What Are Graphs?
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On the Archimedean Or Semiregular Polyhedra
ON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA Mark B. Villarino Depto. de Matem´atica, Universidad de Costa Rica, 2060 San Jos´e, Costa Rica May 11, 2005 Abstract We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler’s polyhedral formula. Contents 1 Introduction 2 1.1 RegularPolyhedra .............................. 2 1.2 Archimedean/semiregular polyhedra . ..... 2 2 Proof techniques 3 2.1 Euclid’s proof for regular polyhedra . ..... 3 2.2 Euler’s polyhedral formula for regular polyhedra . ......... 4 2.3 ProofsofArchimedes’theorem. .. 4 3 Three lemmas 5 3.1 Lemma1.................................... 5 3.2 Lemma2.................................... 6 3.3 Lemma3.................................... 7 4 Topological Proof of Archimedes’ theorem 8 arXiv:math/0505488v1 [math.GT] 24 May 2005 4.1 Case1: fivefacesmeetatavertex: r=5. .. 8 4.1.1 At least one face is a triangle: p1 =3................ 8 4.1.2 All faces have at least four sides: p1 > 4 .............. 9 4.2 Case2: fourfacesmeetatavertex: r=4 . .. 10 4.2.1 At least one face is a triangle: p1 =3................ 10 4.2.2 All faces have at least four sides: p1 > 4 .............. 11 4.3 Case3: threefacesmeetatavertes: r=3 . ... 11 4.3.1 At least one face is a triangle: p1 =3................ 11 4.3.2 All faces have at least four sides and one exactly four sides: p1 =4 6 p2 6 p3. 12 4.3.3 All faces have at least five sides and one exactly five sides: p1 =5 6 p2 6 p3 13 1 5 Summary of our results 13 6 Final remarks 14 1 Introduction 1.1 Regular Polyhedra A polyhedron may be intuitively conceived as a “solid figure” bounded by plane faces and straight line edges so arranged that every edge joins exactly two (no more, no less) vertices and is a common side of two faces. -
Chapter 7 Planar Graphs
Chapter 7 Planar graphs In full: 7.1–7.3 Parts of: 7.4, 7.6–7.8 Skip: 7.5 Prof. Tesler Math 154 Winter 2020 Prof. Tesler Ch. 7: Planar Graphs Math 154 / Winter 2020 1 / 52 Planar graphs Definition A planar embedding of a graph is a drawing of the graph in the plane without edges crossing. A graph is planar if a planar embedding of it exists. Consider two drawings of the graph K4: V = f1, 2, 3, 4g E = f1, 2g , f1, 3g , f1, 4g , f2, 3g , f2, 4g , f3, 4g 1 2 1 2 3 4 3 4 Non−planar embedding Planar embedding The abstract graph K4 is planar because it can be drawn in the plane without crossing edges. Prof. Tesler Ch. 7: Planar Graphs Math 154 / Winter 2020 2 / 52 How about K5? Both of these drawings of K5 have crossing edges. We will develop methods to prove that K5 is not a planar graph, and to characterize what graphs are planar. Prof. Tesler Ch. 7: Planar Graphs Math 154 / Winter 2020 3 / 52 Euler’s Theorem on Planar Graphs Let G be a connected planar graph (drawn w/o crossing edges). Define V = number of vertices E = number of edges F = number of faces, including the “infinite” face Then V - E + F = 2. Note: This notation conflicts with standard graph theory notation V and E for the sets of vertices and edges. Alternately, use jV(G)j - jE(G)j + jF(G)j = 2. Example face 3 V = 4 E = 6 face 1 F = 4 face 4 (infinite face) face 2 V - E + F = 4 - 6 + 4 = 2 Prof. -
Graph Theory
1 Graph Theory “Begin at the beginning,” the King said, gravely, “and go on till you come to the end; then stop.” — Lewis Carroll, Alice in Wonderland The Pregolya River passes through a city once known as K¨onigsberg. In the 1700s seven bridges were situated across this river in a manner similar to what you see in Figure 1.1. The city’s residents enjoyed strolling on these bridges, but, as hard as they tried, no residentof the city was ever able to walk a route that crossed each of these bridges exactly once. The Swiss mathematician Leonhard Euler learned of this frustrating phenomenon, and in 1736 he wrote an article [98] about it. His work on the “K¨onigsberg Bridge Problem” is considered by many to be the beginning of the field of graph theory. FIGURE 1.1. The bridges in K¨onigsberg. J.M. Harris et al., Combinatorics and Graph Theory , DOI: 10.1007/978-0-387-79711-3 1, °c Springer Science+Business Media, LLC 2008 2 1. Graph Theory At first, the usefulness of Euler’s ideas and of “graph theory” itself was found only in solving puzzles and in analyzing games and other recreations. In the mid 1800s, however, people began to realize that graphs could be used to model many things that were of interest in society. For instance, the “Four Color Map Conjec- ture,” introduced by DeMorgan in 1852, was a famous problem that was seem- ingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions are colored differently. -
