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Q(A) - Balance Super Edge Magic Graphs Results
International Journal of Pure and Applied Mathematical Sciences. ISSN 0972-9828 Volume 10, Number 2 (2017), pp. 157-170 © Research India Publications http://www.ripublication.com Q(A) - Balance Super Edge Magic Graphs Results M. Rameshpandi and S. Vimala 1Assistant Professor, Department of Mathematics, Pasumpon Muthuramalinga Thevar College, Usilampatti, Madurai. 2Assistant Professor, Department of Mathematics, Mother Teresa Women’s University, Kodaikanal. Abstract Let G be (p,q)-graph in which the edges are labeled 1,2,3,4,…q so that the vertex sums are constant, mod p, then G is called an edge magic graph. Magic labeling on Q(a) balance super edge–magic graphs introduced [5]. In this paper extend my discussion of Q(a)- balance super edge magic labeling(BSEM) to few types of special graphs. AMS 2010: 05C Keywords: edge magic graph, super edge magic graph, Q(a) balance edge- magic graph. 1.INTRODUCTION Magic graphs are related to the well-known magic squares, but are not quite as famous. Magic squares are an arrangement of numbers in a square in such a way that the sum of the rows, columns and diagonals is always the same. All graphs in this paper are connected, (multi-)graphs without loop. The graph G has vertex – set V(G) and edge – set E(G). A labeling (or valuation) of a graph is a map that carries graph elements to numbers(usually to the positive or non- negative integers). Edge magic graph introduced by Sin Min Lee, Eric Seah and S.K Tan in 1992. Various author discussed in edge magic graphs like Edge magic(p,3p-1)- graphs, Zykov sums of graphs, cubic multigraphs, Edge-magicness of the composition of a cycle with a null graph. -
On the Group-Theoretic Properties of the Automorphism Groups of Various Graphs
ON THE GROUP-THEORETIC PROPERTIES OF THE AUTOMORPHISM GROUPS OF VARIOUS GRAPHS CHARLES HOMANS Abstract. In this paper we provide an introduction to the properties of one important connection between the theories of groups and graphs, that of the group formed by the automorphisms of a given graph. We provide examples of important results in graph theory that can be understood through group theory and vice versa, and conclude with a treatment of Frucht's theorem. Contents 1. Introduction 1 2. Fundamental Definitions, Concepts, and Theorems 2 3. Example 1: The Orbit-Stabilizer Theorem and its Application to Graph Automorphisms 4 4. Example 2: On the Automorphism Groups of the Platonic Solid Skeleton Graphs 4 5. Example 3: A Tight Bound on the Product of the Chromatic Number and Independence Number of Vertex-Transitive Graphs 6 6. Frucht's Theorem 7 7. Acknowledgements 9 8. References 9 1. Introduction Groups and graphs are two highly important kinds of structures studied in math- ematics. Interestingly, the theory of groups and the theory of graphs are deeply connected. In this paper, we examine one particular such connection: that which emerges from the observation that the automorphisms of any given graph form a group under composition. In section 2, we provide a framework for understanding the material discussed in the paper. In sections 3, 4, and 5, we demonstrate how important results in group theory illuminate some properties of automorphism groups, how the geo- metric properties of particular embeddings of graphs can be used to determine the structure of the automorphism groups of all embeddings of those graphs, and how the automorphism group can be used to determine fundamental truths about the structure of the graph. -
Finite Projective Geometries 243
FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig. -
Cayley Graphs of PSL(2) Over Finite Commutative Rings Kathleen Bell Western Kentucky University, [email protected]
Western Kentucky University TopSCHOLAR® Masters Theses & Specialist Projects Graduate School Spring 2018 Cayley Graphs of PSL(2) over Finite Commutative Rings Kathleen Bell Western Kentucky University, [email protected] Follow this and additional works at: https://digitalcommons.wku.edu/theses Part of the Algebra Commons, and the Other Mathematics Commons Recommended Citation Bell, Kathleen, "Cayley Graphs of PSL(2) over Finite Commutative Rings" (2018). Masters Theses & Specialist Projects. Paper 2102. https://digitalcommons.wku.edu/theses/2102 This Thesis is brought to you for free and open access by TopSCHOLAR®. It has been accepted for inclusion in Masters Theses & Specialist Projects by an authorized administrator of TopSCHOLAR®. For more information, please contact [email protected]. CAYLEY GRAPHS OF P SL(2) OVER FINITE COMMUTATIVE RINGS A Thesis Presented to The Faculty of the Department of Mathematics Western Kentucky University