Chromatic equivalence classes of certain generalized polygon trees, III

AbstractLet P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, if P(G)=P(H). A set of graphs S is called a chromatic equivalence class if for any graph H that is chromatically equivalent with a graph G in S, then H∈S. Peng et al. (Discrete Math. 172 (1997) 103–114), studied the chromatic equivalence classes of certain generalized polygon trees. In this paper, we continue that study and present a solution to Problem 2 in Koh and Teo (Discrete Math. 172 (1997) 59–78)

Chromatic equivalence classes of certain generalized polygon trees

Let P(G) denote the chromatic polynomial of a graph G. Two graphs G and H are chromatically equivalent, written G ∼ H, if P(G) = P(H). Let g denote the family of all generalized polygon trees with three interior regions. Xu (1994) showed that g is a union of chromatic equivalence classes under the equivalence relation '∼'. In this paper, we determine infinitely many chromatic equivalence classes in g under '∼'. As a byproduct, we obtain a family of chromatically unique graphs established by Peng (1995)

Chromatic Equivalence Classes and Chromatic Defining Numbers of Certain Graphs

There are two parts in this dissertation: the chromatic equivalence classes and the chromatic defining numbers of graphs. In the first part the chromaticity of the family of generalized polygon trees with intercourse number two, denoted by Cr (a, b; c, d), is studied. It is known that Cr( a, b; c, d) is a chromatic equivalence class if min {a, b, c, d} ≥ r+3. We consider Cr( a, b; c, d) when min{ a, b, c, d} ≤ r + 2. The necessary and sufficient conditions for Cr(a, b; c, d) with min {a, b, c, d} ≤ r + 2 to be a chromatic equivalence class are given. Thus, the chromaticity of Cr (a, b; c, d) is completely characterized. In the second part the defining numbers of regular graphs are studied. Let d(n, r, X = k) be the smallest value of defining numbers of all r-regular graphs of order n and the chromatic number equals to k. It is proved that for a given integer k and each r ≥ 2(k - 1) and n ≥ 2k, d(n, r, X = k) = k - 1. Next, a new lower bound for the defining numbers of r-regular k-chromatic graphs with k < r < 2( k - 1) is found. Finally, the value of d( n , r, X = k) when k < r < 2(k - 1) for certain values of n and r is determined

Analysis of analysis: importance of different musical parameters for Schenkerian analysis

While criteria for Schenkerian analysis have been much discussed, such discussions have generally not been informed by data. Kirlin [Kirlin, Phillip B., 2014 “A Probabilistic Model of Hierarchical Music Analysis.” Ph.D. thesis, University of Massachusetts Amherst] has begun to fill this vacuum with a corpus of textbook Schenkerian analyses encoded using data structures suggested byYust [Yust, Jason, 2006 “Formal Models of Prolongation.” Ph.D. thesis, University of Washington] and a machine learning algorithm based on this dataset that can produce analyses with a reasonable degree of accuracy. In this work, we examine what musical features (scale degree, harmony, metrical weight) are most significant in the performance of Kirlin's algorithm.Accepted manuscrip

Chromatic polynomials

In this thesis, we shall investigate chromatic polynomials of graphs, and some related polynomials. In Chapter 1, we study the chromatic polynomial written in a modified form, and use these results to characterise the chromatic polynomials of polygon trees. In Chapter 2, we consider the chromatic polynomial written as a sum of the chromatic polynomials of complete graphs; in particular, we determine for which graphs the coefficients are symmetrical, and show that the coefficients exhibit a skewed property. In Chapter 3, we dualise many results about chromatic polynomials to flow polynomials, including the results in Chapter 1, and a result about a zero-free interval. Finally, in Chapter 4, we investigate the zeros of the Tutte Polynomial; in particular their observed proximity to certain hyperbole in the xy-plane

Chromatic polynomials

In this thesis, we shall investigate chromatic polynomials of graphs, and some related polynomials. In Chapter 1, we study the chromatic polynomial written in a modified form, and use these results to characterise the chromatic polynomials of polygon trees. In Chapter 2, we consider the chromatic polynomial written as a sum of the chromatic polynomials of complete graphs; in particular, we determine for which graphs the coefficients are symmetrical, and show that the coefficients exhibit a skewed property. In Chapter 3, we dualise many results about chromatic polynomials to flow polynomials, including the results in Chapter 1, and a result about a zero-free interval. Finally, in Chapter 4, we investigate the zeros of the Tutte Polynomial; in particular their observed proximity to certain hyperbole in the xy-plane

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Alliance polynomial and hyperbolicity in regular graphs

