DOCSLIB.ORG

D-Self Center Graphs and Graph Operations Pp.: 1–4

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
D-Self Center Graphs and Graph Operations Pp.: 1–4 Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University Talk d-self center graphs and graph operations pp.: 1{4 d-self Center Graphs and Graph Operations Yasser Alizadeh Ehsan Estaji ∗ Hakim Sabzevari University Hakim Sabzevari University Abstract Let G be a simple connected graph. The graph G is called d-self center if it’s vertices are of eccentricity d. In this paper, some self center composite graphs are investigated. Some mathematical properties of self center graphs is investigated. It is proved that a self center graph is 2-connected. Some infinite family of asymmetric self center graphs is constructed. Keywords: eccentricity, d-self center graph, composite graphs Mathematics Subject Classification [2010]: 05C12, 05C76, 05C90 1 Introduction All considered graphs are simple and connected. Distance between two vertices is defined as usual length of shortest path connecting them. Eccentricity of vertex v is denoted by ε(v) is the maximum distance between v and other vertices. The maximum and minimum eccentricity among all vertices of G are called diameter of G, diam(G) and radius of G, rad(G) respectively. The Center of G, C(G) is the set of vertices of rad(G). Let G be a simple connected graph. The graph G is called d-self center if it’s vertices are of eccentricity d. Center of graph G is the set of vertices of minimum eccentricity. Then the Center of a self center graph contains all vertices of the graph. In a series of paper, topological indices based on eccentricity of vertices were studied and for some family of molecular graphs such indices were calculated. For more information, we refer the reader to [1, 2, 3, 4, 5, 6, 8, 7, 10]. In this paper, self center graphs under some graph operations is studied. It is proved that a self center graph is 2-connected. By graph operations, some asymmetric self center graphs is constructed. 2 Main Results n As example complete graphs Kn, cycles Cn and sierpinski graphs Sk are three well-known family of self center graphs. A graph G is called vertex- transitive if for given vertices u and v there is an auto-morphism of G, f such that f (u) = v. For example the complete graphs and cycle graphs and Petersen graph are vertex transitive graphs. Since distance between vertices and eccentricity are invariant under auto morphism of graphs, then the ∗Speaker 670 www.SID.ir Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University Talk d-self center graphs and graph operations pp.: 2{4 n vertex transitive graphs are self center but the reverse is not true. The Sierpinski graphs Sk are a family of self center graphs but non vertex transitive. A regular graph that is edge- transitive but not vertex-transitive is called a semi-symmetric graph. The Gray graph (with 54 vertices), the Tutte 12-cage (with 126 vertices) are two namely semi-symmetric and self center graph and the Folkman graph (with 20 vertices) and the Ljubljana graph (with 112 vertices) are other semi-symmetric graph but non self center graph. It seems an interesting problem to characterize the self center semi-symmetric graphs. A self center graph with n 3 vertices is a block graph or 2-connected graph. ≥ Theorem 2.1. Let G be a self center graph with n 3 vertices. Then G is 2-connected. ≥ There are some self center graphs such as cycles that are not 3-connected graph. Let Aut(G) be the group automorphism of graph G. Orbit of vertex v is denoted by Orb(v) and defined as Orb(v) = f (v) f Aut(G) . The vertices of Orb (v) have the same eccentricity. | ∈ A graph G is vertex transitive if and only if G has exactly one orbit. The following example illustrated in Figure 1. is a graph with 7 orbits but all vertices have a same eccentricity then the graph is self center. Figure 1: asymmetric graph with 7 orbits Proposition 2.2. [9] If G is a 2-self center graph on n 5 vertices then G has at least 2n 5 ≥ − edges. 