24,768 research outputs found
The Cover Time of Random Walks on Graphs
A simple random walk on a graph is a sequence of movements from one vertex to
another where at each step an edge is chosen uniformly at random from the set
of edges incident on the current vertex, and then transitioned to next vertex.
Central to this thesis is the cover time of the walk, that is, the expectation
of the number of steps required to visit every vertex, maximised over all
starting vertices. In our first contribution, we establish a relation between
the cover times of a pair of graphs, and the cover time of their Cartesian
product. This extends previous work on special cases of the Cartesian product,
in particular, the square of a graph. We show that when one of the factors is
in some sense larger than the other, its cover time dominates, and can become
within a logarithmic factor of the cover time of the product as a whole. Our
main theorem effectively gives conditions for when this holds. The techniques
and lemmas we introduce may be of independent interest. In our second
contribution, we determine the precise asymptotic value of the cover time of a
random graph with given degree sequence. This is a graph picked uniformly at
random from all simple graphs with that degree sequence. We also show that with
high probability, a structural property of the graph called conductance, is
bounded below by a constant. This is of independent interest. Finally, we
explore random walks with weighted random edge choices. We present a weighting
scheme that has a smaller worst case cover time than a simple random walk. We
give an upper bound for a random graph of given degree sequence weighted
according to our scheme. We demonstrate that the speed-up (that is, the ratio
of cover times) over a simple random walk can be unboundedComment: 179 pages, PhD thesi
Dominating sequences in grid-like and toroidal graphs
A longest sequence of distinct vertices of a graph such that each
vertex of dominates some vertex that is not dominated by its preceding
vertices, is called a Grundy dominating sequence; the length of is the
Grundy domination number of . In this paper we study the Grundy domination
number in the four standard graph products: the Cartesian, the lexicographic,
the direct, and the strong product. For each of the products we present a lower
bound for the Grundy domination number which turns out to be exact for the
lexicographic product and is conjectured to be exact for the strong product. In
most of the cases exact Grundy domination numbers are determined for products
of paths and/or cycles.Comment: 17 pages 3 figure
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