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8660 results in Discrete Mathematics, Information Theory and Coding

Page 74 of 433

  • Density of Real Zeros of the Tutte Polynomial

  • Part of
  • SEONGMIN OK, THOMAS J. PERRETT
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 3 / May 2018
    Published online by Cambridge University Press:
    09 March 2018, pp. 398-410

  • The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the real zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This is the first density result for the real zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.

  • A Sharp Dirac–Erdős Type Bound for Large Graphs

  • Part of
  • H. A. KIERSTEAD, A. V. KOSTOCHKA, A. McCONVEY
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 3 / May 2018
    Published online by Cambridge University Press:
    09 March 2018, pp. 387-397

  • Let k ⩾ 3 be an integer, hk(G) be the number of vertices of degree at least 2k in a graph G, and ℓk(G) be the number of vertices of degree at most 2k − 2 in G. Dirac and Erdős proved in 1963 that if hk(G) − ℓk(G) ⩾ k2 + 2k − 4, then G contains k vertex-disjoint cycles. For each k ⩾ 2, they also showed an infinite sequence of graphs Gk(n) with hk(Gk(n)) − ℓk(Gk(n)) = 2k − 1 such that Gk(n) does not have k disjoint cycles. Recently, the authors proved that, for k ⩾ 2, a bound of 3k is sufficient to guarantee the existence of k disjoint cycles, and presented for every k a graph G0(k) with hk(G0(k)) − ℓk(G0(k)) = 3k − 1 and no k disjoint cycles. The goal of this paper is to refine and sharpen this result. We show that the Dirac–Erdős construction is optimal in the sense that for every k ⩾ 2, there are only finitely many graphs G with hk(G) − ℓk(G) ⩾ 2k but no k disjoint cycles. In particular, every graph G with |V(G)| ⩾ 19k and hk(G) − ℓk(G) ⩾ 2k contains k disjoint cycles.

  • On Zeros of a Polynomial in a Finite Grid

  • Part of
  • ANURAG BISHNOI, PETE L. CLARK, ADITYA POTUKUCHI, JOHN R. SCHMITT
  • Journal:
    Combinatorics, Probability and Computing / Volume 27 / Issue 3 / May 2018
    Published online by Cambridge University Press:
    15 February 2018, pp. 310-333

  • A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain ‘Condition (D)’ on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further generalized Alon–Füredi theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon–Füredi. We then discuss the relationship between Alon–Füredi and results of DeMillo–Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon–Füredi theorem and its generalization in terms of Reed–Muller-type affine variety codes is shown, which gives us the minimum Hamming distance of these codes. Then we apply the Alon–Füredi theorem to quickly recover – and sometimes strengthen – old and new results in finite geometry, including the Jamison–Brouwer–Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

  • 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.

Page 74 of 433