This paper presents a survey of concepts in commutative algebra that have applications to topology and graph theory. The primary algebraic focus will be on Stanley-Reisner rings, classes of polynomial rings that can describe simplicial complexes. Stanley-Reisner rings are defined via square-free monomial ideals. The paper will present many aspects of the theory of these ideals and discuss how they relate to important constructions in commutative algebra, such as finite generation of ideals, graded rings and modules, localization and associated primes, primary decomposition of ideals and Hilbert series. In particular, the primary decomposition and Hilbert series for certain types of monomial ideals will be analyzed through explicit examples of simplicial complexes and graphs.

Classical vertex coloring problems ask for the minimum number of colors needed to color the vertices of a graph, such that adjacent vertices use different colors. Vertex coloring does have quite a few practical applications in communication theory, industry engineering and computer science. Such examples can be found in the book of Hansen and Marcotte. Deciding whether a graph is 3-colorable or not is a well-known NP-complete problem, even for triangle-free graphs. Intuitively, large girth may help reduce the chromatic number. However, in 1959, Erdos used the probabilitic method to prove that for any two positive integers g and k, there exist graphs of girth at least g and chromatic number at least k. Thus, restricting girth alone does not help bound the chromatic number. However, if we forbid certain tree structure in addition to girth restriction, then it is possible to bound the chromatic number. Randerath determined several such tree structures, and conjectured that if a graph is fork-free and triangle-free, then it is 3-colorable, where a fork is a star K1,4 with two branches subdivided once. The main result of this thesis is that Randerath’s conjecture is true for graphs with odd girth at least 7. We also give a proof that Randerath’s conjecture holds for graphs with maximum degree 4.

The k-core of a graph is its maximal subgraph with minimum degree at least k. The study of k-cores in random graphs was initiated by Bollobás in 1984 in connection to k-connected subgraphs of random graphs. Subsequently, k-cores and their properties have been extensively investigated in random graphs and hypergraphs, with the determination of the threshold for the emergence of a giant k-core, due to Pittel, Spencer and Wormald, as one of the most prominent results. In this thesis, we obtain an asymptotic formula for the number of 2-connected graphs, as well as 2-edge-connected graphs, with given number of vertices and edges in the sparse range by exploiting properties of random 2-cores. Our results essentially cover the whole range for which asymptotic formulae were not described before. This is joint work with G. Kemkes and N. Wormald. By defining and analysing a core-type structure for uniform hypergraphs, we obtain an asymptotic formula for the number of connected 3-uniform hypergraphs with given number of vertices and edges in a sparse range. This is joint work with N. Wormald. We also examine robustness aspects of k-cores of random graphs. More specifically, we investigate the effect that the deletion of a random edge has in the k-core as follows: we delete a random edge from the k-core, obtain the k-core of the resulting graph, and compare its order with the original k-core. For this investigation we obtain results for the giant k-core for Erdős-Rényi random graphs as well as for random graphs with minimum degree at least k and given number of vertices and edges.

This thesis investigates three areas of the theory of graph homomorphisms: cores of graphs, the homomorphism order, and quantum homomorphisms. A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that, for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets which each induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions. Next we examine the restriction of the homomorphism order of graphs to line graphs. Our main focus is in comparing this restriction to the whole order. The primary tool we use in our investigation is that, as a consequence of Vizing's theorem, this partial order can be partitioned into intervals which can then be studied independently. We denote the line graph of X by L(X). We show that for all n ≥ 2, for any line graph Y strictly greater than the complete graph Kₙ, there exists a line graph X sitting strictly between Kₙ and Y. In contrast, we prove that there does not exist any connected line graph which sits strictly between L(Kₙ) and Kₙ, for n odd. We refer to this property as being ``n-maximal", and we show that any such line graph must be a core and the line graph of a regular graph of degree n. Finally, we introduce quantum homomorphisms as a generalization of, and framework for, quantum colorings. Using quantum homomorphisms, we are able to define several other quantum parameters in addition to the previously defined quantum chromatic number. We also define two other parameters, projective rank and projective packing number, which satisfy a reciprocal relationship similar to that of fractional chromatic number and independence number, and are closely related to quantum homomorphisms. Using the projective packing number, we show that there exists a quantum homomorphism from X to Y if and only if the quantum independence number of a certain product graph achieves |V(X)|. This parallels a well known classical result, and allows us to construct examples of graphs whose independence and quantum independence numbers differ. Most importantly, we show that if there exists a quantum homomorphism from a graph X to a graph Y, then ϑ̄(X) ≤ ϑ̄(Y), where ϑ̄ denotes the Lovász theta function of the complement. We prove similar monotonicity results for projective rank and the projective packing number of the complement, as well as for two variants of ϑ̄. These immediately imply that all of these parameters lie between the quantum clique and quantum chromatic numbers, in particular yielding a quantum analog of the well known ``sandwich theorem". We also briefly investigate the quantum homomorphism order of graphs.