3 Composite graphs In this section some self centered graph arised from graph operation are presented. We start by join of graphs. Theorem 3.1. Let G1 and G2 be two simple connected graphs. Then G1 + G2 is self center graph if and only if G1 and G2 are self center graphs. Proposition 3.2. For any n 4 there is a family of 2-self center graphs on n vertices. ≥ It is enough to consider K + K where m + p = n and m, p 2. We have a similar m p ≥ statement about cartesian product of graphs. Theorem 3.3. Let G and H be two simple connected graphs. Then G G2 is self center graph if 1 × and only if G1 and G2 are self center graphs. 671 www.SID.ir Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University Talk d-self center graphs and graph operations pp.: 3{4 n Corollary 3.4. Gi is self center iff each Gi is self center for 1 i n. i=1 ≤ ≤ Q Corollary 3.5. Ther is an infinite family of non vertex transitive self centered . Consider the powers of a non-vertex transitive self center graph such as the Gray graph or the Tutte 12-cage graph. For any two simple connected graph with at least two vertices, we can construct a self center graph by symmetric difference operation of the graphs. Theorem 3.6. Let G and H be two simple connected graph with at least two vertices. Then the symmetric difference G H is 2-self center. ⊕ It is easy to see that the disjunction of two complete graphs is a complete graph and vice versa. In the case radius of both G and H is at least 2 the disjunction G H is 2-self ∨ center. Theorem 3.7. Let G and H be two simple connected graph with radius at least 2. Then the disjunction G H is 2-self center. ∨ References [1] Y. Alizadeh, M. Azari, T. Dosliˇ c,´ Computing the eccentricity-related invariants of single-defeat carbon nanocones, J. Comput. Theoret. Nanosci., to appear. [2] M. Azari, A. Iranmanesh, Computing the eccentric-distance sum for graph opera- tions, Disc. Appl. Math. 161 (2013) 2827-2840. [3] N. De, On Eccentric Connectivity Index and Polynomial of Thorn Graph, Applied Mathematics, 3 (2012) 931-934. [4] N. De, Sk. Md. A. Nayeem, A. Pal, Total Eccentricity Index of the Generalized Hierarchical Product of Graphs, Int. J. Appl. Comput. Math 2014. [5] T. Dosliˇ c,´ A. Graovac, O. Ori, Eccentric Connectivity Index of Hexagonal Belts and Chains, MATCH Commun. Math. Comput. Chem. 65 (2011) 745-752. [6] T. Dosliˇ c,´ M. Saheli, Augmented Eccentric Connectivity Index, Miskolc Mathematical Notes. 12 (2011) 149-157. [7] M. Ghorbani, M.A Hosseinzadeh, new version of Zagreb indices, Filomat. 26:1 (2012) 93-100. [8] A. Iranmenesh, Y. Alizadeh. Eccentric Connectivity Index of HAC5C7[p,q] and HAC5C6C7[p,q] Nanotubes, MATCH Commun. Math. Comput. Chem. 69 (2013) 175-182. [9] Z. Stanic, Some notes on minimal self centered graphs, AKCE J. Graphs. Combin. 7 (2010) 97-102 672 www.SID.ir Archive of SID 46th Annual Iranian Mathematics Conference 25-28 August 2015 Yazd University Talk d-self center graphs and graph operations pp.: 4{4 [10] Z. Yarahmadi, S. Moradi, The Center and Periphery of Composite Graphs, IJMC. 5 (2014) 35-44. Email: [email protected] Email: [email protected] 673 www.SID.ir .
Load more

Recommended publications
  • Configurations of Points and Lines
    Configurations of Points and Lines "RANKO'RüNBAUM 'RADUATE3TUDIES IN-ATHEMATICS 6OLUME !MERICAN-ATHEMATICAL3OCIETY http://dx.doi.org/10.1090/gsm/103 Configurations of Points and Lines Configurations of Points and Lines Branko Grünbaum Graduate Studies in Mathematics Volume 103 American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 01A55, 01A60, 05–03, 05B30, 05C62, 51–03, 51A20, 51A45, 51E30, 52C30. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-103 Library of Congress Cataloging-in-Publication Data Gr¨unbaum, Branko. Configurations of points and lines / Branko Gr¨unbaum. p. cm. — (Graduate studies in mathematics ; v. 103) Includes bibliographical references and index. ISBN 978-0-8218-4308-6 (alk. paper) 1. Configurations. I. Title. QA607.G875 2009 516.15—dc22 2009000303 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society.