Archimedean Solids
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat). -
An Introduction to Graph Colouring
An Introduction to Graph Colouring Evelyne Smith-Roberge University of Waterloo March 29, 2017 Recap... Last week, we covered: I What is a graph? I Eulerian circuits I Hamiltonian Cycles I Planarity Reminder: A graph G is: I a set V (G) of objects called vertices together with: I a set E(G), of what we call called edges. An edge is an unordered pair of vertices. We call two vertices adjacent if they are connected by an edge. Today, we'll get into... I Planarity in more detail I The four colour theorem I Vertex Colouring I Edge Colouring Recall... Planarity We said last week that a graph is planar if it can be drawn in such a way that no edges cross. These areas, including the infinite area surrounding the graph, are called faces. We denote the set of faces of a graph G by F (G). Planarity You'll notice the edges of planar graphs cut up our space into different sections. ) Planarity You'll notice the edges of planar graphs cut up our space into different sections. ) These areas, including the infinite area surrounding the graph, are called faces. We denote the set of faces of a graph G by F (G). Degree of a Vertex Last week, we defined the degree of a vertex to be the number of edges that had that vertex as an endpoint. In this graph, for example, each of the vertices has degree 3. Degree of a Face Similarly, we can define the degree of a face. The degree of a face is the number of edges that make up the boundary of that face. -
Graph Mining: Social Network Analysis and Information Diffusion
Graph Mining: Social network analysis and Information Diffusion Davide Mottin, Konstantina Lazaridou Hasso Plattner Institute Graph Mining course Winter Semester 2016 Lecture road Introduction to social networks Real word networks’ characteristics Information diffusion GRAPH MINING WS 2016 2 Paper presentations ▪ Link to form: https://goo.gl/ivRhKO ▪ Choose the date(s) you are available ▪ Choose three papers according to preference (there are three lists) ▪ Choose at least one paper for each course part ▪ Register to the mailing list: https://lists.hpi.uni-potsdam.de/listinfo/graphmining-ws1617 GRAPH MINING WS 2016 3 What is a social network? ▪Oxford dictionary A social network is a network of social interactions and personal relationships GRAPH MINING WS 2016 4 What is an online social network? ▪ Oxford dictionary An online social network (OSN) is a dedicated website or other application, which enables users to communicate with each other by posting information, comments, messages, images, etc. ▪ Computer science : A network that consists of a finite number of users (nodes) that interact with each other (edges) by sharing information (labels, signs, edge directions etc.) GRAPH MINING WS 2016 5 Definitions ▪Vertex/Node : a user of the social network (Twitter user, an author in a co-authorship network, an actor etc.) ▪Edge/Link/Tie : the connection between two vertices referring to their relationship or interaction (friend-of relationship, re-tweeting activity, etc.) Graph G=(V,E) symbols meaning V users E connections deg(u) node degree: -
V=Ih0cpr745fm
MITOCW | watch?v=Ih0cPR745fM The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Today we have a lecturer, guest lecture two of two, Costis Daskalakis. COSTIS Glad to be back. So let's continue on the path we followed last time. Let me remind DASKALAKIS: you what we did last time, first of all. So I talked about interesting theorems in topology-- Nash, Sperner, and Brouwer. And I defined the corresponding-- so these were theorems in topology. Define the corresponding problems. And because of these existence theorems, the corresponding search problems were total. And then I looked into the problems in NP that are total, and I tried to identify what in these