Graph homomorphisms, the Tutte polynomial and “q-state Potts uniqueness”

We establish for which weighted graphs H homomorphism functions from multigraphs G to H are specializations of the Tutte polynomial of G, answering a question of Freedman, Lov´asz and Schrijver. We introduce a new property of graphs called “q-state Potts uniqueness” and relate it to chromatic and Tutte uniqueness, and also to “chromatic–flow uniqueness”, recently studied by Duan, Wu and Yu.Ministerio de Educación y Ciencia MTM2005-08441-C02-0

Chromaticity of Certain 2-Connected Graphs

Since the introduction of the concepts of chromatically unique graphs and chromatically equivalent graphs, many families of such graphs have been obtained. In this thesis, we continue with the search of families of chromatically unique graphs and chromatically equivalent graphs. In Chapter 1, we define the concept of graph colouring, the associated chromatic polynomial and some properties of a chromatic polynomial. We also give some necessary conditions for graphs that are chromatically unique or chromatically equivalent. Chapter 2 deals with the chromatic classes of certain existing 2-connected (n, n + 1,)-graphs for z = 0, 1, 2 and 3. Many families of chromatically unique graphs and chromatically equivalent graphs of these classes have been obtained. At the end of the chapter, we re-determine the chromaticity of two families of 2-connected (n, n + 3)-graphs with at least two triangles. Our main results in this thesis are presented in Chapters 3, 4 and 5. In Chapter 3, we classify all the 2-connected (n, n + 4)-graphs wit h at least four triangles . In Chapter 4 , we classify all the 2-connected (n, n + 4)-graphs wit h t hree triangles and one induced 4-cycle. In Chapter 5, we classify all the 2-connected (n, n + 4)graphs with three triangles and at least two induced 4-cycles . In each chapter, we obtain new families of chromatically unique graphs and chromatically equivalent graphs. We end the thesis by classifying all the 2-connected (n, n + 4)-graphs with exactly three triangles. We also determine the chromatic polynomial of all these graphs. The determination of the chromaticity of most classes of these graphs is left as an open problem for future research

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)

Homomorphisms and polynomial invariants of graphs

This paper initiates a general study of the connection between graph homomorphisms and the Tutte polynomial. This connection can be extended to other polynomial invariants of graphs related to the Tutte polynomial such as the transition, the circuit partition, the boundary, and the coboundary polynomials. As an application, we describe in terms of homomorphism counting some fundamental evaluations of the Tutte polynomial in abelian groups and statistical physics. We conclude the paper by providing a homomorphism view of the uniqueness conjectures formulated by Bollobás, Pebody and Riordan.Ministerio de Educación y Ciencia MTM2005-08441-C02-01Junta de Andalucía PAI-FQM-0164Junta de Andalucía P06-FQM-0164

Graphs determined by polynomial invariants

AbstractMany polynomials have been defined associated to graphs, like the characteristic, matchings, chromatic and Tutte polynomials. Besides their intrinsic interest, they encode useful combinatorial information about the given graph. It is natural then to ask to what extent any of these polynomials determines a graph and, in particular, whether one can find graphs that can be uniquely determined by a given polynomial. In this paper we survey known results in this area and, at the same time, we present some new results

An investigation of specific structural techniques used by the Miles Davis Quintet on selected live recordings from 1964-1965

This dissertation intends to deconstruct and analyse specific structural techniques employed by members of Miles Davis\u27 second great quintet, which consisted of Wayne Shorter, Herbie Hancock, Ron Carter, and Tony Williams. The motivation behind the selection of structural techniques as the basis of this dissertation lies in its ability to clearly articulate a large aspect of the group\u27s amazing flexibility as an ensemble. The specific areas of analysis are: tempo feel changes, tempo fluctuations, and texture changes. The recordings in focus are \u27Autumn Leaves\u27 from the album \u27Miles In Berlin\u27, and three selections from the influential live recording of two nights from the Plugged Nickel jazz club in Chicago in December of 1965. An analysis of \u27Autumn Leaves\u27 will look at tempo fluctuations and texture changes. This will be presented in graphic format, and specific musical excerpts will be included to highlight how the ensemble achieves these changes and create interest. From the Plugged Nickel recordings, an analysis of two ballads, \u27Stella By Starlight\u27 and \u27I Fall In Love Too Easily\u27 will look at the group\u27s treatment of tempo feel changes. Additionally, analysis will be conducted of tempo fluctuations that occur during Shorter and Hancock\u27s solos on \u27No Blues\u27 from the second night of recording. Again, these tempo alterations will be presented in a graphic format, and excerpts will be provided to demonstrate specific techniques used by the band in achieving these changes in feel or speed. A discussion of the possible influences on the group around this time - musically and socially- will be included as a way of contextualising the musical approaches of the group. Further, a discussion will be included focusing on two pieces from the author\u27s graduating recital, discussing the level of success in applying techniques similar to those that Davis\u27 quintet applies on the recordings in focus

JOHN MACKEY’S WINE-DARK SEA: SYMPHONY FOR BAND A DISCOURSE AND ANALYSIS OF JOHN MACKEY’S SYMPHONY FOR BAND

John Mackey’s Wine-Dark Sea: Symphony for Band(2014) is a work of epic proportions and was the winner of the William D. Revelli Composition Contest of the National Band Association in 2015. Wine-Dark Sea: Symphony for Bandhas received much acclaim and many performances including a recording by the University of Texas Wind Ensemble in 2016. The purposes of this dissertation are 1) to provide historical information on the genesis of the work through interviews with its composer, John Mackey, and commissioning director, Jerry Junkin; 2) to provide an analysis of how the programmatic elements of Homer’s Odysseyinteract with the musical aspects of the work. The first chapter discusses biographical information essential to the understanding of John Mackey’s music. Chapter two includes information specific to the creation of Wine-Dark Sea: Symphony for Band. Chapters three through five provide analytical information alongside programmatic information to provide a clear understanding of how the music and programmatic elements combine to create the work. Chapter six concludes the document with some performance suggestions for the conductor. An appendix of information including graphs of how dynamic range corresponds to programmatic elements and interviews with the composer, John Mackey, and the commissioner, Jerry Junkin, are also provided