This thesis studies the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We begin by finding the eigenvalues of matrices describing the quantum walk for regular graphs. We also show that if two graphs are isomorphic, then the corresponding matrices produced by the procedure of Emms et al. are cospectral. We then look at the entries of the cube of the transition matrix and find an expression for the matrices produced by the procedure of Emms et al. in terms of the adjacency matrix and incidence matrices of the graph.

It is not hard to see that the number of equiangular lines in a complex space of dimension $d$ is at most $d^{2}$. A set of $d^{2}$ equiangular lines in a $d$-dimensional complex space is of significant importance in Quantum Computing as it corresponds to a measurement for which its statistics determine completely the quantum state on which the measurement is carried out. The existence of $d^{2}$ equiangular lines in a $d$-dimensional complex space is only known for a few values of $d$, although physicists conjecture that they do exist for any value of $d$. The main results in this thesis are: \begin{enumerate} \item Abelian covers of complete graphs that have certain parameters can be used to construct sets of $d^2$ equiangular lines in $d$-dimen\-sion\-al space; \item we exhibit infinitely many parameter sets that satisfy all the known necessary conditions for the existence of such a cover; and \item we find the decompose of the space into irreducible modules over the Terwilliger algebra of covers of complete graphs. \end{enumerate} A few techniques are known for constructing covers of complete graphs, none of which can be used to construct covers that lead to sets of $d^{2}$ equiangular lines in $d$-dimensional complex spaces. The third main result is developed in the hope of assisting such construction.

Graph searching problems are described as games played on graphs, between a set of searchers and a fugitive. Variants of the game restrict the abilities of the searchers and the fugitive and the corresponding search number (the least number of searchers that have a winning strategy) is related to several well-known parameters in graph theory. One popular variant is called the Cops and Robber game, where the searchers (cops) and the fugitive (robber) move in rounds, and in each round they move to an adjacent vertex. This game, defined in late 1970's, has been studied intensively. The most famous open problem is Meyniel's conjecture, which states that the cop number (the minimum number of cops that can always capture the robber) of a connected graph on n vertices is O(sqrt n). We consider a version of the Cops and Robber game, where the robber is faster than the cops, but is not allowed to jump over the cops. This version was first studied in 2008. We show that when the robber has speed s, the cop number of a connected n-vertex graph can be as large as Omega(n^(s/s+1)). This improves the Omega(n^(s-3/s-2)) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, to appear). We also conjecture a general upper bound O(n^(s/s+1)) for the cop number, generalizing Meyniel's conjecture. Then we focus on the version where the robber is infinitely fast, but is again not allowed to jump over the cops. We give a mathematical characterization for graphs with cop number one. For a graph with treewidth tw and maximum degree Delta, we prove the cop number is between (tw+1)/(Delta+1) and tw+1. Using this we show that the cop number of the m-dimensional hypercube is between c1 n / m sqrt(m) and c2 n / m for some constants c1 and c2. If G is a connected interval graph on n vertices, then we give a polynomial time 3-approximation algorithm for finding the cop number of G, and prove that the cop number is O(sqrt(n)). We prove that given n, there exists a connected chordal graph on n vertices with cop number Omega(n/log n). We show a lower bound for the cop numbers of expander graphs, and use this to prove that the random G(n,p) that is not very sparse, asymptotically almost surely has cop number between d1 / p and d2 log (np) / p for suitable constants d1 and d2. Moreover, we prove that a fixed-degree regular random graph with n vertices asymptotically almost surely has cop number Theta(n).

A clique of a graph G is a set of pairwise adjacent nodes of G. Similarly, a biclique (U, V ) of a bipartite graph G is a pair of disjoint, independent vertex sets such that each node in U is adjacent to every node in V in G. We consider the problems of identifying the maximum clique of a graph, known as the maximum clique problem, and identifying the biclique (U, V ) of a bipartite graph that maximizes the product |U | · |V |, known as the maximum edge biclique problem. We show that finding a clique or biclique of a given size in a graph is equivalent to finding a rank one matrix satisfying a particular set of linear constraints. These problems can be formulated as rank minimization problems and relaxed to convex programming by replacing rank with its convex envelope, the nuclear norm. Both problems are NP-hard yet we show that our relaxation is exact in the case that the input graph contains a large clique or biclique plus additional nodes and edges. For each problem, we provide two analyses of when our relaxation is exact. In the first, the diversionary edges are added deterministically by an adversary. In the second, each potential edge is added to the graph independently at random with fixed probability p. In the random case, our bounds match the earlier bounds of Alon, Krivelevich, and Sudakov, as well as Feige and Krauthgamer for the maximum clique problem. We extend these results and techniques to the k-disjoint-clique problem. The maximum node k-disjoint-clique problem is to find a set of k disjoint cliques of a given input graph containing the maximum number of nodes. Given input graph G and nonnegative edge weights w, the maximum mean weight k-disjoint-clique problem seeks to identify the set of k disjoint cliques of G that maximizes the sum of the average weights of the edges, with respect to w, of the complete subgraphs of G induced by the cliques. These problems may be considered as a way to pose the clustering problem. In clustering, one wants to partition a given data set so that the data items in each partition or cluster are similar and the items in different clusters are dissimilar. For the graph G such that the set of nodes represents a given data set and any two nodes are adjacent if and only if the corresponding items are similar, clustering the data into k disjoint clusters is equivalent to partitioning G into k-disjoint cliques. Similarly, given a complete graph with nodes corresponding to a given data set and edge weights indicating similarity between each pair of items, the data may be clustered by solving the maximum mean weight k-disjoint-clique problem. We show that both instances of the k-disjoint-clique problem can be formulated as rank constrained optimization problems and relaxed to semidefinite programs using the nuclear norm relaxation of rank. We also show that when the input instance corresponds to a collection of k disjoint planted cliques plus additional edges and nodes, this semidefinite relaxation is exact for both problems. We provide theoretical bounds that guarantee exactness of our relaxation and provide empirical examples of successful applications of our algorithm to synthetic data sets, as well as data sets from clustering applications.