problems make them total and tried to identify combinatorial argument that guarantees the existence of solutions in these problems. Motivated by the argument, which turned out to be a parity argument on directed graphs, I defined the class PPAD, and I introduced the problem of ArithmCircuitSAT, which is PPAD complete, and from which I promised to show a bunch of PPAD hardness deductions this time. So let me remind you the salient points from this list before I keep going. So first of all, the PPAD class has a combinatorial flavor. In the definition of the class, what I'm doing is I'm defining a graph on all possible n-bit strings, so an exponentially large set by providing two circuits, P and N. -
What Is a Polyhedron?
3. POLYHEDRA, GRAPHS AND SURFACES 3.1. From Polyhedra to Graphs What is a Polyhedron? Now that we’ve covered lots of geometry in two dimensions, let’s make things just a little more difficult. We’re going to consider geometric objects in three dimensions which can be made from two-dimensional pieces. For example, you can take six squares all the same size and glue them together to produce the shape which we call a cube. More generally, if you take a bunch of polygons and glue them together so that no side gets left unglued, then the resulting object is usually called a polyhedron.1 The corners of the polygons are called vertices, the sides of the polygons are called edges and the polygons themselves are called faces. So, for example, the cube has 8 vertices, 12 edges and 6 faces. Different people seem to define polyhedra in very slightly different ways. For our purposes, we will need to add one little extra condition — that the volume bound by a polyhedron “has no holes”. For example, consider the shape obtained by drilling a square hole straight through the centre of a cube. Even though the surface of such a shape can be constructed by gluing together polygons, we don’t consider this shape to be a polyhedron, because of the hole. We say that a polyhedron is convex if, for each plane which lies along a face, the polyhedron lies on one side of that plane. So, for example, the cube is a convex polyhedron while the more complicated spec- imen of a polyhedron pictured on the right is certainly not convex. -
1 Bipartite Matching and Vertex Covers
ORF 523 Lecture 6 Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Any typos should be emailed to a a [email protected]. In this lecture, we will cover an application of LP strong duality to combinatorial optimiza- tion: • Bipartite matching • Vertex covers • K¨onig'stheorem • Totally unimodular matrices and integral polytopes. 1 Bipartite matching and vertex covers Recall that a bipartite graph G = (V; E) is a graph whose vertices can be divided into two disjoint sets such that every edge connects one node in one set to a node in the other. Definition 1 (Matching, vertex cover). A matching is a disjoint subset of edges, i.e., a subset of edges that do not share a common vertex. A vertex cover is a subset of the nodes that together touch all the edges. (a) An example of a bipartite (b) An example of a matching (c) An example of a vertex graph (dotted lines) cover (grey nodes) Figure 1: Bipartite graphs, matchings, and vertex covers 1 Lemma 1. The cardinality of any matching is less than or equal to the cardinality of any vertex cover. This is easy to see: consider any matching. Any vertex cover must have nodes that at least touch the edges in the matching. Moreover, a single node can at most cover one edge in the matching because the edges are disjoint. As it will become clear shortly, this property can also be seen as an immediate consequence of weak duality in linear programming. Theorem 1 (K¨onig). If G is bipartite, the cardinality of the maximum matching is equal to the cardinality of the minimum vertex cover. -
An Enduring Error