    [Show full text]
  • Pdf 143.41 K
    ISSN: 1017-060X (Print) ISSN: 1735-8515 (Online) Bulletin of the Iranian Mathematical Society Vol. 43 (2017), No. 7, pp. 2281{2292 . Title: On the fixed number of graphs Author(s): I. Javaid, M. Murtaza, M. Asif and F. Iftikhar Published by the Iranian Mathematical Society http://bims.ims.ir Bull. Iranian Math. Soc. Vol. 43 (2017), No. 7, pp. 2281{2292 Online ISSN: 1735-8515 ON THE FIXED NUMBER OF GRAPHS I. JAVAID∗, M. MURTAZA, M. ASIF AND F. IFTIKHAR (Communicated by Ali Reza Ashrafi) Abstract. A set of vertices S of a graph G is called a fixing set of G, if only the trivial automorphism of G fixes every vertex in S. The fixing number of a graph is the smallest cardinality of a fixing set. The fixed number of a graph G is the minimum k, such that every k-set of vertices of G is a fixing set of G. A graph G is called a k-fixed graph, if its fixing number and fixed number are both k. In this paper, we study the fixed number of a graph and give a construction of a graph of higher fixed number from a graph of lower fixed number. We find the bound on k in terms of the diameter d of a distance-transitive k-fixed graph. Keywords: Fixing set, stabilizer, fixing number, fixed number. MSC(2010): Primary: 05C25; Secondary: 05C60. 1. Introduction Let G = (V (G);E(G)) be a connected graph of order n. The degree of a vertex v in G, denoted by degG(v), is the number of edges that are incident to v in G.
    [Show full text]
  • Graph Theory with Applications
    GRAPH THEORY WITH APPLICATIONS J. A. Bondy and U. S. R. Murty Department of Combina tories and Optimization, University of Waterloo, Ontario, Canada NORfH-HOLLAND New York • Amsterdam • Oxford @J.A. Bondy and U.S.R. Muny 1976 First published in Great Britain 1976 by The Macmillan Press Ltd. First published in the U.S.A. 1976 by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, N.Y. 10017 Fifth Printing, 1982. Sole Distributor in the U.S.A: Elsevier Science Publishing Co., Inc. Library of Congress Cataloging in Publication Data Bondy, John Adrian. Graph theory with applications. Bibliography: p. lncludes index. 1. Graph theory. 1. Murty, U.S.R., joint author. II. Title. QA166.B67 1979 511 '.5 75-29826 ISBN 0.:444-19451-7 AU rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. Printed in the United States of America To our parents Preface This book is intended as an introduction to graph theory. Our aim bas been to present what we consider to be the basic material, together with a wide variety of applications, both to other branches of mathematics and to real-world problems. Included are simple new proofs of theorems of Brooks, Chvâtal, Tutte and Vizing. The applications have been carefully selected, and are treated in some depth. We have chosen to omit ail so-called 'applications' that employ just the language of graphs and no theory. The applications appearing at the end of each chapter actually make use of theory developed earlier in the same chapter.
    [Show full text]
  • Orbit Polynomial of Graphs Versus Polynomial with Integer Coefficients
    S S symmetry Article Orbit Polynomial of Graphs versus Polynomial with Integer Coefficients Modjtaba Ghorbani 1,* , Maryam Jalali-Rad 1 and Matthias Dehmer 2,3,4 1 Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-136, Iran; [email protected] 2 Department of Computer Science, Swiss Distance University of Applied Sciences, 3900 Brig, Switzerland; [email protected] 3 Department of Biomedical Computer Science and Mechatronics, UMIT, A-6060 Hall in Tyrol, Austria 4 College of Artficial Intelligence, Nankai University, Tianjin 300071, China * Correspondence: [email protected]; Tel.: +98-21-22970029 Abstract: Suppose ai indicates the number of orbits of size i in graph G. A new counting polynomial, i namely an orbit polynomial, is defined as OG(x) = ∑i aix . Its modified version is obtained by subtracting the orbit polynomial from 1. In the present paper, we studied the conditions under which an integer polynomial can arise as an orbit polynomial of a graph. Additionally, we surveyed graphs with a small number of orbits and characterized several classes of graphs with respect to their orbit polynomials. Keywords: orbit; group action; polynomial roots; orbit-stabilizer theorem 1. Introduction Citation: Ghorbani, M.; Jalali-Rad, M.; Dehmer, M. Orbit Polynomial of By having the orbits and their structures in a graph, we can infer many algebraic prop- Graphs versus Polynomial with erties about the automorphism group and thus about the similar vertices. For example, the Integer Coefficients. Symmetry 2021, length of orbits of a network gives provides important information about each individual 13, 710. https://doi.org/10.3390/ component in the network.
