Combinatorics and Optimization
http://hdl.handle.net/10012/9928
2018-11-16T12:47:09Z
2018-11-16T12:47:09Z
On Eulerian orientations of even-degree hypercubes
Levit, Maxwell
Chandran, L. Sunil
Cheriyan, Joseph
http://hdl.handle.net/10012/14037
2018-10-23T02:31:32Z
2018-09-01T00:00:00Z
On Eulerian orientations of even-degree hypercubes
Levit, Maxwell; Chandran, L. Sunil; Cheriyan, Joseph
It is well known that every Eulerian orientation of an Eulerian 2k-edge connected (undirected) graph is strongly k-edge connected. A long-standing goal in the area is to obtain analogous results for other types of connectivity, such as node connectivity. We show that every Eulerian orientation of the hypercube of degree 2k is strongly k-node connected.
The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.orl.2018.09.002 © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/
2018-09-01T00:00:00Z
Discrete Quantum Walks on Graphs and Digraphs
Zhan, Hanmeng
http://hdl.handle.net/10012/13952
2018-09-27T02:31:07Z
2018-09-26T00:00:00Z
Discrete Quantum Walks on Graphs and Digraphs
Zhan, Hanmeng
This thesis studies various models of discrete quantum walks on graphs and digraphs via a spectral approach.
A discrete quantum walk on a digraph $X$ is determined by a unitary matrix $U$, which acts on complex functions of the arcs of $X$. Generally speaking, $U$ is a product of two sparse unitary matrices, based on two direct-sum decompositions of the state space. Our goal is to relate properties of the walk to properties of $X$, given some of these decompositions.
We start by exploring two models that involve coin operators, one due to Kendon, and the other due to Aharonov, Ambainis, Kempe, and Vazirani. While $U$ is not defined as a function in the adjacency matrix of the graph $X$, we find exact spectral correspondence between $U$ and $X$. This leads to characterization of rare phenomena, such as perfect state transfer and uniform average vertex mixing, in terms of the eigenvalues and eigenvectors of $X$. We also construct infinite families of graphs and digraphs that admit the aforementioned phenomena.
The second part of this thesis analyzes abstract quantum walks, with no extra assumption on $U$. We show that knowing the spectral decomposition of $U$ leads to better understanding of the time-averaged limit of the probability distribution. In particular, we derive three upper bounds on the mixing time, and characterize different forms of uniform limiting distribution, using the spectral information of $U$.
Finally, we construct a new model of discrete quantum walks from orientable embeddings of graphs. We show that the behavior of this walk largely depends on the vertex-face incidence structure. Circular embeddings of regular graphs for which $U$ has few eigenvalues are characterized. For instance, if $U$ has exactly three eigenvalues, then the vertex-face incidence structure is a symmetric $2$-design, and $U$ is the exponential of a scalar multiple of the skew-symmetric adjacency matrix of an oriented graph. We prove that, for every regular embedding of a complete graph, $U$ is the transition matrix of a continuous quantum walk on an oriented graph.
2018-09-26T00:00:00Z
Results on Chromatic Polynomials Inspired by a Correlation Inequality of G.E. Farr
McKay, Ghislain
http://hdl.handle.net/10012/13924
2018-09-26T02:34:33Z
2018-09-25T00:00:00Z
Results on Chromatic Polynomials Inspired by a Correlation Inequality of G.E. Farr
McKay, Ghislain
In1993 Graham Farr gave a proof of a correlation inequality involving colourings of graphs. His work eventually led to a conjecture that number of colourings of a graph with certain properties gave a log-concave sequence. We restate Farr's work in terms of the bivariate chromatic polynomial of Dohmen, Poenitz, Tittman and give a simple, self-contained proof of Farr's inequality using a basic combinatorial approach. We attempt to prove Farr's conjecture through methods in stable polynomials and computational verification, ultimately leading to a stronger conjecture.
2018-09-25T00:00:00Z
Ordinary and Generalized Circulation Algebras for Regular Matroids
Olson-Harris, Nicholas
http://hdl.handle.net/10012/13826
2018-09-20T02:31:14Z
2018-09-19T00:00:00Z
Ordinary and Generalized Circulation Algebras for Regular Matroids
Olson-Harris, Nicholas
Let E be a finite set, and let R(E) denote the algebra of polynomials in indeterminates (x_e)_{e in E}, modulo the squares of these indeterminates. Subalgebras of R(E) generated by homogeneous elements of degree 1 have been studied by many authors and can be understood combinatorially in terms of the matroid represented by the linear equations satisfied by these generators. Such an algebra is related to algebras associated to deletions and contractions of the matroid by a short exact sequence, and can also be written as the quotient of a polynomial algebra by certain powers of linear forms.
We study such algebras in the case that the matroid is regular, which we term circulation algebras following Wagner. In addition to surveying the existing results on these algebras, we give a new proof of Wagner's result that the structure of the algebra determines the matroid, and construct an explicit basis in terms of basis activities in the matroid. We then consider generalized circulation algebras in which we mod out by a fixed power of each variable, not necessarily equal to 2. We show that such an algebra is isomorphic to the circulation algebra of a "subdivided" matroid, a variation on a result of Nenashev, and derive from this generalized versions of many of the results on ordinary circulation algebras, including our basis result. We also construct a family of short exact sequences generalizing the deletion-contraction decomposition.
2018-09-19T00:00:00Z