.15 G. Brown: Leonard triples associated with hypercubes 12.20 – 12.40 J. J. Li: On 2-arc transitive graphs of order odd 12.45 – 14.00 LUNCH 15.00 – 18.00 Discussion / Afternoon activities 18.00 – 20.30 DINNER Friday, August 6 07.30 – 09.30 BREAKFAST 09.30 – 10.10 C. Li: Finite edge-transitive graphs 10.15 – 10.35 M. Giudici: Locally distance transitive graphs 10.40 – 11.00 J. Pan: Finite vertex-transitive edge-transitive metacirculants 11.00 – 11.30 COFFEEBREAK 11.30 – 11.50 C. Sarti: Wilson’s graph operations on Wada dessins 11.55 – 12.15 E. Hanson: A characterization of Leonard pairs using the notion of a tail 12.20 – 12.40 C. Zhang: TBA 12.45 – 13.00 Closing remarks 13.00 – 14.00 LUNCH 11 SYGN 2010 ABSTRACTS Symmetric graphs of diameter 2 with complete normal quotients Carmen Amarra University of Western Australia, Australia [email protected] A graph has diameter 2 if it is not a complete graph and if every pair of nonadjacent ver- tices is joined by a path of length 2. Our general problem is to examine the overall structure of graphs which are both arc-transitive and of diameter 2 using normal quotients. We identify the basic graphs in the class of arc-transitive diameter 2 graphs to be those Γ which satisfy one of the following (relative to some group G of automorphisms of Γ): (i) Γ does not have a nontrivial G -normal quotient, or (ii) all nontrivial G -normal quotients of Γ are complete graphs. In this talk we discuss basic graphs which are of type (ii), in particular the case where there are at least 3 nontrivial normal quotients. For this case we can identify all the graphs that arise, using the classification of the transitive finite linear groups. Leonard triples associated with hypercubes and their antipodal quotients George Brown University of Wisconsin - Madison, USA [email protected] Let be the unital associative algebra over C with generators x,y,z and relations x y + y x = 2zA, y z + z y = 2x and z x + xz = 2y . We find the finite-dimensional irreducible - modules and show that x,y,z act on these modules as bipartite or almost bipartite LeonardA triples. We define an operator s on finite-dimensional sl2-modules that gives them an - structure. A Let d denote a nonnegative integer and let Qd denote the graph of the d -dimensional hypercube. It is known that the Terwilliger algebra of Qd has an sl2-module structure. When d is even, we show that applying s to the Terwilliger algebra of Qd produces the Terwilliger algebra of the alternateQ-polynomial structure ofQd . When d is odd, we show that applying s to the Terwilliger algebra of Qd produces the Terwilliger algebra of the antipodal quotient of Qd . 12 SYGN 2010 Regular (flag-transitive) polytopes with few flags Marston Conder University of Auckland, New Zealand [email protected] An abstract n-polytope is a partially-ordered set endowed with a rank function (plus a unique minimum and unique maximum element), such that all maximal chains have length n + 2. Elements of ranks 0, 1, 2 and n 1 are the vertices, edges, 2-faces and facets. The polytope is called regular if its automorphism− group has a single orbit on flags.
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