72 research outputs found
MATH 314: Linear Algebra-A Peer Review of Teaching Project Benchmark Portfolio
My intention with this portfolio is to present my approach to teaching Math 314. This is a linear algebra course for senior-year undergraduate students in different STEM programs. The course serves as a transition from computational courses, such as calculus, to more theoretical ones, and exposes students, possibly for the first time, to abstract mathematical objects (such as general vector spaces) and simple proofs. Several lecture-based strategies are discussed and evaluated in connection with student learning outcomes. I also present samples of course materials and student work
The bondage number of random graphs
A dominating set of a graph is a subset of its vertices such that every
vertex not in is adjacent to at least one member of . The domination
number of a graph is the number of vertices in a smallest dominating set of
. The bondage number of a nonempty graph is the size of a smallest set
of edges whose removal from results in a graph with domination number
greater than the domination number of . In this note, we study the bondage
number of binomial random graph . We obtain a lower bound that matches
the order of the trivial upper bound. As a side product, we give a one-point
concentration result for the domination number of under certain
restrictions
A probabilistic version of the game of Zombies and Survivors on graphs
We consider a new probabilistic graph searching game played on graphs,
inspired by the familiar game of Cops and Robbers. In Zombies and Survivors, a
set of zombies attempts to eat a lone survivor loose on a given graph. The
zombies randomly choose their initial location, and during the course of the
game, move directly toward the survivor. At each round, they move to the
neighbouring vertex that minimizes the distance to the survivor; if there is
more than one such vertex, then they choose one uniformly at random. The
survivor attempts to escape from the zombies by moving to a neighbouring vertex
or staying on his current vertex. The zombies win if eventually one of them
eats the survivor by landing on their vertex; otherwise, the survivor wins. The
zombie number of a graph is the minimum number of zombies needed to play such
that the probability that they win is strictly greater than 1/2. We present
asymptotic results for the zombie numbers of several graph families, such as
cycles, hypercubes, incidence graphs of projective planes, and Cartesian and
toroidal grids
Rainbow Hamilton Cycles in Random Geometric Graphs
Let be chosen independently and uniformly at random from
the unit -dimensional cube . Let be given and let . The random geometric graph has
vertex set and an edge whenever . We show
that if each edge of is colored independently from one of colors
and has the smallest value such that has minimum degree at least two,
then contains a rainbow Hamilton cycle a.a.s
The domination number of on-line social networks and random geometric graphs
We consider the domination number for on-line social networks, both in a
stochastic network model, and for real-world, networked data. Asymptotic
sublinear bounds are rigorously derived for the domination number of graphs
generated by the memoryless geometric protean random graph model. We establish
sublinear bounds for the domination number of graphs in the Facebook 100 data
set, and these bounds are well-correlated with those predicted by the
stochastic model. In addition, we derive the asymptotic value of the domination
number in classical random geometric graphs
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