Bowling Green, Kentucky In Partial Fulfillment Of the Requirements for the Degree Master of Science By Kathleen Bell May 2018 ACKNOWLEDGMENTS I have been blessed with the support of many people during my time in graduate school; without them, this thesis would not have been completed. To my parents: thank you for prioritizing my education from the beginning. It is because of you that I love learning. To my fianc´e,who has spent several \dates" making tea and dinner for me while I typed math: thank you for encouraging me and believing in me. To Matthew, who made grad school fun: thank you for truly being a best friend. I cannot imagine the past two years without you. To the math faculty at WKU: thank you for investing your time in me. -
Algorithms for Generating Star and Path of Graphs Using BFS Dr
Web Site: www.ijettcs.org Email: [email protected], [email protected] Volume 1, Issue 3, September – October 2012 ISSN 2278-6856 Algorithms for Generating Star and Path of Graphs using BFS Dr. H. B. Walikar2, Ravikumar H. Roogi1, Shreedevi V. Shindhe3, Ishwar. B4 2,4Prof. Dept of Computer Science, Karnatak University, Dharwad, 1,3Research Scholars, Dept of Computer Science, Karnatak University, Dharwad Abstract: In this paper we deal with BFS algorithm by 1.8 Diamond Graph: modifying it with some conditions and proper labeling of The diamond graph is the simple graph on nodes and vertices which results edges illustrated Fig.7. [2] and on applying it to some small basic 1.9 Paw Graph: class of graphs. The BFS algorithm has to modify The paw graph is the -pan graph, which is also accordingly. Some graphs will result in and by direct application of BFS where some need modifications isomorphic to the -tadpole graph. Fig.8 [2] in the algorithm. The BFS algorithm starts with a root vertex 1.10 Gem Graph: called start vertex. The resulted output tree structure will be The gem graph is the fan graph illustrated Fig.9 [2] in the form of Structure or in Structure. 1.11 Dart Graph: Keywords: BFS, Graph, Star, Path. The dart graph is the -vertex graph illustrated Fig.10.[2] 1.12 Tetrahedral Graph: 1. INTRODUCTION The tetrahedral graph is the Platonic graph that is the 1.1 Graph: unique polyhedral graph on four nodes which is also the A graph is a finite collection of objects called vertices complete graph and therefore also the wheel graph . -
The Spectrum of Platonic Graphs Over Finite Fields
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 307 (2007) 1074–1081 www.elsevier.com/locate/disc The spectrum of Platonic graphs over finite fields Michelle DeDeoa, Dominic Lanphierb,∗, Marvin Mineic aDepartment of Mathematics, University of North Florida, Jacksonville, FL 32224, USA bDepartment of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA cDepartment of Mathematics, University of California, Irvine, CA 92697, USA Received 15 September 2005; received in revised form 26 June 2006; accepted 20 July 2006 Available online 13 October 2006 Abstract We give a decomposition theorem for Platonic graphs over finite fields and use this to determine the spectrum of these graphs. We also derive estimates for the isoperimetric numbers of the graphs. © 2006 Elsevier B.V. All rights reserved. Keywords: Cayley graph; Ramanujan graph; Isoperimetric number 1. Introduction r Let p be an odd prime, let r 1, and let Fq be the finite field with q = p elements. We consider a generalization of the Platonic graphs studied in [2,7–9]. In particular, using the notation from [7], we define our graphs G∗(q) as follows. Definition 1. Let the vertex set of G∗(q) be ∗ V(G (q)) ={()|, ∈ Fq ,() = (00)}/±1, and let () be adjacent to () if and only if det =±1. For q = p, the graph G∗(q) is called the qth Platonic graph. The name comes from the fact that when p = 3or 5, the graphs G∗(p) correspond to 1-skeletons of two of the Platonic solids, in other words, the tetrahedron and the icosahedron as in [2]. -
Package 'Igraph'
Package ‘igraph’ February 28, 2013 Version 0.6.5-1 Date 2013-02-27 Title Network analysis and visualization Author See AUTHORS file. Maintainer Gabor Csardi <[email protected]> Description Routines for simple graphs and network analysis. igraph can handle large graphs very well and provides functions for generating random and regular graphs, graph visualization,centrality indices and much more. Depends stats Imports Matrix Suggests igraphdata, stats4, rgl, tcltk, graph, Matrix, ape, XML,jpeg, png License GPL (>= 2) URL http://igraph.sourceforge.net SystemRequirements gmp, libxml2 NeedsCompilation yes Repository CRAN Date/Publication 2013-02-28 07:57:40 R topics documented: igraph-package . .5 aging.prefatt.game . .8 alpha.centrality . 10 arpack . 11 articulation.points . 15 as.directed . 