Voltage and Derived Graphs and Their Relation to the Free Product of Graphs
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
LDPC Codes from Voltage Graphs
ISIT 2008, Toronto, Canada, July 6 - 11, 2008 LDPC codes from voltage graphs Christine A. Kelley Judy L. Walker Department of Mathematics Department of Mathematics University of Nebraska-Lincoln University of Nebraska–Lincoln Lincoln, NE 68588, USA. Lincoln, NE 68588, USA. Email: [email protected] Email: [email protected] Abstract— Several well-known structure-based constructions we review the terminology of voltage graphs. In Section III, of LDPC codes, for example codes based on permutation and we show how graph-based codes can be obtained algebraically circulant matrices and in particular, quasi-cyclic LDPC codes, from voltage graphs, and illustrate this using the Sridhara- can be interpreted via algebraic voltage assignments. We explain this connection and show how this idea from topological graph Fuja-Tanner (SFT) LDPC codes and array-based LDPC codes theory can be used to give simple proofs of many known as examples. In Section IV, the notion of abelian-inevitable properties of these codes. In addition, the notion of abelian- cycles is introduced and the isomorphism classes of subgraphs inevitable cycle is introduced and the subgraphs giving rise to (or equivalently, matrix substructures) that give rise to these these cycles are classified. We also indicate how, by using more cycles in the LDPC Tanner graph are identified and classified. sophisticated voltage assignments, new classes of good LDPC codes may be obtained. The results presented here correct a result used in [16], [20], and, subsequently, extend the work from [16]. Ongoing work I. INTRODUCTION addressing how this method may be used to construct new Graph-based codes have attracted widespread interest due families of LDPC and other graph-based codes is outlined in to their efficient decoding algorithms and remarkable perfor- Section V. -
Applications of the Quantum Algorithm for St-Connectivity
Applications of the quantum algorithm for st-connectivity KAI DELORENZO1 , SHELBY KIMMEL1 , AND R. TEAL WITTER1 1Middlebury College, Computer Science Department, Middlebury, VT Abstract We present quantum algorithms for various problems related to graph connectivity. We give simple and query-optimal algorithms for cycle detection and odd-length cycle detection (bipartiteness) using a reduction to st-connectivity. Furthermore, we show that our algorithm for cycle detection has improved performance under the promise of large circuit rank or a small number of edges. We also provide algo- rithms for detecting even-length cycles and for estimating the circuit rank of a graph. All of our algorithms have logarithmic space complexity. 1 Introduction Quantum query algorithms are remarkably described by span programs [15, 16], a linear algebraic object originally created to study classical logspace complexity [12]. However, finding optimal span program algorithms can be challenging; while they can be obtained using a semidefinite program, the size of the program grows exponentially with the size of the input to the algorithm. Moreover, span programs are designed to characterize the query complexity of an algorithm, while in practice we also care about the time and space complexity. One of the nicest span programs is for the problem of undirected st-connectivity, in which one must decide whether two vertices s and t are connected in a given graph. It is “nice” for several reasons: It is easy to describe and understand why it is correct. • It corresponds to a quantum algorithm that uses logarithmic (in the number of vertices and edges of • the graph) space [4, 11]. -
The Normalized Laplacian Spectrum of Subdivisions of a Graph
The normalized Laplacian spectrum of subdivisions of a graph Pinchen Xiea,b, Zhongzhi Zhanga,c, Francesc Comellasd aShanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China bDepartment of Physics, Fudan University, Shanghai 200433, China cSchool of Computer Science, Fudan University, Shanghai 200433, China dDepartment of Applied Mathematics IV, Universitat Polit`ecnica de Catalunya, 08034 Barcelona Catalonia, Spain Abstract Determining and analyzing the spectra of graphs is an important and excit- ing research topic in mathematics science and theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spec- tra of the normalized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact val- ues of their multiplicative degree-Kirchhoff index, Kemeny's constant and number of spanning trees. Keywords: Normalized Laplacian spectrum, Subdivision graph, Degree-Kirchhoff index, Kemeny's constant, Spanning trees 1. Introduction Spectral analysis of graphs has been the subject of considerable research effort in mathematics and computer science [1, 2, 3], due to its wide appli- cations in this area and in general [4, 5]. In the last few decades a large body of scientific literature has established that important structural and dynamical properties -
Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory
Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 973920, 27 pages doi:10.1155/2012/973920 Research Article Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory Jianguo Tang,1, 2 Kun She,1 and William Zhu3 1 School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China 2 School of Computer Science and Engineering, XinJiang University of Finance and Economics, Urumqi 830012, China 3 Lab of Granular Computing, Zhangzhou Normal University, Zhangzhou 363000, China Correspondence should be addressed to William Zhu, [email protected] Received 4 February 2012; Revised 30 April 2012; Accepted 18 May 2012 Academic Editor: Mehmet Sezer Copyright q 2012 Jianguo Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Constructing structures with other mathematical theories is an important research field of rough sets. As one mathematical theory on sets, matroids possess a sophisticated structure. This paper builds a bridge between rough sets and matroids and establishes the matroidal structure of rough sets. In order to understand intuitively the relationships between these two theories, we study this problem from the viewpoint of graph theory. Therefore, any partition of the universe can be represented by a family of complete graphs or cycles. Then two different kinds of matroids are constructed and some matroidal characteristics of them are discussed, respectively. The lower and the upper approximations are formulated with these matroidal characteristics. -
Permutation of Elements in Double Semigroups
PERMUTATION OF ELEMENTS IN DOUBLE SEMIGROUPS MURRAY BREMNER AND SARA MADARIAGA Abstract. Double semigroups have two associative operations ◦, • related by the interchange relation: (a • b) ◦ (c • d) ≡ (a ◦ c) • (b ◦ d). Kock [13] (2007) discovered a commutativity property in degree 16 for double semigroups: as- sociativity and the interchange relation combine to produce permutations of elements. We show that such properties can be expressed in terms of cycles in directed graphs with edges labelled by permutations. We use computer alge- bra to show that 9 is the lowest degree for which commutativity occurs, and we give self-contained proofs of the commutativity properties in degree 9. 1. Introduction Definition 1.1. A double semigroup is a set S with two associative binary operations •, ◦ satisfying the interchange relation for all a,b,c,d ∈ S: (⊞) (a • b) ◦ (c • d) ≡ (a ◦ c) • (b ◦ d). The symbol ≡ indicates that the equation holds for all values of the variables. We interpret ◦ and • as horizontal and vertical compositions, so that (⊞) ex- presses the equivalence of two decompositions of a square array: a b a b a b (a ◦ b) • (c ◦ d) ≡ ≡ ≡ ≡ (a • c) ◦ (b • d). c d c d c d This interpretation of the operations extends to any double semigroup monomial, producing what we call the geometric realization of the monomial. The interchange relation originated in homotopy theory and higher categories; see Mac Lane [15, (2.3)] and [16, §XII.3]. It is also called the Godement relation by arXiv:1405.2889v2 [math.RA] 26 Mar 2015 some authors; see Simpson [21, §2.1]. -
G-Parking Functions and Tree Inversions
G-PARKING FUNCTIONS AND TREE INVERSIONS DAVID PERKINSON, QIAOYU YANG, AND KUAI YU Abstract. A depth-first search version of Dhar’s burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the spanning tree. Specializing to the complete graph solves a problem posed by R. Stanley. 1. Introduction Let G = (V, E) be a connected simple graph with vertex set V = {0,...,n} and edge set E. Fix a root vertex r ∈ V and let SPT(G) denote the set of spanning trees of G rooted at r. We think of each element of SPT(G) as a directed graph in which all paths lead away from the root. If i, j ∈ V and i lies on the unique path from r to j in the rooted spanning tree T , then i is an ancestor of j and j is a descendant of i in T . If, in addition, there are no vertices between i and j on the path from the root, then i is the parent of its child j, and (i, j) is a directed edge of T . Definition 1. An inversion of T ∈ SPT(G) is a pair of vertices (i, j), such that i is an ancestor of j in T and i > j. It is a κ-inversion if, in addition, i is not the root and i’s parent is adjacent to j in G. The number of κ-inversions of T is the tree’s κ-number, denoted κ(G, T ). -