An increasing number of models have been proposed to explain the link structure observed in complex networks. The central problem addressed in this thesis is: how do we select the best model? The model-selection method we implement is based on supervised learning. We train a classifier on six complex network models incorporating various link attachment mechanisms, including preferential attachment, copying and spatial. For the classification we represent graphs as feature vectors, integrating common complex network statistics with raw counts of small connected subgraphs commonly referred to as graphlets. The outcome of each experiment strongly indicates that models which incorporate the preferential attachment mechanism fit the network structure of Facebook the best. The experiments also suggest that graphlet structure is better at distinguishing different network models than more traditional complex network statistics. To further the understanding of our experimental results, we compute the expected number of triangles, 3-paths and 4-cycles which appear in our selected models. This analysis shows that the spatial preferential attachment model generates 3-paths, triangles and 4-cycles in abundance, giving a closer match to the observed network structure of the Facebook networks used in our model selection experiment. The other models generate some of these subgraphs in abundance but not all three at once. In general, we show that our selected models generate vastly different amounts of triangles, 3-paths and 4-cycles, verifying our experimental conclusion that graphlets are distinguishing features of these complex network models.

In this thesis, we study a variant of the maximum cardinality matching problem known as the maximum charge problem. Given a graph with arbitrary positive integer capacities assigned on every vertex and every edge, the goal is to maximize the assignment of positive feasible charges on the edges obeying the capacity constraints, so as to maximize the total sum of the charges. We use the primal-dual approach. We propose a combinatorial algorithm for solving the dual of the restricted primal and show that the primal-dual algorithm runs in a polynomial time. / ix, 96 leaves : ill. ; 29 cm

A new automatic compression scheme that adapts to any image set is presented in this thesis. The proposed scheme requires no a priori knowledge on the properties of the image set. This scheme is obtained using a unified graph-theoretical framework that allows for compression strategies to be compared both theoretically and experimentally. This strategy achieves optimal lossless compression by computing a minimum spanning tree of a graph constructed from the image set. For lossy compression, this scheme is near-optimal and a performance guarantee relative to the optimal one is provided. Experimental results demonstrate that this compression strategy compares favorably to the previously proposed strategies, with improvements up to 7% in the case of lossless compression and 72% in the case of lossy compression. This thesis also shows that the choice of underlying compression algorithm is important for compressing image sets using the proposed scheme. / x, 77 leaves ; 29 cm.

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<p>This thesis is a study on the confluence of algebraic structures and graph theory. Its aim is to consider groupoids from factorisations of complete graphs. We are especially interested in the cases where the factors are isomorphic. We analyse the loops obtained from homogeneous factorisations and ask if homogeneity is reflected in the kind of loops that are obtained. In particular, we are interested to see if we obtain either groups or quasi-associative Cayley sets from these loops. November 2008.</p>

Let G = (V, E) be a graph. For any real valued function f : V >R and SCV, let f (s) = z ues f(u). The weight of f is defined as f(V). A signed k-subdominating function (signed kSF) of G is defined as a function f : V > {-I, I} such that f(N[v]) > 1 for at least k vertices of G, where N[v] denotes the closed neighborhood of v. The signed k-subdomination number of a graph G, denoted by yks-11(G), is equal to min{f(V) I f is a signed kSF of G}. If instead of the range {-I, I}, we require the range {-I, 0, I}, then we obtain the concept of a minus k-subdominating function. Its associated parameter, called the minus k-subdomination number of G, is denoted by ytks-101(G). In chapter 2 we survey recent results on signed and minus k-subdomination in graphs. In Chapter 3, we compute the signed and minus k-subdomination numbers for certain complete multipartite graphs and their complements, generalizing results due to Holm [30]. In Chapter 4, we give a lower bound on the total signed k-subdomination number in terms of the minimum degree, maximum degree and the order of the graph. A lower bound in terms of the degree sequence is also given. We then compute the total signed k-subdomination number of a cycle, and present a characterization of graphs G with equal total signed k-subdomination and total signed l-subdomination numbers. Finally, we establish a sharp upper bound on the total signed k-subdomination number of a tree in terms of its order n and k where 1 < k < n, and characterize trees attaining these bounds for certain values of k. For this purpose, we first establish the total signed k-subdomination number of simple structures, including paths and spiders. In Chapter 5, we show that the decision problem corresponding to the computation of the total minus domination number of a graph is NP-complete, even when restricted to bipartite graphs or chordal graphs. For a fixed k, we show that the decision problem corresponding to determining whether a graph has a total minus domination function of weight at most k may be NP-complete, even when restricted to bipartite or chordal graphs. Also in Chapter 5, linear time algorithms for computing Ytns-11(T) and Ytns-101(T) for an arbitrary tree T are presented, where n = n(T). In Chapter 6, we present cubic time algorithms to compute Ytks-11(T) and Ytks-101l(T) for a tree T. We show that the decision problem corresponding to the computation of Ytks-11(G) is NP-complete, and that the decision problem corresponding to the computation of Ytks-101 (T) is NP-complete, even for bipartite graphs. In addition, we present cubic time algorithms to computeYks-11(T) and Yks-101(T) for a tree T, solving problems appearing in [25]. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 2003.