On the Embeddings of the Semi-Strong Product Graph
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JMM 2017 Student Poster Session Abstract Book
Abstracts for the MAA Undergraduate Poster Session Atlanta, GA January 6, 2017 Organized by Eric Ruggieri College of the Holy Cross and Chasen Smith Georgia Southern University Organized by the MAA Committee on Undergraduate Student Activities and Chapters and CUPM Subcommittee on Research by Undergraduates Dear Students, Advisors, Judges and Colleagues, If you look around today you will see over 300 posters and nearly 500 student presenters, representing a wide array of mathematical topics and ideas. These posters showcase the vibrant research being conducted as part of summer programs and during the academic year at colleges and universities from across the United States and beyond. It is so rewarding to see this session, which offers such a great opportunity for interaction between students and professional mathematicians, continue to grow. The judges you see here today are professional mathematicians from institutions around the world. They are advisors, colleagues, new Ph.D.s, and administrators. We have acknowledged many of them in this booklet; however, many judges here volunteered on site. Their support is vital to the success of the session and we thank them. We are supported financially by Tudor Investments and Two Sigma. We are also helped by the mem- bers of the Committee on Undergraduate Student Activities and Chapters (CUSAC) in some way or other. They are: Dora C. Ahmadi; Jennifer Bergner; Benjamin Galluzzo; Kristina Cole Garrett; TJ Hitchman; Cynthia Huffman; Aihua Li; Sara Louise Malec; Lisa Marano; May Mei; Stacy Ann Muir; Andy Nieder- maier; Pamela A. Richardson; Jennifer Schaefer; Peri Shereen; Eve Torrence; Violetta Vasilevska; Gerard A. -
Graph Theory, Prentice -Hall of India
Annexure 145 www.ccsenet.org/jmr Journal of Mathematics Research Vol. 3, No. 3; August 2011 Product Cordial Labeling in the Context of Tensor Product of Graphs S K Vaidya (Corresponding author) Department of Mathematics, Saurashtra University Rajkot-360 005. GUJARAT, India E-mail: [email protected] N B Vyas Atmiya Institute of Technology and Science Rajkot-360 005. GUJARAT, India E-mail: [email protected] Received: March 10, 2011 Accepted: March 28, 2011 doi:10.5539/jmr.v3n3p83 Abstract For the graph G1 and G2 the tensor product is denoted by G1(T p)G2 which is the graph with vertex set V(G1(T p)G2) = V(G1) × V(G2) and edge set E(G1(T p)G2) = {(u1, v1), (u2, v2)/u1u2E(G1) and v1v2E(G2)}. The graph Pm(T p)Pn is disconnected for ∀m, n while the graphs Cm(T p)Cn and Cm(T p)Pn are disconnected for both m and n even. We prove that these graphs are product cordial graphs. In addition to this we show that the graphs obtained by joining the connected components of respective graphs by a path of arbitrary length also admit product cordial labeling. Keywords: Cordial labeling, Porduct cordial labeling, Tensor product AMS Subject classification (2010): 05C78. 1. Introduction We begin with simple, finite and undirected graph G = (V(G), E(G)). For standard terminology and notations we follow (West, D. B, 2001). The brief summary of definitions and relevant results are given below. 1.1 Definition: If the vertices of the graph are assigned values subject to certain condition(s) then it is known as graph labeling. -
Graph Operations and Upper Bounds on Graph Homomorphism Counts
Graph Operations and Upper Bounds on Graph Homomorphism Counts Luke Sernau March 9, 2017 Abstract We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any n-vertex, d-regular graph G and any graph H (possibly with loops), n n d d hom(G, H) ≤ max hom(Kd,d,H) 2 , hom(Kd+1,H) +1 , n o where hom(G, H) is the number of homomorphisms from G to H. By exploiting properties of the graph tensor product and graph exponentiation, we also find new infinite families of H for which the bound stated above on hom(G, H) holds for all n-vertex, d-regular G. In particular we show that if HWR is the complete looped path on three vertices, also known as the Widom-Rowlinson graph, then n d hom(G, HWR) ≤ hom(Kd+1,HWR) +1 for all n-vertex, d-regular G. This verifies a conjecture of Galvin. arXiv:1510.01833v3 [math.CO] 8 Mar 2017 1 Introduction Graph homomorphisms are an important concept in many areas of graph theory. A graph homomorphism is simply an adjacency-preserving map be- tween a graph G and a graph H. That is, for given graphs G and H, a function φ : V (G) → V (H) is said to be a homomorphism from G to H if for every edge uv ∈ E(G), we have φ(u)φ(v) ∈ E(H) (Here, as throughout, all graphs are simple, meaning without multi-edges, but they are permitted 1 to have loops). -