Dynamic Cage Survey
Dynamic Cage Survey Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Robert Jajcay Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809, U.S.A. [email protected] Department of Algebra Comenius University Bratislava, Slovakia [email protected] Submitted: May 22, 2008 Accepted: Sep 15, 2008 Version 1 published: Sep 29, 2008 (48 pages) Version 2 published: May 8, 2011 (54 pages) Version 3 published: July 26, 2013 (55 pages) Mathematics Subject Classifications: 05C35, 05C25 Abstract A(k; g)-cage is a k-regular graph of girth g of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions. the electronic journal of combinatorics (2013), #DS16 1 Contents 1 Origins of the Problem 3 2 Known Cages 6 2.1 Small Examples . 6 2.1.1 (3,5)-Cage: Petersen Graph . 7 2.1.2 (3,6)-Cage: Heawood Graph . 7 2.1.3 (3,7)-Cage: McGee Graph . 7 2.1.4 (3,8)-Cage: Tutte-Coxeter Graph . 8 2.1.5 (3,9)-Cages . 8 2.1.6 (3,10)-Cages . 9 2.1.7 (3,11)-Cage: Balaban Graph . 9 2.1.8 (3,12)-Cage: Benson Graph . 9 2.1.9 (4,5)-Cage: Robertson Graph . 9 2.1.10 (5,5)-Cages . -
The Normalized Laplacian Spectrum of Subdivisions of a Graph
The normalized Laplacian spectrum of subdivisions of a graph Pinchen Xiea,c, Zhongzhi Zhangb,c, Francesc Comellasd aDepartment of Physics, Fudan University, Shanghai 200433, China bSchool of Computer Science, Fudan University, Shanghai 200433, China cShanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China dDepartment of Applied Mathematics IV, Universitat Polit`ecnica de Catalunya, 08034 Barcelona Catalonia, Spain Abstract Determining and analyzing the spectra of graphs is an important and exciting research topic in theoretical computer science. The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to random walks. In this paper, we give the spectra of the nor- malized Laplacian of iterated subdivisions of simple connected graphs. As an example of application of these results we find the exact values of their multiplicative degree-Kirchhoff index, Kemeny’s constant and number of spanning trees. Keywords: Normalized Laplacian spectrum, Subdivision graph, Degree-Kirchhoff index, Kemeny’s constant, Spanning trees 1. Introduction arXiv:1510.02394v1 [math.CO] 7 Oct 2015 Spectral analysis of graphs has been the subject of considerable research effort in theoretical computer science [1, 2, 3], due to its wide applications in this area and in general [4, 5]. In the last few decades a large body of scientific literature has established that important structural and dynamical properties -
Classes of Graphs Embeddable in Order-Dependent Surfaces
Classes of graphs embeddable in order-dependent surfaces Colin McDiarmid Sophia Saller Department of Statistics Department of Mathematics University of Oxford University of Oxford [email protected] and DFKI [email protected] 12 June 2021 Abstract Given a function g = g(n) we let Eg be the class of all graphs G such that if G has order n (that is, has n vertices) then it is embeddable in some surface of Euler genus at most g(n), and let eEg be the corresponding class of unlabelled graphs. We give estimates of the sizes of these classes. For example we 3 g show that if g(n) = o(n= log n) then the class E has growth constant γP, the (labelled) planar graph growth constant; and when g(n) = O(n) we estimate the number of n-vertex graphs in Eg and eEg up g to a factor exponential in n. From these estimates we see that, if E has growth constant γP then we must have g(n) = o(n= log n), and the generating functions for Eg and eEg have strictly positive radius of convergence if and only if g(n) = O(n= log n). Such results also hold when we consider orientable and non-orientable surfaces separately. We also investigate related classes of graphs where we insist that, as well as the graph itself, each subgraph is appropriately embeddable (according to its number of vertices); and classes of graphs where we insist that each minor is appropriately embeddable. In a companion paper [43], these results are used to investigate random n-vertex graphs sampled uniformly from Eg or from similar classes. -
Cheminformatics for Genome-Scale Metabolic Reconstructions