    [Show full text]
  • XYZ Decompositions of Graphs Dagstuhl Seminar 08191
    XYZ decom- positions V. Batagelj XYZ decompositions of graphs Dagstuhl Seminar 08191 Motivation XYZ decom- Vladimir Batagelj positions Results University of Ljubljana, FMF, Dept. of Mathematics; and References IMFM Ljubljana, Dept. of Theoretical Computer Science joint work with Franz J. Brandenburg, Walter Didimo, Giuseppe Liotta, Maurizio Patrignani 23rd Leoben-Ljubljana Graph Theory Seminar Ljubljana, 14-15. November 2008 V. Batagelj XYZ decompositions Outline XYZ decom- positions V. Batagelj Dagstuhl Seminar 08191 Motivation 1 XYZ decom- Dagstuhl Seminar 08191 positions 2 Motivation Results 3 XYZ decompositions References 4 Results 5 References V. Batagelj XYZ decompositions Dagstuhl Seminar 08191 XYZ decom- positions V. Batagelj Dagstuhl Seminar 08191 Motivation XYZ decom- positions Results References Dagstuhl Seminar 08191: Graph Drawing with Applications to Bioinformatics and Social Sciences. Dagstuhl, May 4-9, 2008 V. Batagelj XYZ decompositions Motivation XYZ decom- positions For dense graphs the matrix representation is better [3]. Pajek - shadow [0.00,1.00] V. Batagelj uki fra wge jap net ita usa bel lux car swe kod den swi can nor par spa irn Dagstuhl yem mat irq pak ire nau aut bol hun Seminar 08191 lib hai isr nic sau alb kuw aus gua mon fin por zai hon bra arg pol syr cze kmr usr Motivation tri ege yug sri ind cha saf gre tur sie egy bul rum uru tai rum syr XYZ decom- leb upv cyp vnr ice arg tun gab cub liy positions pol mor usr lux alg spanor nig isr mli uga ins bul liy ken mex usa eth brm pan net tha cze fra jor sud gre
    [Show full text]
  • ASYMMETRIC GRAPHS by P
    ASYMMETRIC GRAPHS By P. ERDÖS and A . RÉNYI (Budapest), members Of the Academy Dedicated to T. GALLAI . at the occasion of his 50th birthday Introduction We consider in this paper only non-directed graphs without multiple edges and without loops . The number of vertices of a graph G will be called its order, and will be denoted by N(G). We shall call such a graph symmetric, if there exists a non-identical permutation of its vertices, which leaves the graph invariant. By other words a graph is called symmetric if the group of its automorphisms has degree greater than 1 . A graph which is not symmetric will be called asymmetric. The degree of symmetry of a symmetric graph is evidently measured by the degree of its group of automorphisms . The question which led us to the results contained in the present paper is the following : how can we measure the degree of asymmetry of an asymmetric graph? Evidently any asymmetric graph can be made symmetric by deleting certain of its edges and by adding certain new edges connecting its vertices . We shall call such a transformation of the graph its symmetrization . For each symmetrization of the graph let us take the sum of the number of deleted edges - say r - and the number of new edges - say s ; it is reasonable to define the degree of asymmetry A [G] of a graph G, as the minimum of r+s where the minimum is taken over all possible symmetrizations of the graph G. (In what follows if in order to make a graph symmetric we delete r of its edges and add s new edges, we shall say that we changed r + s edges.) Clearly the asymmetry of a symmetric graph is according to this definition equal to 0, while the asymmetry of any asymmetric graph is a positive integer .
    [Show full text]
  • Semisymmetric Graphs from Polytopes
    Semisymmetric Graphs from Polytopes Barry Monson¤ University of New Brunswick Fredericton, New Brunswick, Canada E3B 5A3 ([email protected]) Toma·z Pisanskiy University of Ljubljana, FMF Jadranska 19, Ljubljana 1111, Slovenia; and Faculty of Education, University of Primorska, Cankarjeva 5, Koper 6000, Slovenia ([email protected]) Egon Schultez Northeastern University Boston MA 02115, USA ([email protected]) Asia Ivi¶c Weissx York University Toronto, Ontario, Canada M3J 1P3 ([email protected]) Abstract Every ¯nite, self-dual, regular (or chiral) 4-polytope of type f3;q;3g has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by drop- ping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope. Key Words: semisymmetric graphs, abstract regular and chiral polytopes. AMS Subject Classi¯cation (1991): Primary: 05C25. Secondary: 51M20. ¤Supported by the NSERC of Canada, grant 4818 ySupported by Ministry of Higher Education, Science and Technolgy of Slovenia grants P1-0294,J1-6062,L1-7230. zSupported by NSA-grant H98230-05-1-0027 xSupported by the NSERC of Canada, grant 8857 1 1 Introduction The theory of symmetric trivalent graphs and the theory of regular polytopes are each abundant sources of beautiful mathematical ideas. In [22], two of the authors established some general and unexpected connections between the two subjects, building upon a rich variety of examples appearing in the literature (see [4], [7], [10], [11], [28] and [29]). Here we develop these connections a little further, with speci¯c focus on semisymmetric graphs.