16 1 2 R topics documented: as.igraph . 18 assortativity . 19 attributes . 21 autocurve.edges . 23 barabasi.game . 24 betweenness . 26 biconnected.components . 28 bipartite.mapping . 29 bipartite.projection . 31 bonpow . 32 canonical.permutation . 34 centralization . 36 cliques . 39 closeness . 40 clusters . 42 cocitation . 43 cohesive.blocks . 44 Combining attributes . 48 communities . 51 community.to.membership . 55 compare.communities . 56 components . 57 constraint . 58 contract.vertices . 59 conversion . 60 conversion between igraph and graphNEL graphs . 62 convex.hull . 63 decompose.graph . 64 degree . 65 degree.sequence.game . 66 dendPlot . 67 dendPlot.communities . 68 dendPlot.igraphHRG . 70 diameter . 72 dominator.tree . 73 Drawing graphs . 74 dyad.census . 80 eccentricity . 81 edge.betweenness.community . 82 edge.connectivity . 84 erdos.renyi.game . 86 evcent . 87 fastgreedy.community . 89 forest.fire.game . 90 get.adjlist . 92 get.edge.ids . 93 get.incidence . 94 get.stochastic . -
A Note on Fibered Quadrangles
Konuralp Journal of Mathematics Volume 3 No. 2 pp. 185{189 (2015) c KJM A NOTE ON FIBERED QUADRANGLES S. EKMEKC¸ I_ & A. BAYAR Abstract. In this work, the fibered versions of the diagonal triangle and the quadrangular set of a complete quadrangle in fibered projective planes are introduced. And then some related theorems with them are given. 1. Introduction Fuzzy set theory was introduced by Zadeh [10] and this theory has been applied in many areas. One of them is projective geometry, see for instance [1,2,5,7,8,9]. A first model of fuzzy projective geometries was introduced by Kuijken, Van Maldeghem and Kerre [7,8]. Also, Kuijken and Van Maldeghem contributed to fuzzy theory by introducing fibered geometries, which is a particular kind of fuzzy geometries [6]. They gave the fibered versions of some classical results in projective planes by using minimum operator. Then the role of the triangular norm in the theory of fibered projective planes and fibered harmonic conjugates and a fibered version of Reidemeister's condition were given in [3]. The fibered version of Menelaus and Ceva's 6-figures was studied in [4]. It is well known that triangles and quadrangles have an important role in projec- tive geometry. A complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting each pair of points. The free completion of a configuration containing either a quadrangle or a quadrilateral is a projective plane. -
1. Math Olympiad Dark Arts
Preface In A Mathematical Olympiad Primer , Geoff Smith described the technique of inversion as a ‘dark art’. It is difficult to define precisely what is meant by this phrase, although a suitable definition is ‘an advanced technique, which can offer considerable advantage in solving certain problems’. These ideas are not usually taught in schools, mainstream olympiad textbooks or even IMO training camps. One case example is projective geometry, which does not feature in great detail in either Plane Euclidean Geometry or Crossing the Bridge , two of the most comprehensive and respected British olympiad geometry books. In this volume, I have attempted to amass an arsenal of the more obscure and interesting techniques for problem solving, together with a plethora of problems (from various sources, including many of the extant mathematical olympiads) for you to practice these techniques in conjunction with your own problem-solving abilities. Indeed, the majority of theorems are left as exercises to the reader, with solutions included at the end of each chapter. Each problem should take between 1 and 90 minutes, depending on the difficulty. The book is not exclusively aimed at contestants in mathematical olympiads; it is hoped that anyone sufficiently interested would find this an enjoyable and informative read. All areas of mathematics are interconnected, so some chapters build on ideas explored in earlier chapters. However, in order to make this book intelligible, it was necessary to order them in such a way that no knowledge is required of ideas explored in later chapters! Hence, there is what is known as a partial order imposed on the book. -
Self-Dual Harmonicity: the Planar Case