CHEMINFORMATICS FOR GENOME-SCALE METABOLIC RECONSTRUCTIONS John W. May European Molecular Biology Laboratory European Bioinformatics Institute University of Cambridge Homerton College A thesis submitted for the degree of Doctor of Philosophy June 2014 Declaration This thesis is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. This dissertation is not substantially the same as any I have submitted for a degree, diploma or other qualification at any other university, and no part has already been, or is currently being submitted for any degree, diploma or other qualification. This dissertation does not exceed the specified length limit of 60,000 words as defined by the Biology Degree Committee. This dissertation has been typeset using LATEX in 11 pt Palatino, one and half spaced, according to the specifications defined by the Board of Graduate Studies and the Biology Degree Committee. June 2014 John W. May to Róisín Acknowledgements This work was carried out in the Cheminformatics and Metabolism Group at the European Bioinformatics Institute (EMBL-EBI). The project was fund- ed by Unilever, the Biotechnology and Biological Sciences Research Coun- cil [BB/I532153/1], and the European Molecular Biology Laboratory. I would like to thank my supervisor, Christoph Steinbeck for his guidance and providing intellectual freedom. I am also thankful to each member of my thesis advisory committee: Gordon James, Julio Saez-Rodriguez, Kiran Patil, and Gos Micklem who gave their time, advice, and guidance. I am thankful to all members of the Cheminformatics and Metabolism Group. -
COLORING GRAPHS USING TOPOLOGY 11 Realized by Tutte Who Called an Example of a Disc G for Which the Boundary Has Chromatic Number 4 a Chromatic Obstacle
COLORING GRAPHS USING TOPOLOGY OLIVER KNILL Abstract. Higher dimensional graphs can be used to color 2- dimensional geometric graphs G. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colorable with 4 colors. The later goal is achieved if all inte- rior edge degrees are even. Using a refinement process which cuts the interior along surfaces S we can adapt the degrees along the boundary S. More efficient is a self-cobordism of G with itself with a host graph discretizing the product of G with an interval. It fol- lows from the fact that Euler curvature is zero everywhere for three dimensional geometric graphs, that the odd degree edge set O is a cycle and so a boundary if H is simply connected. A reduction to minimal coloring would imply the four color theorem. The method is expected to give a reason \why 4 colors suffice” and suggests that every two dimensional geometric graph of arbitrary degree and ori- entation can be colored by 5 colors: since the projective plane can not be a boundary of a 3-dimensional graph and because for higher genus surfaces, the interior H is not simply connected, we need in general to embed a surface into a 4-dimensional simply connected graph in order to color it. This explains the appearance of the chromatic number 5 for higher degree or non-orientable situations, a number we believe to be the upper limit. For every surface type, we construct examples with chromatic number 3; 4 or 5, where the construction of surfaces with chromatic number 5 is based on a method of Fisk. -
Graph Theory Trees
GGRRAAPPHH TTHHEEOORRYY -- TTRREEEESS http://www.tutorialspoint.com/graph_theory/graph_theory_trees.htm Copyright © tutorialspoint.com Trees are graphs that do not contain even a single cycle. They represent hierarchical structure in a graphical form. Trees belong to the simplest class of graphs. Despite their simplicity, they have a rich structure. Trees provide a range of useful applications as simple as a family tree to as complex as trees in data structures of computer science. Tree A connected acyclic graph is called a tree. In other words, a connected graph with no cycles is called a tree. The edges of a tree are known as branches. Elements of trees are called their nodes. The nodes without child nodes are called leaf nodes. A tree with ‘n’ vertices has ‘n-1’ edges. If it has one more edge extra than ‘n-1’, then the extra edge should obviously has to pair up with two vertices which leads to form a cycle. Then, it becomes a cyclic graph which is a violation for the tree graph. Example 1 The graph shown here is a tree because it has no cycles and it is connected. It has four vertices and three edges, i.e., for ‘n’ vertices ‘n-1’ edges as mentioned in the definition. Note − Every tree has at least two vertices of degree one. Example 2 In the above example, the vertices ‘a’ and ‘d’ has degree one. And the other two vertices ‘b’ and ‘c’ has degree two. This is possible because for not forming a cycle, there should be at least two single edges anywhere in the graph.