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9084 results in Discrete Mathematics, Information Theory and Coding

Page 47 of 455

  • Removal lemmas and approximate homomorphisms

  • Part of
  • Jacob Fox, Yufei Zhao
  • Journal:
    Combinatorics, Probability and Computing / Volume 31 / Issue 4 / July 2022
    Published online by Cambridge University Press:
    24 January 2022, pp. 721-736
    • Article
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  • We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists M such that every triangle-free graph G has an $\epsilon$-approximate homomorphism to a triangle-free graph F on at most M vertices (here an $\epsilon$-approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $\epsilon \left\lvert{V(G)}\right\rvert^2$ edges of G are mapped to edges of F). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $\epsilon^{-1}$. We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.

Frameworks, Tensegrities, and Symmetry
  • Robert Connelly, Simon D. Guest
  • Published online:
    13 January 2022
    Print publication:
    27 January 2022
  • This introduction to the theory of rigid structures explains how to analyze the performance of built and natural structures under loads, paying special attention to the role of geometry. The book unifies the engineering and mathematical literatures by exploring different notions of rigidity - local, global, and universal - and how they are interrelated. Important results are stated formally, but also clarified with a wide range of revealing examples. An important generalization is to tensegrities, where fixed distances are replaced with 'cables' not allowed to increase in length and 'struts' not allowed to decrease in length. A special feature is the analysis of symmetric tensegrities, where the symmetry of the structure is used to simplify matters and allows the theory of group representations to be applied. Written for researchers and graduate students in structural engineering and mathematics, this work is also of interest to computer scientists and physicists.

  • 12 - Parameter table

  • Andries E. Brouwer, Technische Universiteit Eindhoven, The Netherlands, H. Van Maldeghem, Universiteit Gent, Belgium
  • Book:
    Strongly Regular Graphs
    Published online:
    06 January 2022
    Print publication:
    13 January 2022, pp 399-424

  • Summary

    In this chapter we give a table with the feasible parameter sets of arbitrary strongly regular graphs on at most 512 vertices, and add comments about the known examples. These include: existence (and number of nonisomorphic examples), the parameters, the spectrum, information about being a descendant of a regular two-graph or being in the switching class of a regular two-graph, possible relation with a projective two-weight code,possible relation with a partial geometry, whether it is a conference graph or transversal design. Miscellaneous comments include references to earlier parts of the book, name of the graph, reason for non-existence, possible relation with a Steiner system, etc.

  • 4 - Buildings

  • Andries E. Brouwer, Technische Universiteit Eindhoven, The Netherlands, H. Van Maldeghem, Universiteit Gent, Belgium
  • Book:
    Strongly Regular Graphs
    Published online:
    06 January 2022
    Print publication:
    13 January 2022, pp 114-138

  • Summary

    In the chapter we introduce (spherical) buildings. We develop the theory in some detail, sometimes providing proofs. We introduce the shadow geometries and discuss some properties of particular instances in detail. To that end we use “chain calculus”, which provides an efficient way to determine the diameter of a given shadow geometry, or the maximal distance between two generic objects of distinct type. We hence deduce that the shadow geometry of type E(6,1) yields a strongly regular graph. We provide an explicit construction of that geometry using a split octonion algebra. We also discuss the Klein correspondence, and we discuss triality, again with the aid of a split octonion algebra, and use this to construct the split Cayley hexagon over any field.We deduce a rank 4 representation of a corresponding strongly regular graph.

Page 47 of 455