For a given graph, G, the crossing number crₐ(G) denotes the minimum number of edge crossings when a graph is drawn on an orientable surface of genus a. The sequence cr₀(G), cr₁(G), ... is said to be the crossing sequence of a G. An equivalent definition exists for non-orientable surfaces. In 1983, Jozef Širáň proved that for every decreasing, convex sequence of non-negative integers, there is a graph G such that this sequence is the crossing sequence of G. This main result of this thesis proves the existence of a graph with non-convex crossing sequence of arbitrary length.

We study a directed network design problem called the $k$-$(S,T)$-connectivity problem; we design and analyze approximation algorithms and give hardness results. For each positive integer $k$, the minimum cost $k$-vertex connected spanning subgraph problem is a special case of the $k$-$(S,T)$-connectivity problem. We defer precise statements of the problem and of our results to the introduction. For $k=1$, we call the problem the $(S,T)$-connectivity problem. We study three variants of the problem: the standard $(S,T)$-connectivity problem, the relaxed $(S,T)$-connectivity problem, and the unrestricted $(S,T)$-connectivity problem. We give hardness results for these three variants. We design a $2$-approximation algorithm for the standard $(S,T)$-connectivity problem. We design tight approximation algorithms for the relaxed $(S,T)$-connectivity problem and one of its special cases. For any $k$, we give an $O(\log k\log n)$-approximation algorithm, where $n$ denotes the number of vertices. The approximation guarantee almost matches the best approximation guarantee known for the minimum cost $k$-vertex connected spanning subgraph problem which is $O(\log k\log\frac{n}{n-k})$ due to Nutov in 2009.

The central topic of this thesis is Schnyder's Theorem. Schnyder's Theorem provides a characterization of planar graphs in terms of their poset dimension, as follows: a graph G is planar if and only if the dimension of the incidence poset of G is at most three. One of the implications of the theorem is proved by giving an explicit mapping of the vertices to R^2 that defines a straightline embedding of the graph. The other implication is proved by introducing the concept of normal labelling. Normal labellings of plane triangulations can be used to obtain a realizer of the incidence poset. We present an exposition of Schnyder’s theorem with his original proof, using normal labellings. An alternate proof of Schnyder’s Theorem is also presented. This alternate proof does not use normal labellings, instead we use some structural properties of a realizer of the incidence poset to deduce the result. Some applications and a generalization of one implication of Schnyder’s Theorem are also presented in this work. Normal labellings of plane triangulations can be used to obtain a barycentric embedding of a plane triangulation, and they also induce a partition of the edge set of a plane triangulation into edge disjoint trees. These two applications of Schnyder’s Theorem and a third one, relating realizers of the incidence poset and canonical orderings to obtain a compact drawing of a graph, are also presented. A generalization, to abstract simplicial complexes, of one of the implications of Schnyder’s Theorem was proved by Ossona de Mendez. This generalization is also presented in this work. The concept of order labelling is also introduced and we show some similarities of the order labelling and the normal labelling. Finally, we conclude this work by showing the source code of some implementations done in Sage.

This thesis studies the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We begin by finding the eigenvalues of matrices describing the quantum walk for regular graphs. We also show that if two graphs are isomorphic, then the corresponding matrices produced by the procedure of Emms et al. are cospectral. We then look at the entries of the cube of the transition matrix and find an expression for the matrices produced by the procedure of Emms et al. in terms of the adjacency matrix and incidence matrices of the graph.

It is not hard to see that the number of equiangular lines in a complex space of dimension $d$ is at most $d^{2}$. A set of $d^{2}$ equiangular lines in a $d$-dimensional complex space is of significant importance in Quantum Computing as it corresponds to a measurement for which its statistics determine completely the quantum state on which the measurement is carried out. The existence of $d^{2}$ equiangular lines in a $d$-dimensional complex space is only known for a few values of $d$, although physicists conjecture that they do exist for any value of $d$. The main results in this thesis are: \begin{enumerate} \item Abelian covers of complete graphs that have certain parameters can be used to construct sets of $d^2$ equiangular lines in $d$-dimen\-sion\-al space; \item we exhibit infinitely many parameter sets that satisfy all the known necessary conditions for the existence of such a cover; and \item we find the decompose of the space into irreducible modules over the Terwilliger algebra of covers of complete graphs. \end{enumerate} A few techniques are known for constructing covers of complete graphs, none of which can be used to construct covers that lead to sets of $d^{2}$ equiangular lines in $d$-dimensional complex spaces. The third main result is developed in the hope of assisting such construction.

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.