SELF-DUAL HARMONICITY: THE PLANAR CASE P.L. ROBINSON Abstract. We present a manifestly geometrically self-dual version of the Fano harmonicity axiom for the projective plane. Introduction In a recent sequence of papers, we have explored manifestly self-dual formulations of three- dimensional projective space: [2] begins the sequence, basing projective space on lines and their abstract incidence, from which points and planes are derived as secondary elements; [4] pursues the complementary approach, taking points and planes to be primary elements from which lines are derived. The most recent paper [6] addresses the axioms (or assumptions) of harmonicity and projectvity: traditionally, harmonicity asserts (with Fano) that ‘the diagonal points of a complete quadrangle are noncollinear’ while projectivity asserts that ‘if a projectivity leaves each of three distinct points of a line invariant, it leaves every point of the line invariant’; see [7] Section 18 and [7] Section 35 respectively. In keeping with our aim to present projective space in a formulation that is manifestly self- dual, it is desirable to express projectivity and harmonicity in manifestly self-dual forms. The axiom of projectivity is so expressed in [3] and [6]: indeed, [7] Section 103 contains a self-dual expression of projectivity in its discussion of reguli. In [5] and [6] our versions of the axiom of harmonicity are self-dual only in being the conjunction of a pair of dual statements, each of which implies the other; we might say that these versions are self-dual in rather a logical sense. Our purpose in the present paper and its sequel is to offer expressions of harmonicity that are more geometrically self-dual: the present paper deals with harmonicity in the plane; its sequel deals with the more elaborate spatial harmonicity. -
Operation-Based Notation for Archimedean Graph
Operation-Based Notation for Archimedean Graph Hidetoshi Nonaka Research Group of Mathematical Information Science Division of Computer Science, Hokkaido University N14W9, Sapporo, 060 0814, Japan ABSTRACT Table 1. The list of Archimedean solids, where p, q, r are the number of vertices, edges, and faces, respectively. We introduce three graph operations corresponding to Symbol Name of polyhedron p q r polyhedral operations. By applying these operations, thirteen Archimedean graphs can be generated from Platonic graphs that A(3⋅4)2 Cuboctahedron 12 24 14 are used as seed graphs. A4610⋅⋅ Great Rhombicosidodecahedron 120 180 62 A468⋅⋅ Great Rhombicuboctahedron 48 72 26 Keyword: Archimedean graph, Polyhedral graph, Polyhedron A 2 Icosidodecahedron 30 60 32 notation, Graph operation. (3⋅ 5) A3454⋅⋅⋅ Small Rhombicosidodecahedron 60 120 62 1. INTRODUCTION A3⋅43 Small Rhombicuboctahedron 24 48 26 A34 ⋅4 Snub Cube 24 60 38 Archimedean graph is a simple planar graph isomorphic to the A354 ⋅ Snub Dodecahedron 60 150 92 skeleton or wire-frame of the Archimedean solid. There are A38⋅ 2 Truncated Cube 24 36 14 thirteen Archimedean solids, which are semi-regular polyhedra A3⋅102 Truncated Dodecahedron 60 90 32 and the subset of uniform polyhedra. Seven of them can be A56⋅ 2 Truncated Icosahedron 60 90 32 formed by truncation of Platonic solids, and all of them can be A 2 Truncated Octahedron 24 36 14 formed by polyhedral operations defined as Conway polyhedron 4⋅6 A 2 Truncated Tetrahedron 12 18 8 notation [1-2]. 36⋅ The author has recently developed an interactive modeling system of uniform polyhedra including Platonic solids, Archimedean solids and Kepler-Poinsot solids, based on graph drawing and simulated elasticity, mainly for educational purpose [3-5]. -
Arxiv:1911.05648V2 [Math.CO] 28 Feb 2020 a Graph G Is K-Regular When the Degrees of All Vertices Are Equal to K.A Regular Graph Is One That Is K-Regular for Some K
2-nearly Platonic graphs are unique v07e D. Froncek, M.R. Khorsandi, S.R. Musawi, J. Qiu Abstract A 2-nearly Platonic graph of type (kjd) is a k-regular planar graph with f faces, f − 2 of which are of degree d and the remaining two are of degrees m1; m2, both different from d. Such a graph is called balanced if m1 = m2. We show that all 2-nearly Platonic graphs are necessarily balanced. This proves a recent conjecture by Keith, Froncek, and Kreher. Keywords: Planar graphs, regular graphs, Platonic graphs Mathematics Subject Classification: 05C10 1 Introduction Throughout this paper, all graphs we consider are finite, simple, connected, planar, undirected and non-trivial graph. A graph is said to be planar, or embeddable in the plane, if it can be drawn in the plane such that each common point of two edges is a vertex. This drawing of a planar graph G is called a planar embedding of G and can itself be regarded as a graph isomorphic to G. Sometimes, we call a planar embedding of a graph a plane graph. By this definition, it is clear that we need some matters of the topology of the plane. Immediately, after deleting the points of a plane graph from the plane, we have some maximal open sets (or regions) of points in the plane called faces of the plane graph. There exist exactly one unbounded region that we call it an outerface of the plane graph and other faces are called as internal faces. The boundary of a face is the set of points of vertices and edges touching the face.