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9085 results in Discrete Mathematics, Information Theory and Coding

Page 88 of 455

  • 11 - Nontrees

  • Charles R. Johnson, College of William and Mary, Virginia, Carlos M. Saiago, Universidade Nova de Lisboa, Portugal
  • Book:
    Eigenvalues, Multiplicities and Graphs
    Published online:
    09 February 2018
    Print publication:
    12 February 2018, pp 211-231

  • Summary

    Introduction and Observations

    The problem of multiplicity lists for Hermitian matrices is remarkably different when the underlying graph G is not a tree. Some highly algebraic aspects of the problem are the same. For example,

  • • Maximum multiplicity in H(G) corresponds to minimum rank in H(G).

  • • Multiplicity can change by at most 1 when a vertex is removed.

  • • Multiplicity can change under perturbation by at most the rank of the perturbation.

  • • Many of the results on perturbation of diagonal entries (Section 4.1) are the same.

    • However, as seen in Chapter 2, there is little vestige of the technology of Parter vertices for general graphs, and it maywell be that all vertices are downers, even when multiple eigenvalues are present. Two of the construction techniques we have described in Chapter 7 are almost not available, and the third, the IFT, is more problematic to carry out. The similarities above are too weak to be very helpful, and minimum rank for general graphs is known to be very subtle, while it has a lovely answer for trees. Thus, determination of possible multiplicities is even more difficult for general graphs and often involves looking at very particular cases.

    There are some things that have been said, or can be said, and we review a selection of them here.

    The Complete Graph

    The complete graph Kn on n ≥ 2 vertices is relatively easy to analyze, as it allows many multiplicity lists. However, it requires different techniques to verify that various spectra occur, and these techniques are useful for other dense graphs that we pursue in Section 11.7. One list that Kn obviously does not allow is just one eigenvalue of multiplicity n. An Hermitian matrix with this list would be similar to, and thus, equal to, a multiple of I. It would be diagonal and thus is allowed only by the trivial graph with n vertices and no edges.

  • Preface

  • Charles R. Johnson, College of William and Mary, Virginia, Carlos M. Saiago, Universidade Nova de Lisboa, Portugal
  • Book:
    Eigenvalues, Multiplicities and Graphs
    Published online:
    09 February 2018
    Print publication:
    12 February 2018, pp xiii-xvi

  • Summary

    Among the n eigenvalues of an n-by-n matrix may be several repetitions (the number of which counts toward the total of n). For general matrices over a general field, these multiplicities may be algebraic (the number of appearances as a root of the characteristic polynomial) or geometric (the dimension of the corresponding eigenspace). These multiplicities are quite important in the analysis of matrix structure because of numerical calculation, a variety of applications, and for theoretical interest. We are primarily concerned with geometric multiplicities and, in particular but not exclusively, with real symmetric or complex Hermitian matrices, for which the two notions of multiplicity coincide.

    It has been known for some time, and is not surprising, that the arrangement of nonzero entries of a matrix, conveniently described by the graph of the matrix, limits the possible geometric multiplicities of the eigenvalues. Much less limited by this information are either the algebraic multiplicities or the numerical values of the (distinct) eigenvalues. So, it is natural to study exactly how the graph of a matrix limits the possible geometric eigenvalue multiplicities.

    Organized study of “eigenvalues, multiplicities and graphs” really began in the 1990s, though two earlier papers, [P] and [Wie], play an important role, including motivational. There had also been considerable interest in the eigenvalues of particular matrices with a given graph, such as the adjacency or Laplacian matrix. It was recognized early that the theory is most rich in case the graph is minimally connected, i.e., a tree. For this reason, the theory is relatively well developed for trees. However, in recent papers and in the preparation of this monograph, there has been an opportunity to identify more clearly which ideas have analogs for nontrees and for which ideas trees are essential. We have also recently noticed that for trees, and sometimes for general graphs, ideas about real symmetric/complex Hermitian matrices carry over to geometric multiplicities in general matrices over a field, sometimes under a diagonalizability hypothesis. This is an important advance that we have included herein (Chapter 12); the proofs are necessarily very different, and we have also included earlier proofs for the symmetric case, which are of interest for themselves and support other work.

Eigenvalues, Multiplicities and Graphs
  • Charles R. Johnson, Carlos M. Saiago
  • Published online:
    09 February 2018
    Print publication:
    12 February 2018
  • The arrangement of nonzero entries of a matrix, described by the graph of the matrix, limits the possible geometric multiplicities of the eigenvalues, which are far more limited by this information than algebraic multiplicities or the numerical values of the eigenvalues. This book gives a unified development of how the graph of a symmetric matrix influences the possible multiplicities of its eigenvalues. While the theory is richest in cases where the graph is a tree, work on eigenvalues, multiplicities and graphs has provided the opportunity to identify which ideas have analogs for non-trees, and those for which trees are essential. It gathers and organizes the fundamental ideas to allow students and researchers to easily access and investigate the many interesting questions in the subject.

  • Ramsey Goodness of Bounded Degree Trees

  • Part of
  • IGOR BALLA, ALEXEY POKROVSKIY, BENNY SUDAKOV
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 3 / May 2018
    Published online by Cambridge University Press:
    16 January 2018, pp. 289-309

  • Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red–blue colouring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If a graph G is connected, it is well known and easy to show that R(G, H) ≥ (|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ(H) is the size of the smallest colour class in a χ(H)-colouring of H. A graph G is called H-good if R(G, H) = (|G|−1)(χ(H)−1)+σ(H). The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then.

    In this paper we show that if n≥ Ω(|H| log4 |H|) then every n-vertex bounded degree tree T is H-good. The dependency between n and |H| is tight up to log factors. This substantially improves a result of Erdős, Faudree, Rousseau, and Schelp from 1985, who proved that n-vertex bounded degree trees are H-good when n ≥ Ω(|H|4).

Page 88 of 455