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. -
Coverings of Cubic Graphs and 3-Edge Colorability
Discussiones Mathematicae Graph Theory 41 (2021) 311–334 doi:10.7151/dmgt.2194 COVERINGS OF CUBIC GRAPHS AND 3-EDGE COLORABILITY Leonid Plachta AGH University of Science and Technology Krak´ow, Poland e-mail: [email protected] Abstract Let h: G˜ → G be a finite covering of 2-connected cubic (multi)graphs where G is 3-edge uncolorable. In this paper, we describe conditions under which G˜ is 3-edge uncolorable. As particular cases, we have constructed regular and irregular 5-fold coverings f : G˜ → G of uncolorable cyclically 4-edge connected cubic graphs and an irregular 5-fold covering g : H˜ → H of uncolorable cyclically 6-edge connected cubic graphs. In [13], Steffen introduced the resistance of a subcubic graph, a char- acteristic that measures how far is this graph from being 3-edge colorable. In this paper, we also study the relation between the resistance of the base cubic graph and the covering cubic graph. Keywords: uncolorable cubic graph, covering of graphs, voltage permuta- tion graph, resistance, nowhere-zero 4-flow. 2010 Mathematics Subject Classification: 05C15, 05C10. 1. Introduction 1.1. Motivation and statement of results We will consider proper edge colorings of G. Let ∆(G) be the maximum degree of vertices in a graph G. Denote by χ′(G) the minimum number of colors needed for (proper) edge coloring of G and call it the chromatic index of G. Recall that according to Vizing’s theorem (see [14]), either χ′(G) = ∆(G), or χ′(G) = ∆(G)+1. In this paper, we shall consider only subcubic graphs i.e., graphs G such that ∆(G) ≤ 3. -
Efficient Quantum Circuits for Szegedy Quantum Walks
Efficient quantum circuits for Szegedy quantum walks T. Loke and J.B. Wang∗ School of Physics, The University of Western Australia, Perth WA 6009, Australia Abstract A major advantage in using Szegedy's formalism over discrete-time and continuous-time quantum walks lies in its ability to define a unitary quantum walk on directed and weighted graphs. In this paper, we present a general scheme to construct efficient quantum circuits for Szegedy quantum walks that correspond to classical Markov chains possessing transforma- tional symmetry in the columns of the transition matrix. In particular, the transformational symmetry criteria do not necessarily depend on the sparsity of the transition matrix, so this scheme can be applied to non-sparse Markov chains. Two classes of Markov chains that are amenable to this construction are cyclic permutations and complete bipartite graphs, for which we provide explicit efficient quantum circuit implementations. We also prove that our scheme can be applied to Markov chains formed by a tensor product. We also briefly discuss the implementation of Markov chains based on weighted interdependent net- works. In addition, we apply this scheme to construct efficient quantum circuits simulating the Szegedy walks used in the quantum Pagerank algorithm for some classes of non-trivial graphs, providing a necessary tool for experimental demonstration of the quantum Pagerank algorithm. 1. Introduction In the last two decades, quantum walks have produced a wide variety of quantum algo- rithms to address different problems, such as the graph isomorphism problem [1{3], ranking nodes in a network [4{6], and quantum simulation [7{11]. -
Expander Graphs
