Early Graduate Research (EGR)
Math 610: Early Graduate Research (EGR) has been offered in the spring since 2008 by various instructors.
This page describes the version led by Leslie Hogben in 2008, 2011, 2014, and 2017, and the EGR planned for 2019.
During the 3-4 weeks of the semester I teach some of the
necessary background material and students read related research
papers. The group then splits into teams and each team begins
research on a problem, meeting weekly to discuss findings. Toward the end of the semester teams present their work in the Discrete Mathematics Seminar and paper(s) are written and submitted.
Section: The graph theory EGR described here is Section C of Math 610.
Prerequisite: The prerequisite is some knowledge of graph
theory, and first and second year graduate students are particularly
encouraged to enroll.
Class meeting time/place:
The official class time is 2:10-5:00 PM Mondays in Carver 409.
The plan is for
all of us to meet during these 3 hours the first few weeks and then
split into two or more research groups (depending on how many students
there are) with each student involed in one group and attending two hours of research meetings
per week after that. The research group meetings will be
scheduled at mutual convenience (including the 2-5PM Monday, but other
times are also possible).
Topic: During Spring 2019 students will work on questions related to graph
searching, the time needed to complete the search, and a combination of
resources used to accomplish the task and time needed to accomplish the
task (throttling). Parameters that may be studied include zero
forcing, power domination, cops & robbers, metric dimension, etc.
In spring 2017, the research problems were
aspects of the inverse eigenvalue of
a graph and related parameters. Groups worked on graphs with
maximum nullity equal to zero forcing number, the minimum number of
eigenvalues of a graph, and throttling for positive semidefinite zero
forcing. Each group submitted a paper; one has been published and the other two have been accepted.
In spring 2014 the research focused on minimum rank for loop graphs. During that semester the group subdivided into smaller groups working
on different parts but came together to produce one paper, which has been published.
In spring 2011, the research problems were on positive semidefinite
zero forcing number. Zero forcing number is a graph parameter
related to minimum rank and also used by physicists in quantum control;
positive semidefinite zero forcing number is a variant.
During that semester the group subdivided into smaller groups working
on different parts but came together to produce one paper that has been published. This group
also spun off two subgroups, each of which published a paper.
In spring 2008, the research problems were on various
aspects of the minimum rank problem for symmetric matrices described by
a graph, with one group working on small graphs (and developing computational software) and the other working
on minimum rank over fields other than the real numbers. Both
groups published papers.
Families of graphs with maximum nullity equal to zero forcing number
J.S. Alameda, E. Curl, A. Grez, L. Hogben, O'N. Kingston, A. Schulte, D. Young, M. Young. Special Matrices 6 (2018) 56 – 67 [PDF preprint] (2017 EGR)
Applications of analysis to the determination of the minimum number of distinct eigenvalues of a graph
B. Bjorkman, L. Hogben, S. Ponce, C. Reinhart, T. Tranel. To appear in Pure Appl. Funct. Anal. [PDF preprint] (2017 EGR)
Throttling positive semidefinite zero forcing propagation time on graphs
J. Carlson, L. Hogben, J. Kritschgau, K. Lorenzen, M.S. Ross, S. Selken, V. Valle Martinez. To appear in Discrete Applied Mathematics [PDF preprint] (2017 EGR)
Minimum rank of graphs with loops.
C. Bozeman, AV. Ellsworth, L. Hogben, J.C.-H. Lin, G. Maurer, K. Nowak, A. Rodriguez, J.
Strickland. Electonic Journal of Linear Algebra 27 (2014) 907 – 934. [PDF] (2014 EGR)
Computing positive semidefinite minimum rank for small graphs. S. Osborne, N. Warnberg. Involve: a journal of mathematics 7 (2014) 595–609. (2011 EGR)
Positive semidefinite zero forcing number.
J.Ekstrand, C. Erickson, H.T. Hall, D. Hay, L. Hogben, R. Johnson, N.
Kingsley, S. Osborne, T. Peters, J. Roat, A. Ross, D. Row, N. Warnberg,
M. Young. Linear Algebra and its Applications, 439 (2013): 1862 – 1874. [PDF preprint] (2011 EGR)
Note on positive semidefinite maximum nullity and positive semidefinite zero forcing number of partial 2-trees. J.Ekstrand, C. Erickson, D. Hay, L. Hogben, J. Roat. Electonic Journal of Linear Algebra, 23 (2012) 79-87 [PDF] (2011 EGR)
for determining the minimum rank of a small graph L. DeLoss,
J. Grout, L. Hogben, T. McKay, J. Smith, G. Tims. Linear Algebra and its Applications, 432 (2010) 2995–3001. [PDF
preprint] (2008 EGR)
optimal matrices and field independence of the minimum rank of a graph L. DeAlba,
J. Grout, L. Hogben, R. Mikkelson, K. Rasmussen. Electronic
Algebra 18 (2009): 403-419. (2008 EGR)