One of the open problems in graph theory is the characterization of any graph by a polynomial. Research in this area has been largely driven by the advantages offered by the use of computers which make working with graphs: it is simpler to represent a graph by a polynomial (a vector) that by the adjacency matrix (a matrix). We introduce the alliance polynomial of a graph. The alliance polynomial of a graph G with order n and maximum degree δ_1 is the polynomial A(G; x) = ∑_(k=〖-δ〗_1)^(δ_1)▒〖Ak(G) x^(n+k) 〗, where A{_k}(G) is the number of exact defensive k-alliances in G. Also, we develop and implement an algorithm that computes in an efficient way the alliance polynomial. We obtain some properties of A(G; x) and its coefficients for: • Path, cycle, complete and star graphs. In particular, we prove that they are characterized by their alliance polynomials. • Cubic graphs (graphs with all of their vertices of degree 3), since they are a very interesting class of graphs with many applications. We prove that they verify unimodality. Also, we compute the alliance polynomial for cubic graphs of small order, which satisfy uniqueness. • Regular graphs (graphs with the same degree for all vertices). In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of connected ∆-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not connected ∆-regular. If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in the δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) := inf{δ >= 0 : X is δ-hyperbolic }. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. We obtain information about the hyperbolicity constant of cubic graphs. These graphs are also very important in the study of Gromov hyperbolicity, since for any graph G with bounded maximum degree there exists a cubic graph G* such that G is the hyperbolic if and only if G* is hyperbolic. We find some characterizations for the cubic graphs which have small hyperbolicity constants. Besides, we obtain bounds for the hyperbolicity constant of the complement graph of a cubic graph; our main result of this kind says that for any finite cubic graph G which is not isomorphic either to K_4 or to K_3,3, the inequalities 5k/4 <= δ (G ̅) <=3k/2 hold, if k is the length of every edge in G. --------------------Uno de los problemas abiertos en la teoría de grafos es la caracterización de cualquier grafo por un polinomio. La investigación en este área ha sido impulsada en gran parte por las ventajas que ofrece el uso de las computadoras que hacen que trabajar con grafos sea más simple. En esta Tesis introducimos el polinomio de alianza de un grafo. El polinomio de alianza de un grafo G con orden n y grado m´aximo δ_1 es el polinomio A(G; x) = ∑_(k=〖-δ〗_1)^(δ_1)▒〖Ak(G) x^(n+k) , donde A{_k}(G) es el n´umero de k alianzas defensivas exactas en G. También desarrollamos e implementamos un algoritmo que calcula de manera eficiente el polinomio de alianza. En este trabajo obtenemos algunas propiedades de A(G; x) y sus coeficientes para: • Grafos caminos, ciclos, completos y estrellas. En particular, hemos demostrado que se caracterizan mediante sus polinomios de alianza. • Grafos cúbicos (grafos con todos sus vértices de grado 3), ya que son una clase muy interesante de grafos con muchas aplicaciones. Hemos demostrado que sus polinomios de alianza verifican unimodalidad. Además, calculamos el polinomio de alianza para grafos cúbicos de orden pequeño, los cuales satisfacen unicidad. • Grafos regulares (grafos con todos sus vértices de igual grado). En particular, se caracteriza el grado de los grafos regulares por el n´umero de coeficientes distintos de cero de su polinomio de alianza. Además, se demuestra que la familia de polinomios de alianza de grafos conexos _-regulares con grado pequeño es muy especial, ya que no contiene polinomios de alianza de grafos conexos que no sean _-regulares. Si X es un espacio métrico geodésico y x1, x2, x3 ∈ X, un triángulo geodésico T = {x1, x2, x3} es la unión de tres geodésicas [x1x2], [x2x3] and [x3x1] de X. El espacio X es δ-hiperbólico (en el sentido de Gromov) si todo lado de todo triángulo geodésico T de X está contenido en la δ-vecindad de la unóon de los otros dos lados. Se denota por δ(X) la constante de hiperbolicidad óptima de X, es decir, δ(X) := inf{δ > 0 : X es δ-hiperbólico }. El estudio de los grafos hiperbólicos es un tema interesante dado que la hiperbolicidad de un espacio métrico geodésico es equivalente a la hiperbolicidad de un grafo más sencillo asociado al espacio. Hemos obtenido información acerca de la constante de hiperbolicidad de los grafos cúbicos; dichos grafos son muy importantes en el estudio de la hiperbolicidad, ya que para cualquier grafo G con grado máximo acotado existe un grafo cúbico G∗ tal que G es hiperbólico si y sólo si G∗ es hiperbólico. En esta memoria conseguimos caracterizar los grafos cúbicos con constante de hiperbolicidad pequeña. Además, se obtienen cotas para la constante de hiperbolicidad del grafo complemento de un grafo cúbico; nuestro principal resultado dice que para cualquier grafo cúbico finito G no isomorfo a K4 o K3,3, se cumple la relación 5k/4 <= δ (G ̅) <=3k/2, donde k es la longitud de todas las aristas en G.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Domingo de Guzmán Pestana Galván; Secretario: Eva Tourís Lojo; Vocal: Sergio Bermudo Navarret