Version June 5, 2008 Branko Grünbaum: An enduring error 1. Introduction. Mathematical truths are immutable, but mathematicians do make errors, especially when carrying out non-trivial enumerations. Some of the errors are "innocent" –– plain mis- takes that get corrected as soon as an independent enumeration is carried out. For example, Daublebsky [14] in 1895 found that there are precisely 228 types of configurations (123), that is, collections of 12 lines and 12 points, each incident with three of the others. In fact, as found by Gropp [19] in 1990, the correct number is 229. Another example is provided by the enumeration of the uniform tilings of the 3-dimensional space by Andreini [1] in 1905; he claimed that there are precisely 25 types. However, as shown [20] in 1994, the correct num- ber is 28. Andreini listed some tilings that should not have been included, and missed several others –– but again, these are simple errors easily corrected. Much more insidious are errors that arise by replacing enumeration of one kind of ob- ject by enumeration of some other objects –– only to disregard the logical and mathematical distinctions between the two enumerations. It is surprising how errors of this type escape detection for a long time, even though there is frequent mention of the results. One example is provided by the enumeration of 4-dimensional simple polytopes with 8 facets, by Brückner [7] in 1909. He replaces this enumeration by that of 3-dimensional "diagrams" that he inter- preted as Schlegel diagrams of convex 4-polytopes, and claimed that the enumeration of these objects is equivalent to that of the polytopes. -
The Power of Shaking Hands
The power of shaking hands Dion Gijswijt Although very simple to prove, the handshaking lemma can be a powerful Observe that the number of neighbours of a Hamiltonian path P between s and u is equal to d(u) − 1. It follows that such a Hamiltonian path has odd tool in the hands of a combinatorialist. Here, I will show you some degree in H if and only if u = t. By the handshaking lemma, H has an even number of odd degree nodes, which means that G has an even number colourful applications and pose some challenges to you, dear reader. of Hamiltonian paths between s and t. Let me start by posing a simple question. Perhaps you already know the answer. As an example, consider the graph in Figure 1. Nodes s and t have degree 2 If not, take a minute or two to see if you can solve it! and all other nodes have degree 3. One Hamiltonian path between s and t is indicated. By Smith’s theorem, there are an even number of such paths, hence There are seven people at a party. Is it possible that each of them there must be at least one other such path. Can you find it? shakes hands with exactly three others? In the language of graph theory, we are asking for a graph1 with 7 nodes in which every node has degree 3. The following simple observation will be a central idea in this article. s t Lemma 1 (handshaking lemma). Let G be a graph. -
A Multigraph Approach to Social Network Analysis
1 Introduction Network data involving relational structure representing interactions between actors are commonly represented by graphs where the actors are referred to as vertices or nodes, and the relations are referred to as edges or ties connecting pairs of actors. Research on social networks is a well established branch of study and many issues concerning social network analysis can be found in Wasserman and Faust (1994), Carrington et al. (2005), Butts (2008), Frank (2009), Kolaczyk (2009), Scott and Carrington (2011), Snijders (2011), and Robins (2013). A common approach to social network analysis is to only consider binary relations, i.e. edges between pairs of vertices are either present or not. These simple graphs only consider one type of relation and exclude the possibility for self relations where a vertex is both the sender and receiver of an edge (also called edge loops or just shortly loops). In contrast, a complex graph is defined according to Wasserman and Faust (1994): If a graph contains loops and/or any pairs of nodes is adjacent via more than one line the graph is complex. [p. 146] In practice, simple graphs can be derived from complex graphs by collapsing the multiple edges into single ones and removing the loops. However, this approach discards information inherent in the original network. In order to use all available network information, we must allow for multiple relations and the possibility for loops. This leads us to the study of multigraphs which has not been treated as extensively as simple graphs in the literature. As an example, consider a network with vertices representing different branches of an organ- isation.