scholarly journals Regions Without Complex Zeros for Chromatic Polynomials on Graphs with Bounded Degree

2008 ◽  
Vol 17 (2) ◽  
pp. 225-238 ◽  
Author(s):  
ROBERTO FERNÁNDEZ ◽  
ALDO PROCACCI
Keyword(s):  
Finite Graph ◽  
Chromatic Polynomial ◽  
Maximal Degree ◽  
Bounded Degree ◽  
Chromatic Polynomials ◽  
Image Position ◽  
Complex Zeros

We prove that the chromatic polynomial$P_\mathbb{G}(q)$of a finite graph$\mathbb{G}$of maximal degree Δ is free of zeros for |q| ≥C*(Δ) withThis improves results by Sokal and Borgs. Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.

Bounds For The Real Zeros of Chromatic Polynomials

2008 ◽  
Vol 17 (6) ◽  
pp. 749-759 ◽  
Author(s):  
F. M. DONG ◽  
K. M. KOH
Keyword(s):  
Maximum Degree ◽  
Short Proof ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
The Real ◽  
Real Zeros ◽  
Complex Zeros ◽  
Special Case

Sokal in 2001 proved that the complex zeros of the chromatic polynomialPG(q) of any graphGlie in the disc |q| < 7.963907Δ, where Δ is the maximum degree ofG. This result answered a question posed by Brenti, Royle and Wagner in 1994 and hence proved a conjecture proposed by Biggs, Damerell and Sands in 1972. Borgs gave a short proof of Sokal's result. Fernández and Procacci recently improved Sokal's result to |q| < 6.91Δ. In this paper, we shall show that all real zeros ofPG(q) are in the interval [0,5.664Δ). For the special case that Δ = 3, all real zeros ofPG(q) are in the interval [0,4.765Δ).

Extended Chromatic Polynomials

1972 ◽  
Vol 24 (3) ◽  
pp. 492-501 ◽  
Author(s):  
Andrew Sobczyk ◽  
James O. Gettys
Keyword(s):  
Positive Integer ◽  
Finite Graph ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
Vertex Set

Let G be a finite graph with non-empty vertex set (G) and edge set (G) (see [2]). Let λ be a positive integer. Tutte [5] defines a λ-colouring of G as a mapping of (G) into the set Iλ = {1, 2, 3, …, λ} with the property that two ends of any edge are mapped onto distinct integers. The elements of Iλ are commonly called “colours.” If P(G, λ) represents the number of λ-colourings of G, it is well known that P(G, λ) can be expressed as a polynomial in λ. For this reason P(G, λ) is usually referred to as the chromatic polynomial of G.The chromatic polynomial P(G, λ) was first suggested as an approach to the four-colour conjecture. To quote Tutte [5]: ”… many people are specially interested in the value λ = 4.

Chromatic Solutions, II

1982 ◽  
Vol 34 (4) ◽  
pp. 952-960 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Power Series ◽  
Research Report ◽  
Chromatic Polynomials ◽  
Image Position

This paper is a continuation of the Waterloo Research Report CORR 81-12, (see [1]) referred to in what follows as I. That Report is entitled “Chromatic Solutions”. It is largely concerned with a power series h in a variable z2, in which the coefficients are polynomials in a “colour number” λ. By definition the coefficient of z2r, where r > 0, is the sum of the chromatic polynomials of the rooted planar triangulations of 2r faces. (Multiple joins are allowed in these triangulations.) Thus for a positive integral λ the coefficient is the number of λ-coloured rooted planar triangulations of 2r faces. The use of the symbol z2 instead of a simple letter t is for the sake of continuity with earlier papers.In I we consider the case(1)where n is an integer exceeding 4.

Chromatic Solutions

1982 ◽  
Vol 34 (3) ◽  
pp. 741-758 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Chromatic Polynomial ◽  
Basic Theory ◽  
Root Vertex ◽  
Independent Variables ◽  
Image Position ◽  
Formal Power ◽  
Two Independent Variables

Early in the Seventies I sought the number of rooted λ-coloured triangulations of the sphere with 2p faces. In these triangulations double joins, but not loops, were permitted. The investigation soon took the form of a discussion of a certain formal power series l(y, z, λ) in two independent variables y and z.The basic theory of l is set out in [1]. There l is defined as the coefficient of x2 in a more complicated power series g(x, y, z, λ). But the definition is equivalent to the following formula.1Here T denotes a general rooted triangulation. n(T) is the valency of its root-vertex, and 2p(T) is the number of its faces. P(T, λ) is the chromatic polynomial of the graph of T.

scholarly journals Completely bounded isomorphisms of injective von Neumann algebras

