Lecture Notes on Graph Theory
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Grade 7/8 Math Circles Graph Theory the Seven Bridges of Königsberg
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles Graph Theory October 13/14, 2015 The Seven Bridges of K¨onigsberg In the mid-1700s the was a city named K¨onigsberg. Today, the city is called Kaliningrad and is in modern day Russia. However, in the 1700s the city was a part of Prussia and had many Germanic influences. The city sits on the Pregel River. This divides the city into two main areas with the river running between them. In the river there are also two islands that are a part of the city. In 1736, a mathematician by the name of Leonhard Euler visited the city and was fascinated by the bridges. Euler wondered whether or not you could walk through the city and cross each bridge exactly once. Take a few minutes to see if you can find a way on the map of K¨onigsberg below. 1 The Three Utilities Problem Suppose you have a neighbourhood with only three houses. Now, each house needs to be connected to a set of three utilities (gas, water, and electricity) in order for a family to live there. Your challenge is to connect all three houses to each of the utilities. Below is a diagram for you to draw out where each utility line/pipe should go. The catch is that the lines you draw to connect each house to a utility can't cross. Also, you can't draw lines through another house or utility plant. In other words, you can't draw House 1's water line through House 2 or through the electricity plant. -
Enumeration of Unlabeled Graph Classes a Study of Tree Decompositions and Related Approaches
Enumeration of unlabeled graph classes A study of tree decompositions and related approaches Jessica Shi Advisor: Jérémie Lumbroso Independent Work Report Fall, 2015 Abstract In this paper, we study the enumeration of certain classes of graphs that can be fully charac- terized by tree decompositions; these classes are particularly significant due to the algorithmic improvements derived from tree decompositions on classically NP-complete problems on these classes [12, 7, 17, 35]. Previously, Chauve et al. [6] and Iriza [26] constructed grammars from the split decomposition trees of distance hereditary graphs and 3-leaf power graphs. We extend upon these results to obtain an upper bound grammar for parity graphs. Also, Nakano et al. [25] used the vertex incremental characterization of distance hereditary graphs to obtain upper bounds. We constructively enumerate (6;2)-chordal bipartite graphs, (C5, bull, gem, co-gem)-free graphs, and parity graphs using their vertex incremental characterization and extend upon Nakano et al.’s results to analytically obtain upper bounds of O7n and O11n for (6;2)-chordal bipartite graphs and (C5, bull, gem, co-gem)-free graphs respectively. 1. Introduction 1.1. Context The technique of decomposing graphs into trees has been an object of significant interest due to its applications on classical problems in graph theory. Indeed, many graph theoretic problems are inherently difficult due to the lack of recursive structure in graphs, and recursion has historically offered efficient solutions to challenging problems. In this sense, classifying graphs in terms of trees is of particular interest, since it associates a recursive structure to these graphs. -
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 ). -
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. -
Planar Graphs Lecture 10A: Properties
CS 163 & CS 265: Graph Algorithms Week 10: Planar graphs Lecture 10a: Properties David Eppstein University of California, Irvine Winter Quarter, 2021 This work is licensed under a Creative Commons Attribution 4.0 International License Examples of planar and nonplanar graphs The three utilities problem Input: three houses and three utilities, placed in the plane https://commons.wikimedia.org/wiki/File:3_utilities_problem_blank.svg How to draw lines connecting each house to each utility, without crossings (and without passing through other houses or utilities)? There is no solution! If you try it, you will get stuck::: https://commons.wikimedia.org/wiki/File:3_utilities_problem_plane.svg But this is not a satisfactory proof of impossibility Other surfaces have solutions If we place the houses and utilities on a torus (for instance, a coffee cup including the handle) we will be able to solve it https://commons.wikimedia.org/wiki/File:3_utilities_problem_torus.svg So solution depends somehow on the topology of the plane Planar graphs A planar graph is a graph that can be drawn without crossings in the plane The graph we are trying to draw in this case is K3;3, a complete bipartite graph with three houses and three utilities It is not a planar graph; the best we can do is to have only one crossing Examples of planar graphs: trees and grid graphs Examples of planar graphs: convex polyhedra Mostly but not entirely planar: road networks https://commons.wikimedia.org/wiki/File:High_Five.jpg [Eppstein and Gupta 2017] Properties Some properties of planar graphs I Euler's formula: If a connected planar graph has n = V vertices, m = E edges, and divides the plane into F faces (regions, including the region outside the drawing) then V − E + F = 2 I Simple planar graphs have ≤ 3n − 6 edges ) their degeneracy is ≤ 5 I Four-color theorem: Can be colored with ≤ 4 colors (Appel and Haken [1976]; degeneracy-based greedy gives ≤ 6) p I Planar separator theorem: treewidth is O( n) (Lipton and Tarjan [1979]; ) faster algorithms for many problems, e.g. -
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 . -
Planar Graphs and Coloring
Math 443/543 Graph Theory Notes 5: Planar graphs and coloring David Glickenstein October 10, 2014 1 Planar graphs The Three Houses and Three Utilities Problem: Given three houses and three utilities, can we connect each house to all three utilities so that the utility lines do not cross. We can represent this problem with a graph, connecting each house to each utility. We notice that this graph is bipartite: De…nition 1 A bipartite graph is one in which the vertices can be partitioned into two sets X and Y such that every edge joins one vertex in X with one vertex in Y: The partition (X; Y ) is called a bipartition.A complete bipartite graph is one such that each vertex of X is joined with every vertex of Y: The complete bipartite graph such that the order of X is m and the order of Y is n is denoted Km;n or K (m; n) : It is easy to see that the relevant graph in the problem above is K3;3: Now, we wish to embed this graph in the plane such that no two edges cross except at a vertex. De…nition 2 A planar graph is a graph that can be drawn in the plane such that no two edges cross except at a vertex. A planar graph drawn in the plane in a way such that no two edges cross except at a vertex is called a plane graph. Note that it says “can be drawn”not “is drawn.”The problem is that even if a graph is drawn with crossings, it could potentially be drawn in another way so that there are no crossings.