The Tau Constant of a Metrized Graph and its Behavior under Graph Operations

Zubeyir Cinkir

doi:10.37236/568

The Tau Constant of a Metrized Graph and its Behavior under Graph Operations

The Electronic Journal of Combinatorics ◽

10.37236/568 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 1

Author(s):

Zubeyir Cinkir

Keyword(s):

Lower Bound ◽

Algebraic Curves ◽

Arithmetic Properties ◽

Graph Operations ◽

Edge Contraction ◽

Arbitrary Graphs

This paper concerns the tau constant, which is an important invariant of a metrized graph, and which has applications to arithmetic properties of algebraic curves. We give several formulas for the tau constant, and show how it changes under graph operations including deletion of an edge, contraction of an edge, and union of graphs along one or two points. We show how the tau constant changes when edges of a graph are replaced by arbitrary graphs. We prove Baker and Rumely's lower bound conjecture on the tau constant for several classes of metrized graphs.

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The Tau Constant and the Edge Connectivity of a Metrized Graph

The Electronic Journal of Combinatorics ◽

10.37236/2934 ◽

2012 ◽

Vol 19 (4) ◽

Cited By ~ 1

Author(s):

Zubeyir Cinkir

Keyword(s):

Lower Bound ◽

Algebraic Curves ◽

Regular Graphs ◽

Graph Invariants ◽

Edge Connectivity ◽

Kirchhoff Index ◽

Arithmetic Properties

The tau constant is an important invariant of a metrized graph. It has connections to other graph invariants such as Kirchhoff index, and it has applications to arithmetic properties of algebraic curves. We show how the tau constant of a metrized graph changes under successive edge contractions and deletions. We prove identities which we call "contraction", "deletion", and "contraction-deletion" identities on a metrized graph. By establishing a lower bound for the tau constant in terms of the edge connectivity, we prove that Baker and Rumely's lower bound conjecture on the tau constant holds for metrized graphs with edge connectivity 5 or more. We show that proving this conjecture for 3-regular graphs is enough to prove it for all graphs.

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Distinct Distances on Algebraic Curves in the Plane

Combinatorics Probability Computing ◽

10.1017/s0963548316000225 ◽

2016 ◽

Vol 26 (1) ◽

pp. 99-117 ◽

Cited By ~ 4

Author(s):

JÁNOS PACH ◽

FRANK DE ZEEUW

Keyword(s):

Lower Bound ◽

Algebraic Curve ◽

Algebraic Curves ◽

Parallel Lines ◽

Plane Algebraic Curve ◽

Concentric Circles

LetSbe a set ofnpoints in${\mathbb R}^{2}$contained in an algebraic curveCof degreed. We prove that the number of distinct distances determined bySis at leastcdn4/3, unlessCcontains a line or a circle.We also prove the lower boundcd′ min{m2/3n2/3,m2,n2} for the number of distinct distances betweenmpoints on one irreducible plane algebraic curve andnpoints on another, unless the two curves are parallel lines, orthogonal lines, or concentric circles. This generalizes a result on distances between lines of Sharir, Sheffer and Solymosi in [19].

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On some percolation results of J. M. Hammersley

Journal of Applied Probability ◽

10.1017/s0021900200107673 ◽

1979 ◽

Vol 16 (03) ◽

pp. 526-540 ◽

Cited By ~ 3

Author(s):

J. G. Oxley ◽

D. J. A. Welsh

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Percolation Theory ◽

Fkg Inequality ◽

Percolation Probability ◽

Finite Set ◽

The Fkg Inequality ◽

Arbitrary Graphs

We examine how much classical percolation theory on lattices can be extended to arbitrary graphs or even clutters of subsets of a finite set. In the process we get new short proofs of some theorems of J. M. Hammersley. The FKG inequality is used to get an upper bound for the percolation probability and we also derive a lower bound. In each case we characterise when these bounds are attained.

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On some percolation results of J. M. Hammersley

Journal of Applied Probability ◽

10.2307/3213082 ◽

1979 ◽

Vol 16 (3) ◽

pp. 526-540 ◽

Cited By ~ 8

Author(s):

J. G. Oxley ◽

D. J. A. Welsh

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Percolation Theory ◽

Fkg Inequality ◽

Percolation Probability ◽

Finite Set ◽

The Fkg Inequality ◽

Arbitrary Graphs

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Value Sets of Sparse Polynomials

Canadian Mathematical Bulletin ◽

10.4153/s0008439519000316 ◽

2019 ◽

Vol 63 (1) ◽

pp. 187-196

Author(s):

Igor E. Shparlinski ◽

José Felipe Voloch

Keyword(s):

Lower Bound ◽

Finite Field ◽

Individual Values ◽

Value Set ◽

Arithmetic Properties ◽

Sparse Polynomial ◽

Sparse Polynomials ◽

Value Sets

AbstractWe obtain a new lower bound on the size of the value set $\mathscr{V}(f)=f(\mathbb{F}_{p})$ of a sparse polynomial $f\in \mathbb{F}_{p}[X]$ over a finite field of $p$ elements when $p$ is prime. This bound is uniform with respect to the degree and depends on some natural arithmetic properties of the degrees of the monomial terms of $f$ and the number of these terms. Our result is stronger than those that can be extracted from the bounds on multiplicities of individual values in $\mathscr{V}(f)$.

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SIMULTANEOUS ARITHMETIC PROGRESSIONS ON ALGEBRAIC CURVES

International Journal of Number Theory ◽

10.1142/s1793042111004198 ◽

2011 ◽

Vol 07 (04) ◽

pp. 921-931 ◽

Cited By ~ 1

Author(s):

RYAN SCHWARTZ ◽

JÓZSEF SOLYMOSI ◽

FRANK DE ZEEUW

Keyword(s):

Lower Bound ◽

Elliptic Curve ◽

Arithmetic Progression ◽

Upper Bound ◽

Algebraic Curve ◽

Algebraic Curves ◽

Arithmetic Progressions ◽

Real Algebraic Curve

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (xi, yσ(i)), where [Formula: see text] and [Formula: see text] are arithmetic progressions and σ is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over ℚ. We show that 4319 is such a bound for curves over ℝ. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the "crossing inequality" gives a lower bound. Together these bound the length of an s.a.p. on the curve. We also extend this method to bound the k for which a real algebraic curve can contain k points from a k × k grid. Lastly, these results are extended to complex algebraic curves.

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