We study the representations of the symmetric group $S_n$ found by acting on labeled graphs and trees with $n$ vertices. Our main results provide combinatorial interpretations that give the number of times the irreducible representations associated with the integer partitions $(n)$ and $(1^n)$ appear in the representations. We describe a new sign reversing involution with fixed points that provide a combinatorial interpretation for the number of times the irreducible associated with the integer partition $(n-1, 1)$ appears in the representations.

The following presents a brief introduction to tropical geometry, especially tropical curves, and explains a connection to graph theory. We also give a brief summary of the Riemann-Roch property for graphs, established by Baker and Norine (2007), as well as the tools used in their proof. Various generalizations are described, including a more thorough description of the extension to strongly connected directed graphs by Asadi and Backman (2011). Building from their constructions, an algorithm to determine if a directed graph has Row Riemann-Roch Property is given and thoroughly explained.

For a graph G = (V;E), a pebble distribution is defined as a mapping of the vertex set in to the integers, where each vertex begins with f(v) pebbles. A pebbling move takes two pebbles from some vertex adjacent to v and places one pebble on v. A rubbling move takes one pebble from each of two vertices that are adjacent to v and places one pebble on v. A vertex x is reachable under a pebbling distribution f if there exists some sequence of rubbling and pebbling moves that places a pebble on x. A pebbling distribution where every vertex is reachable is called a rubbling configuration. The t-restricted optimal rubbling number of G is the minimum number of pebbles required for a rubbling configuration where no vertex is initially assigned more than t pebbles. Here we present results on the 1-restricted optimal rubbling number and the 2- restricted optimal rubbling number.

A labeling of a graph is an assignment of a natural number to each vertex of a graph. Graceful labelings are very important types of labelings. The study of graceful labelings is very difficult and little has been shown about such labelings. Vertex-relaxed graceful labelings of graphs are a class of labelings that include graceful labelings, and their study gives an approach to the study of graceful labelings. In this thesis we generalize the congruence approach of Rosa to obtain new criteria for vertex-relaxed graceful labelings of graphs. To do this, we generalize Faulhaber’s Formula, which is a famous result about sums of powers of integers.

In this thesis, we provide an algorithm for constructing strongly eutactic lattices from vertex transitive graphs. We show that such construction produces infinitely many strongly eutactic lattices in arbitrarily large dimensions. We demonstrate our algorithm on the example of the famous Petersen graph using Maple computer algebra system. We also discuss some additional examples of strongly eutactic lattices obtained from notable vertex transitive graphs. Further, we study the properties of the lattices generated by product graphs, complement graphs, and line graphs of vertex transitive graphs. This thesis is based on the research paper written by the author jointly with L. Fukshansky, D. Needell and J. Park.

The Fibonomial coefficients are a generalization of the binomial coefficients with a rather nice combinatorial interpretation. While the ordinary binomial coefficients count lattice paths in a grid, the Fibonomial coefficients count the number of ways to draw a lattice path in a grid and then Fibonacci-tile the regions above and below the path in a particular way. We may forgo a literal tiling interpretation and, instead of the Fibonacci numbers, use an arbitrary function to count the number of ways to "tile" the regions of the grid delineated by the lattice path. When the function is a combinatorial sequence such as the Lucas numbers or the q-numbers, the total number of "tilings" is some multiple of a generalized binomial coefficient corresponding to the sequence chosen.

The associahedron has appeared in numerous contexts throughout the field of mathematics. By representing the associahedron as a poset of tubings, Michael Carr and Satyan L. Devadoss were able to create a gener- alized version of the associahedron in the graph-associahedron. We seek to create an alternative generalization of the associahedron by considering a particle-collision model. By extending this model to what we dub the 2- associahedron, we seek to further understand the space of generalizations of the associahedron.

One question relating to partially ordered sets (posets) is that of partitioning or dividing the poset's elements into the fewest number of chains that span the poset. In 1950, Dilworth established that the width of the poset - the size of the largest set composed only of incomparable elements - is the minimum number of chains needed to partition that poset. Such a bound in on-line partitioning has been harder to establish, and work has evalutated classes of posets based on their width. This paper reviews the theorems that established val(2)=5 and illustrates them with examples. It also covers some of the work on establishing bounds for on-line partitioning with the Greedy Algorithm. The paper concludes by contributing a bound on incomparable elements in graded, (t+t)-free, finite width posets.

Reed's conjecture is a proposed upper bound for the chromatic number of a graph. Reed's conjecture has already been proven for several families of graphs. In this paper, I show how one of those families of graphs can be extended to include additional graphs and also show that Reed's conjecture holds for a family of graphs known as cycle-power graphs, and also for their complements.

<p>As stressed by the Society for Industrial and Applied Mathematics (SIAM): Applied mathematics, in partnership with computational science, is essential in solving many real-world problems. Combinatorial optimization focuses on problems arising from discrete structures such as graphs and polyhedra. This thesis deals with extremal graphs and strings and focuses on two problems: the Erdos' problem on multiplicities of complete subgraphs and the maximum number of distinct squares in a string.<br />The first part of the thesis deals with strengthening the bounds for the minimum proportion of monochromatic t cliques and t cocliques for all 2-colourings of the edges of the complete graph on n vertices. Denote by k_t(G) the number of cliques of order t in a graph G. Let k_t(n) = min{k_t(G)+k_t(\overline{G})} where \overline{G} denotes the complement of G of order n. Let c_t(n) = {k_t(n)} / {\tbinom{n}{t}} and c_t be the limit of c_t(n) for n going to infinity. A 1962 conjecture of Erdos stating that c_t = 2^{1-\tbinom{t}{2}} was disproved by Thomason in 1989 for all t > 3. Tighter counterexamples have been constructed by Jagger, Stovicek and Thomason in 1996, by Thomason for t < 7 in 1997, and by Franek for t=6 in 2002. We present a computational framework to investigate tighter upper bounds for small t yielding the following improved upper bounds for t=6,7 and 8: c_6 \leq 0.7445 \times 2^{1- \tbinom{6}{2}}, c_7\leq 0.6869\times 2^{1- \tbinom{7}{2}}, and c_8 \leq 0.7002\times 2^{1- \tbinom{8}{2}}. The constructions are based on a large but highly regular variant of Cayley graphs for which the number of cliques and cocliques can be expressed in closed form. Considering the quantity e_t=2^{\tbinom{t}{2}-1} c_t, the new upper bound of 0.687 for e_7 is the first bound for any e_t smaller than the lower bound of 0.695 for e_4 due to Giraud in 1979.<br />The second part of the thesis deals with extremal periodicities in strings: we consider the problem of the maximum number of distinct squares in a string. The importance of considering as key variables both the length n and the size d of the alphabet is stressed. Let (d,n)-string denote a string of length n with exactly d distinct symbols. We investigate the function \sigma_d(n) = max {s(x) | x} where s(x) denotes the number of distinct primitively rooted squares in a (d,n)-string x. We discuss a computational framework for computing \sigma_d(n) based on the notion of density and exploiting the tightness of the available lower bound. The obtained computational results substantiate the hypothesized upper bound of n-d for \sigma_d(n). The structural similarities with the approach used for investigating the Hirsch bound for the diameter of a polytope of dimension d having n facets is underlined. For example, the role played by (d,2d)-polytope was presented in 1967 by Klee and Walkup who showed the equivalency between the Hirsch conjecture and the d-step conjecture.</p> / Doctor of Philosophy (PhD)