Zeros of Jones Polynomials of Graphs

Journal of Mathematical Inequalities, 2010

New lower bounds for eigenvalues of a simple graph are derived. Upper and lower bounds for eigenvalues of bipartite graphs are presented in terms of traces and degree of vertices. Finally a non-trivial lower bound for the algebraic connectivity of a connected graph is given.

2020

A signed edge domination function (or SEDF) of a simple graph $G=(V,E)$ is a function $f: E\rightarrow \{1,-1\}$ such that $\sum_{e'\in N[e]}f(e')\ge 1$ holds for each edge $e\in E$, where $N[e]$ is the set of edges in $G$ which have at least one common end with $e$. Let $\gamma_s'(G)$ denote the minimum value of $f(G)$ among all SEDFs $f$, where $f(G)=\sum_{e\in E}f(e)$. In 2005, Xu conjectured that $\gamma_s'(G)\le n-1$. This conjecture has been proved for the two cases $v_{odd}(G)=0$ and $v_{even}(G)=0$, where $v_{odd}(G)$ (resp. $v_{even}(G)$) is the number of odd (resp. even) vertices in $G$. This article proves Xu's conjecture for $v_{even}(G)\in \{1, 2\}$. We also show that for any simple graph $G$ of order $n$, $\gamma_s'(G)\le n+v_{odd}(G)/2$ and $\gamma_s'(G)\le n-2+v_{even}(G)$ when $v_{even}(G)>0$, and thus $\gamma_s'(G)\le (4n-2)/3$. Our result improves the known results $\gamma_s'(G)\le 11n/6-1$ and $\gamma_s'(G)\le \lceil 3n/...

Journal of Combinatorial Theory, Series A, 2013

From a combinatorial perspective, we establish three inequalities on coefficients of R-and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of (q − 1)-coefficients of R-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of R-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is to understand Deodhar's inequality in a connection with a sum of R-polynomials and edges of Bruhat graphs. Contents 1. Introduction 1 2. Notation 4 3. R-polynomials 4 4. Some nonnegativity of R-polynomials 5 5. Rational smoothness and singularities 8 6. Deodhar's inequality revisited 10 7. KL polynomials 12 8. Existence of a strict inequality of KL polynomials 14 References 15

If G is a graph without isolated vertices, and if rand s are positive integers, then the (r, s)-domination number 'Yr,s(G) of G is the cardinality of a smallest vertex set D such that every vertex not in D is within distance r from some vertex in D, while every vertex in D is within distance s from another vertex in D. This generalizes the total domination number 'Yt(G) = 'Yl,l(G). Let T(G) denote the set of all spanning trees of a connected graph G. We prove that 'Yr,s(T(G)) is a set of consecutive integers for every connected graph G of order at least two when s 2' : 2r + 1. This is not true if 1 :::; s :::; 2r-1, and for s = 2r the problem is open. We prove that 'Yr,2r(T(G)) is a set of consecutive integers for r = 1 and we conjecture this also holds for r 2' : 2. We also prove that 'Yr,s(T(G)) is a set of consecutive integers for every 2-connected graph G and for any two positive integers rand s. Let G be a simple undirected graph with vertices V(G) and edges E(G). The neighbourhood of a vertex v in G is NG(v) = {u E V(G) : uv E E(G)} and the closed neighbourhood is N G [v] = N G (v) U { v }. For a connected graph G, let d G (v, u) denote the distance between vertices v and u in G. If S is a set of vertices of G and v is a vertex of G, then d G (v, S) denotes the distance between v and S, the shortest distance between v and a vertex of S. Let rand s be two positive integers. A vertex set D of a graph G is an (r,-)-set of G if dG(v, D) :::; r for every v E V(G)-D. Similarly, a subset D of V(G) is

Special Matrices, 2019

We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q(Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ġ ) – xI (x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.

Journal of Mathematical Analysis and Applications, 1996

2001

For a given graph G, let P (G,λ) be the chromatic polynomial of G, where λ is considered to be a real number. In this paper, we study the bounds for P (G,λ)/P (G,λ − 1) and P (G,λ)/P (G − x, λ), where x is a vertex in G, λ ≥ n and n is the number of vertices of G.

Acta Mathematica Hungarica, 1994

Combinatorica, 2010

A real multivariate polynomial p(x 1 ,. .. , x n) is said to sign-represent a Boolean function f : {0, 1} n → {−1, 1} if the sign of p(x) equals f (x) for all inputs x ∈ {0, 1} n. We give new upper and lower bounds on the degree of polynomials which sign-represent Boolean functions. Our upper bounds for Boolean formulas yield the first known subexponential time learning algorithms for formulas of superconstant depth. Our lower bounds for constant-depth circuits and intersections of halfspaces are the first new degree lower bounds since 1968, improving results of Minsky and Papert. The lower bounds are proved constructively; we give explicit dual solutions to the necessary linear programs. * A preliminary version of these results appeared as [24].

Journal of Combinatorial Theory, Series A, 2017

We say that a permutation π = π 1 π 2 • • • π n ∈ S n has a peak at index i if π i−1 < π i > π i+1. Let P(π) denote the set of indices where π has a peak. Given a set S of positive integers, we define P S (n) = {π ∈ S n : P(π) = S}. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers S and sufficiently large n, |P S (n)| = p S (n)2 n−|S|−1 where p S (x) is a polynomial depending on S. They gave a recursive formula for p S (x) involving an alternating sum, and they conjectured that the coefficients of p S (x) expanded in a binomial coefficient basis centered at max(S) are all nonnegative. In this paper we introduce a new recursive formula for |P S (n)| without alternating sums, and we use this recursion to prove that their conjecture is true.

International Journal of Engineering Sciences & Research Technology, 2014

In this paper we find annular bounds for the zeros of a polynomial . Mathematics Subject Classification: 30 C 10, 30 C 15

Journal of Combinatorial Theory, 1996

be points on the real line. For every k=1, 2, ..., d, the k-alternating polynomial P k is the polynomial of degree k and norm &P k & = max 1 l d [ |P k (* l )|] 1 that attains maximum absolute value at any point * Â [* d , * 1 ]. Because of this optimality property, these polynomials may be thought of as the discrete version of the Chebychev polynomials T k and, for particular values of the given points, P k coincides in fact with the``shifted'' T k . In general, however, those polynomials seem to bear a much more involved structure than Chebychev ones. Some basic properties of the P k are studied, and it is shown how to compute them in general. The results are then applied to the study of the relationship between the (standard or Laplacian) spectrum of a (not necessarily regular) graph or bipartite graph and its diameter, improving previous results. 1996 Academic Press, Inc. Recently, some results relating the diameter of a regular graph and its second eigenvalue (in absolute value) have been given by Chung [3], article no. 0033 48 0095-8956Â96 18.00

Linear Algebra and its Applications, 2010

The spread s(M ) of an n × n complex matrix M is s(M ) = max ij |λ i − λ j |, where the maximum is taken over all pairs of eigenvalues of M , λ i , 1 ≤ i ≤ n, [9] and . Based on this concept, Gregory et al.

Let G be a graph without isolated vertices and let α(G) be its stability number and τ (G) its covering number. The σ v -cover number of a graph, denoted by σ v (G), is the maximum natural number m such that every vertex of G belongs to a maximal independent set with at least m vertices. In the first part of this paper we prove that α

Constructive Approximation, 1994