1989 ◽  
Vol 32 (2) ◽  
pp. 317-327 ◽  
Author(s):  
Erik Christensen ◽  
Allan M. Sinclair
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Von Neumann Algebras ◽  
Linear Structure ◽  
Type I ◽  
Hausdorff Spaces ◽  
Bounded Degree ◽  
Von Neumann ◽  
Separable Predual ◽  
Image Position

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

Phase transition in the random triangle model

1999 ◽  
Vol 36 (04) ◽  
pp. 1101-1115 ◽  
Author(s):  
Olle Häggström ◽  
Johan Jonasson
Keyword(s):  
Phase Transition ◽  
Social Networks ◽  
Ising Model ◽  
Hexagonal Lattice ◽  
Graph Model ◽  
Finite Graph ◽  
Critical Value ◽  
Two Dimensional ◽  
Random Graph Model ◽  
Image Position

The random triangle model was recently introduced as a random graph model that captures the property of transitivity that is often found in social networks, i.e. the property that given that two vertices are second neighbors, they are more likely to be neighbors. For parameters p ∊ [0,1] and q ≥ 1, and a finite graph G = (V, E), it assigns to elements η of {0,1} E probabilities which are proportional to where t(η) is the number of triangles in the open subgraph. In this paper the behavior of the random triangle model on the two-dimensional triangular lattice is studied. By mapping the system onto an Ising model with external field on the hexagonal lattice, it is shown that phase transition occurs if and only if p = (q−1)−2/3 and q &gt; q c for a critical value q c which turns out to equal It is furthermore demonstrated that phase transition cannot occur unless p = p c (q), the critical value for percolation of open edges for given q. This implies that for q ≥ q c , p c (q) = (q−1)−2/3.

scholarly journals Approximation Algorithms for Complex-Valued Ising Models on Bounded Degree Graphs

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 162 ◽  
Author(s):  
Ryan L. Mann ◽  
Michael J. Bremner
Keyword(s):  
Partition Function ◽  
Approximation Scheme ◽  
Ising Models ◽  
Quantum Circuits ◽  
Bounded Degree ◽  
Graph Polynomials ◽  
Bounded Degree Graphs ◽  
Complex Zeros ◽  
Complex Valued ◽  
Output Probability

We study the problem of approximating the Ising model partition function with complex parameters on bounded degree graphs. We establish a deterministic polynomial-time approximation scheme for the partition function when the interactions and external fields are absolutely bounded close to zero. Furthermore, we prove that for this class of Ising models the partition function does not vanish. Our algorithm is based on an approach due to Barvinok for approximating evaluations of a polynomial based on the location of the complex zeros and a technique due to Patel and Regts for efficiently computing the leading coefficients of graph polynomials on bounded degree graphs. Finally, we show how our algorithm can be extended to approximate certain output probability amplitudes of quantum circuits.

scholarly journals Density of Chromatic Roots in Minor-Closed Graph Families

2018 ◽  
Vol 27 (6) ◽  
pp. 988-998 ◽  
Author(s):  
THOMAS J. PERRETT ◽  
CARSTEN THOMASSEN
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Classical Result ◽  
Golden Ratio ◽  
Chromatic Polynomial ◽  
Closed Graph ◽  
Small Interval ◽  
Chromatic Polynomials ◽  
Graph Families ◽  
Chromatic Roots

We prove that the roots of the chromatic polynomials of planar graphs are dense in the interval between 32/27 and 4, except possibly in a small interval around τ + 2 where τ is the golden ratio. This interval arises due to a classical result of Tutte, which states that the chromatic polynomial of every planar graph takes a positive value at τ + 2. Our results lead us to conjecture that τ + 2 is the only such number less than 4.

scholarly journals A note on the conditional chromatic polynomial

1994 ◽  
Vol 36 (3) ◽  
pp. 265-267 ◽  
Author(s):  
V. Voloshin
Keyword(s):  
Finite Graph ◽  
Chromatic Polynomial ◽  
Multiple Edges

In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in λ colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed λ [1, 4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another.

On injective chromatic polynomials of graphs

2015 ◽  
Vol 07 (03) ◽  
pp. 1550035
Author(s):  
Anjaly Kishore ◽  
M. S. Sunitha
Keyword(s):  
Chromatic Number ◽  
Discrete Math ◽  
Chromatic Polynomial ◽  
Complete Graphs ◽  
Common Neighbor ◽  
Chromatic Polynomials ◽  
Wheel Graphs ◽  
Minimum Number

The injective chromatic number χi(G) [G. Hahn, J. Kratochvil, J. Siran and D. Sotteau, On the injective chromatic number of graphs, Discrete Math. 256(1–2) (2002) 179–192] of a graph G is the minimum number of colors needed to color the vertices of G such that two vertices with a common neighbor are assigned distinct colors. The nature of the coefficients of injective chromatic polynomials of complete graphs, wheel graphs and cycles is studied. Injective chromatic polynomial on operations like union, join, product and corona of graphs is obtained.

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