    [Show full text]
  • LEOBEN-LJUBLJANA GRAPH THEORY SEMINAR 2016 Dedicated to Norbert Seifter's 60Th Birthday Special Session on the Occasion Of
    LEOBEN-LJUBLJANA GRAPH THEORY SEMINAR 2016 Dedicated to Norbert Seifter's 60th Birthday Special session on the occasion of Bojan Mohar's 60th Birthday 25th - 28th September 2016, Judenburg, Austria Supported by TU Graz, MU Leoben, Austrian Science Fund, Austrian Mathematical Society and the Town of Judenburg. Titles and abstracts Vladimir Batagelj (Ljubljana - Koper): Widespread index Abstract. On January 12, 2016 Clement Levallois posted at SocNet a message asking for a measure of how widespread is the distribution of a node attribute in a network. In the paper we propose and study two indices for measuring the widespread of a node attribute value in a network: the (simple) widespread index and the domination index. A com- putation of the proposed indices using the program Pajek is illustrated on two networks: the Class network and the US Airports network. Chris Godsil (Waterloo): Graph isomorphism and quantum walks Abstract. It has been suggested that graph isomorphism may be one of the problems that can be solved faster on a quantum computer than on a clasical computer. Conse- quently physicists have devoted some effort to find algorithms for graph isomorphism. I will report on work undertaken by me and my colleagues, using tools from graph spec- tral theory and finite geometry, to show that the algorithms proposed to date do not work. Veronica Hern´andezMartinez (Madrid): On the diameter, minimum and maximum degree and hyperbolicity constant Abstract. In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erd¨os, Pach, Pollack and Tuza.
    [Show full text]
  • The Wiener Dimension of a Graph
    The Wiener dimension of a graph Yaser Alizadeh Department of Mathematics Tarbiat Modares University, Tehran, Iran [email protected] Sandi Klavˇzar∗ Faculty of Mathematics and Physics University of Ljubljana, Slovenia and Faculty of Natural Sciences and Mathematics University of Maribor, Slovenia [email protected] March 13, 2012 Abstract The Wiener dimension of a connected graph is introduced as the number of different distances of its vertices. For any integer d and any integer k, a graph of diameter d and of Wiener dimension k is constructed. An infinite family of non-vertex-transitive graphs with Wiener dimension 1 is also presented and it is proved that a graph of dimension 1 is 2-connected. Keywords: Wiener index; Cartesian product graphs; vertex-transitive graphs AMS subject classification (2010): 05C12, 05C25, 92E10 1 Introduction The distance considered in this paper is the usual shortest path distance. We assume throughout the paper that all graphs are connected unless stated otherwise. For terms not defined here we refer to [13]. Let G be a graph and u ∈ V (G). Then the distance of u is dG(u)= X dG(u, v) . v∈V (G) ∗ This work has been financed by ARRS Slovenia under the grant P1-0297 and within the EU- ROCORES Programme EUROGIGA/GReGAS of the European Science Foundation. The author is also with the Institute of Mathematics, Physics and Mechanics, Ljubljana. 1 In location theory, sets of vertices with the minimum (or maximum) distance in a graph play s special role because they form target sets for locations of facilities.