Basic Facts about Expander Graphs Oded Goldreich Abstract. In this survey we review basic facts regarding expander graphs that are most relevant to the theory of computation. Keywords: Expander Graphs, Random Walks on Graphs. This text has been revised based on [8, Apdx. E.2]. 1 Introduction Expander graphs found numerous applications in the theory of computation, ranging from the design of sorting networks [1] to the proof that undirected connectivity is decidable in determinstic log-space [15]. In this survey we review basic facts regarding expander graphs that are most relevant to the theory of computation. For a wider perspective, the interested reader is referred to [10]. Loosely speaking, expander graphs are regular graphs of small degree that exhibit various properties of cliques.1 In particular, we refer to properties such as the relative sizes of cuts in the graph (i.e., relative to the number of edges), and the rate at which a random walk converges to the uniform distribution (relative to the logarithm of the graph size to the base of its degree). Some technicalities. Typical presentations of expander graphs refer to one of several variants. For example, in some sources, expanders are presented as bi- partite graphs, whereas in others they are presented as ordinary graphs (and are in fact very far from being bipartite). We shall follow the latter convention. Furthermore, at times we implicitly consider an augmentation of these graphs where self-loops are added to each vertex. For simplicity, we also allow parallel edges. We often talk of expander graphs while we actually mean an infinite collection of graphs such that each graph in this collection satisfies the same property (which is informally attributed to the collection). -
On the Generalized Θ-Number and Related Problems for Highly Symmetric Graphs
On the generalized #-number and related problems for highly symmetric graphs Lennart Sinjorgo ∗ Renata Sotirov y Abstract This paper is an in-depth analysis of the generalized #-number of a graph. The generalized #-number, #k(G), serves as a bound for both the k-multichromatic number of a graph and the maximum k-colorable subgraph problem. We present various properties of #k(G), such as that the series (#k(G))k is increasing and bounded above by the order of the graph G. We study #k(G) when G is the graph strong, disjunction and Cartesian product of two graphs. We provide closed form expressions for the generalized #-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs and orthogonality graphs. Our paper provides bounds on the product and sum of the k-multichromatic number of a graph and its complement graph, as well as lower bounds for the k-multichromatic number on several graph classes including the Hamming and Johnson graphs. Keywords k{multicoloring, k-colorable subgraph problem, generalized #-number, Johnson graphs, Hamming graphs, strongly regular graphs. AMS subject classifications. 90C22, 05C15, 90C35 1 Introduction The k{multicoloring of a graph is to assign k distinct colors to each vertex in the graph such that two adjacent vertices are assigned disjoint sets of colors. The k-multicoloring is also known as k-fold coloring, n-tuple coloring or simply multicoloring. We denote by χk(G) the minimum number of colors needed for a valid k{multicoloring of a graph G, and refer to it as the k-th chromatic number of G or the multichromatic number of G. -
Linkedness and Path-Pairability in the Cartesian Product of Graphs
Linkedness and Path-Pairability in the Cartesian Product of Graphs by Gábor Mészáros Submitted to Central European University Department of Mathematics and its Applications In partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics and its Applications Supervisor: Ervin Győri Alfréd Rényi Institute of Mathematics Budapest, Hungary 2015 i Contents List of Figures ii 1. Introduction 1 2. Notation, terminology, and folklore results 3 3. Linkedness 5 3.1. Linkedness and connectivity 6 3.2. Special graph classes 9 3.3. Related graph properties and generalizations 11 4. Path-pairability 16 4.1. Necessary and sufficient conditions 17 4.2. Maximum degree 18 4.3. Diameter 20 5. The Cartesian product of graphs and parameter inheritance 25 5.1. Main results 28 5.2. Results for grid graphs 40 5.3. Path-pairable products 47 6. Additional remarks and open questions 52 6.1. Path-pairable planar graphs 52 6.2. Linkedness and path-pairability of directed graphs 55 6.3. Other graph products 57 References 59 ii List of Figures 1 Example for a 5-connected, not 2-linked graph. 