DEDICATION To my beloved parents. v ACKNOWLEDGEMENTS I want to thank my advisor, R. Craig Albertson, for all his help over the past six years, for always being patient and supportive. I would like to thank my committee members, Betsy Dumont, Rolf Karlstrom and Steve McCormick, for supporting and guiding my research. I would also like to thank Bill Detrich, for his generous offer of a field trip...

A labeling of a graph is a function from the vertices of the graph to some finite set. In 1996, Albertson and Collins defined distinguishing labelings of undirected graphs. Their definition easily extends to directed graphs. Let G be a directed graph associated to the k-block presentation of a Bernoulli scheme X . We determine the automorphism group of G, and thus the distinguishing labelings o...

For a vertex set S ⊆ V (G) in a graph G, the distance multiset, D(S), is the multiset of pairwise distances between vertices of S in G. Two vertex sets are called homometric if their distance multisets are identical. For a graph G, the largest integer h, such that there are two disjoint homometric sets of order h in G, is denoted by h(G). We slightly improve the general bound on this parameter ...

A simple graph G is said to be regular if its vertices have the same number of neighbors. Otherwise, id="M2"> nonregular. So far, various formulas, such as Albertson index, total and degree deviation, been introduced quantify irregularity a graph. In this paper, we present sharp lower bounds for these indices in terms or...

Let \(\varLambda (T)\) denote the set of leaves in a tree T. One natural problem is to look for spanning T given graph G such that as large possible. Recently, similar but stronger notion called robust connectivity was introduced, which defined minimum value \(\frac{|R \cap \varLambda (T)|}{|R|}\) taken over all nonempty subsets \({R\subseteq V(G)}\), where \(T = T(R)\) on chosen maximize \({|R...

A celebrated result of Thomassen states that not only can every planar graph be colored properly with five colors, but no matter how arbitrary palettes of five colors are assigned to vertices, one can choose a color from the corresponding palette for each vertex so that the resulting coloring is proper. This result is referred to as 5-choosability of planar graphs. Albertson asked whether Thoma...

Introduced by Albertson and al. [1], the distinguishing number D(G) of a graph G is the least integer r such that there is a r-labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs not depend on n. In this pa...

A graph G is distinguished if its vertices are labelled by a map φ : V (G) −→ {1, 2, . . . , k} so that no non-trivial graph automorphism preserves φ. The distinguishing number of G is the minimum number k necessary for φ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of Γ on a set X. A labelling φ : X −→ {1, 2, . . . , ...