We prove that the out-distance sequence {f(k)} of a vertex-transitive digraph of finite or infinite degree satisfies f(k + 1) ≤ f(k) for k ≥ 1, where f(k) denotes the number of vertices at directed distance k from a given vertex. As a corollary, we prove that for a connected vertextransitive undirected graph of infinite degree d, we have f(k) = d for all k, 1 ≤ k < diam(G). This answers a quest...

Let n be an integer and q be a prime power. Then for any 3 ≤ n ≤ q − 1, or n = 2 and q odd, we construct a connected q-regular edgebut not vertextransitive graph of order 2qn+1. This graph is defined via a system of equations over the finite field of q elements. For n = 2 and q = 3, our graph is isomorphic to the Gray graph.

............................................................................................................................................ 2 ACKNOWLEDGEMENTS ....................................................................................................................... 4 CONTENTS .............................................................................................................

In addition to regular and chiral polyhedra, which have been extensively studied, the vertextransitive polyhedra of higher genus also present an attractive and worthwhile challenge. While the definition is combinatorial, the problem at hand is the realization in Euclidean 3-space as a highly symmetric, non-self-intersecting polyhedron in the more classical sense (with flat, non-self-intersectin...

A graph Γ is self-complementary if its complement is isomorphic to the graph itself. An isomorphism that maps Γ to its complement Γ is called a complementing isomorphism. The majority of this dissertation is intended to present my research results on the study of self-complementary vertex-transitive graphs. I will provide an introductory mini-course for the backgrounds, and then discuss four pr...

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...

iv Acknowledgements v List of Tables vii List of Graphs vii

................................................................................................................................................. 3 Acknowledgements................................................................................................................................ 6 List of Tables...........................................................................................

The main results in this paper are about the full coalescence time C of a system of coalescing random walks over a finite graph G. Letting m(G) denote the mean meeting time of two such walkers, we give sufficient conditions under which E [C] ≈ 2m(G) and C/m(G) has approximatelly the same law as in the “mean field” setting of a large complete graph. One of our theorems is that mean field behavio...

let $n$ be any positive integer and let $f_n$ be the friendship (or dutch windmill) graph with $2n+1$ vertices and $3n$ edges. here we study graphs with the same adjacency spectrum as the $f_n$. two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. let $g$ be a graph cospectral with $f_n$. here we prove that if $g$ has no cycle of length $4$ or $...