61 2 Planar graphs are not 3-linked. 61 3 Path-pairable graph of order 12 61 4 Cartesian product of a claw and a triangle 62 5 Line-up and final match phases. 62 6 Small path-pairable planar graphs can be easily constructed. 62 7 Linking via edge-disjoint directed cycles. 63 8 Four kinds of products of K2 and P2 63 iii Abstract In this dissertation I summarize my work in the field of linkedness and path- pairability of graphs with primary focus on the inheritance of the mentioned prop- erties in the Cartesian product of graphs. -
Not Every Bipartite Double Cover Is Canonical
Volume 82 BULLETIN of the February 2018 INSTITUTE of COMBINATORICS and its APPLICATIONS Editors-in-Chief: Marco Buratti, Donald Kreher, Tran van Trung Boca Raton, FL, U.S.A. ISSN1182-1278 BULLETIN OF THE ICA Volume 82 (2018), Pages 51{55 Not every bipartite double cover is canonical Tomaˇz Pisanski University of Primorska, FAMNIT, Koper, Slovenia and IMFM, Ljubljana, Slovenia. [email protected]. Abstract: It is explained why the term bipartite double cover should not be used to designate canonical double cover alias Kronecker cover of graphs. Keywords: Kronecker cover, canonical double cover, bipartite double cover, covering graphs. Math. Subj. Class. (2010): 05C76, 05C10. It is not uncommon to use different terminology or notation for the same mathematical concept. There are several reasons for such a phenomenon. Authors independently discover the same object and have to name it. It is• quite natural that they choose different names. Sometimes the same concept appears in different mathematical schools or in• different disciplines. By using particular terminology in a given context makes understanding of such a concept much easier. Sometimes original terminology is not well-chosen, not intuitive and it is difficult• to relate the name of the object to its meaning. A name that is more appropriate for a given concept is preferred. One would expect terminology to be as simple as possible, and as easy to connect it to the concept as possible. However, it should not be too simple. Received: 8 October 2018 51 Accepted: 21 January 2018 In other words it should not introduce ambiguity. Unfortunately, the term bipartite double cover that has been used lately in several places to replace an older term canonical double cover, also known as the Kronecker cover, [1,4], is ambiguous. -
Free Groups and Graphs: the Hanna Neumann Theorem
FREE GROUPS AND GRAPHS: THE HANNA NEUMANN THEOREM LAUREN COTE Abstract. The relationship between free groups and graphs is one of the many examples of the interaction between seemingly foreign branches of math- ematics. After defining and proving elementary relations between graphs and groups, this paper will present a result of Hanna Neumann on the intersection of free subgroups. Although Neumann's orginal upper bound on the rank of the intersection has been improved upon, her conjecture, that the upper bound is half of the one she originally proved, remains an open question. Contents 1. Groups 1 2. Graphs 3 3. Hanna Neumann Theorem 9 References 11 1. Groups Definition 1.1. Recall that a group G is a set S and a binary operation · with the properties that G is closed under ·, · is associative, there exists an identity e such that for all g 2 (G); g · e = e · g = g, and there exist inverses such that for each g 2 G; there exists g−1 2 G such that g · g−1 = g−1 · g = e. When working with groups, it is helpful to review the definitions of homomor- phism, an operation perserving map. For groups G1 = (S1; ·), G2 = (S2;?), the homomorphism φ : G1 ! G2 has the property that φ(x · y) = φ(x) ? φ(y). An isomorphism is a bijective homomorphism. Definition 1.2. This paper will focus on a particular kind of group, those which have the least possible amount of restriction: free groups. Let i : S ! F . A group F is free on a set S if for all groups G and for all f : S ! G, there exists a unique homomorphism : F ! G so that i ◦ = f. -
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 .