    [Show full text]
  • Induced Saturation Number Jason James Smith Iowa State University
    Iowa State University Digital Repository @ Iowa State University Graduate Theses and Dissertations Graduate College 2012 Induced Saturation Number Jason James Smith Iowa State University Follow this and additional works at: http://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Smith, Jason James, "Induced Saturation Number" (2012). Graduate Theses and Dissertations. Paper 12465. This Dissertation is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more information, please contact [email protected]. Induced saturation number by Jason James Smith A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Ryan Martin, Major Professor Leslie Hogben Maria Axenovich Ling Long Jack Lutz Iowa State University Ames, Iowa 2012 Copyright © Jason James Smith, 2012. All rights reserved. ii TABLE OF CONTENTS LIST OF FIGURES . iv ACKNOWLEDGEMENTS . vi ABSTRACT . vii CHAPTER 1. INTRODUCTION . 1 1.1 Saturation in Graphs . .1 1.1.1 Definitions and Notation . .2 1.1.2 Applications . .4 CHAPTER 2. INDUCED SATURATION NUMBER AND BOUNDS . 5 2.1 Bounds for Induced Saturation Number . .5 2.2 Induced Saturation Number for a Few Families of Graphs . 12 n+1 CHAPTER 3. INDUCED SATURATION NUMBER OF P4 IS 3 ..... 15 3.1 Definitions and Notation . 15 3.2 Upper Bound by Construction . 17 3.3 Lower Bound by Induction . 19 3.4 Technical Lemma .
    [Show full text]
  • Fundamental Properties and (5,0)-Nanotubical Fullerenes
    MATCH MATCH Commun. Math. Comput. Chem. 72 (2014) 279-294 Communications in Mathematical and in Computer Chemistry ISSN 0340 - 6253 Wiener Dimension: Fundamental Properties and (5,0)-Nanotubical Fullerenes Yaser Alizadeha, Vesna Andova b1, Sandi Klavˇzar c,d,e, Riste Skrekovskiˇ c,e,f,g aDepartment of Mathematics, Hakim Sabzevari University, Sabzevar, Iran E-mail: [email protected] 2Faculty of Electrical Engineering and Information Technologies, Ss Cyril and Methodius Univ., Ruger Boskovik, P. O. Box 574, 1000 Skopje, Macedonia E-mail: [email protected] cFaculty of Mathematics and Physics, University of Ljubljana, Slovenia dFaculty of Natural Sciences and Mathematics, University of Maribor, Slovenia eInstitute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia E-mail: [email protected] f Faculty of Information Studies, Novo Mesto, Slovenia gFaculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska, Slovenia E-mail: [email protected] (Received January 24, 2014) Abstract The Wiener dimension of a connected graph is introduced as the number of different distances of its vertices. For any integer D and any integer k, a graph of diameter D and of Wiener dimension k is constructed. An infinite family of non-vertex-transitive graphs with Wiener dimension 1 is presented and it is proved that a graph of dimension 1 is 2-connected. It is shown that the (5, 0)-nanotubical fullerene graph on 10k (k ≥ 3) vertices has Wiener dimension k. As a consequence the Wiener index of these fullerenes is obtained. 1Corresponding author -280- 1 Introduction The distance considered in this paper is the usual shortest path distance.
    [Show full text]
  • On Cubic Graphs Admitting an Edge-Transitive Solvable Group
    Journal of Algebraic Combinatorics, 20, 99–113, 2004 c 2004 Kluwer Academic Publishers. Manufactured in The Netherlands. On Cubic Graphs Admitting an Edge-Transitive Solvable Group ALEKSANDER MALNICˇ ∗ DRAGAN MARUSIˇ Cˇ ∗ PRIMOZˇ POTOCNIKˇ † IMFM, Oddelek za matematiko, Univerza v Ljubljani, Jadranska 19, 1111 Ljubljana, Slovenija Received March 14, 2002; Revised August 21, 2003; Accepted October 14, 2003 Abstract. Using covering graph techniques, a structural result about connected cubic simple graphs admitting an edge-transitive solvable group of automorphisms is proved. This implies, among other, that every such graph can be obtained from either the 3-dipole Dip3 or the complete graph K4,byasequence of elementary-abelian covers. Another consequence of the main structural result is that the action of an arc-transitive solvable group on a connected cubic simple graph is at most 3-arc-transitive. As an application, a new infinite family of semisymmetric cubic graphs, arising as regular elementary abelian covering projections of K3,3,isconstructed. Keywords: symmetric graph, edge transitive graph, cubic graph, trivalent graph, covering projection of graphs, solvable group of automorphisms 1. Introduction Throughout this paper graphs are assumed to be finite, and unless specified otherwise, simple, undirected and connected. It transpires that, when investigating edge-transitive cubic (simple) graphs the concept of graph coverings plays a central role. A correct treatment of this concept calls for a more general definition of a graph (see Section 2), with the class of simple graphs as a special case. The study of cubic arc-transitive graphs has its roots in the classical result of Tutte [20], who proved that cubic graphs are at most 5-arc-transitive.
    [Show full text]