53 research outputs found
Labeling of graphs, sumset of squares of units modulo n and resonance varieties of matroids
This thesis investigates problems in a number of different areas of graph theory and its applications in other areas of mathematics. Motivated by the 1-2-3-Conjecture, we consider the closed distinguishing number of a graph G, denoted by dis[G]. We provide new upper bounds for dis[G] by using the Combinatorial Nullstellensatz. We prove that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G] = 2. We show that for each integer t there is a bipartite graph G such that dis[G] \u3e t. Then some polynomial time algorithms and NP-hardness results for the problem of partitioning the edges of a graph into regular and/or locally irregular subgraphs are presented. We then move on to consider Johnson graphs to find resonance varieties of some classes of sparse paving matroids. The last application we consider is in number theory, where we find the number of solutions of the equation x21 + _ _ _ + x2 k = c, where c 2 Zn, and xi are all units in the ring Zn. Our approach is combinatorial using spectral graph theory
Enumeration of Matchings: Problems and Progress
This document is built around a list of thirty-two problems in enumeration of
matchings, the first twenty of which were presented in a lecture at MSRI in the
fall of 1996. I begin with a capsule history of the topic of enumeration of
matchings. The twenty original problems, with commentary, comprise the bulk of
the article. I give an account of the progress that has been made on these
problems as of this writing, and include pointers to both the printed and
on-line literature; roughly half of the original twenty problems were solved by
participants in the MSRI Workshop on Combinatorics, their students, and others,
between 1996 and 1999. The article concludes with a dozen new open problems.
(Note: This article supersedes math.CO/9801060 and math.CO/9801061.)Comment: 1+37 pages; to appear in "New Perspectives in Geometric
Combinatorics" (ed. by Billera, Bjorner, Green, Simeon, and Stanley),
Mathematical Science Research Institute publication #37, Cambridge University
Press, 199
Defensive alliances in graphs: a survey
A set of vertices of a graph is a defensive -alliance in if
every vertex of has at least more neighbors inside of than outside.
This is primarily an expository article surveying the principal known results
on defensive alliances in graph. Its seven sections are: Introduction,
Computational complexity and realizability, Defensive -alliance number,
Boundary defensive -alliances, Defensive alliances in Cartesian product
graphs, Partitioning a graph into defensive -alliances, and Defensive
-alliance free sets.Comment: 25 page
Behavior of Petrie Lines in Certain Edge-Transitive Graphs
We survey the construction and classification of one-, two- and infinitely-ended members of a class of highly symmetric, highly connected infinite graphs. In addition, we pose a conjecture concerning the relationship between the Petrie lines and ends of some infinitely-ended members of this class
Contributions at the Interface Between Algebra and Graph Theory
In this thesis, we make some contributions at the interface between algebra and graph theory.
In Chapter 1, we give an overview of the topics and also the definitions and preliminaries.
In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two techniques: a bound on the number of zeros of
lacunary polynomials and a bound on the so-called domination number of a graph.
In Chapter 3, we deal with the determinant of bipartite graphs. The nullity of a graph G is the multiplicity of 0 in the spectrum of G. Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and
Huckel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, examining the determinant of a graph is a way
to attack this problem. In this Chapter, we show that the determinant of a bipartite graph with at least two perfect matchings and with all cycle lengths divisible by four, is zero.
In Chapter 4, we first introduce an application of spectral graph theory in proving trigonometric identities. This is a very simple double counting argument that gives very short proofs for some of
these identities (and perhaps the only existed proof in some cases!). In the rest of Chapter 4, using some properties of the
well-known Chebyshev polynomials, we prove some theorems that allow us to evaluate the number of spanning trees in join of graphs, Cartesian product of graphs, and nearly regular graphs. In the last section of Chapter 4, we obtain the number of spanning
trees in an (r,s)-semiregular graph and its line graph. Note that the same results, as in the last section, were proved by I. Sato using zeta functions. But our proofs are much shorter based on some well-known facts from spectral graph theory. Besides, we
do not use zeta functions in our arguments.
In Chapter 5, we present the conclusion and also some possible projects
Lower algebraic K-theory of certain reflection groups
For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the
property that all interior angles between incident faces are integral
submultiples of Pi, there is a naturally associated Coxeter group generated by
reflections in the faces. Furthermore, this Coxeter group is a lattice inside
the isometry group of hyperbolic 3-space, with fundamental domain the original
polyhedron P. In this paper, we provide a procedure for computing the lower
algebraic K-theory of the integral group ring of such Coxeter lattices in terms
of the geometry of the polyhedron P. As an ingredient in the computation, we
explicitly calculate some of the lower K-groups of the dihedral groups and the
product of dihedral groups with the cyclic group of order two.Comment: 35 pages, 2 figure
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