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.

EGR 2019

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.

Past EGRs

In spring 2017, the research problems  were on various 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. 

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)

Techniques 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)

Universally optimal matrices and field independence of the minimum rank of a graph L. DeAlba, J. Grout, L. Hogben, R. Mikkelson, K. Rasmussen. Electronic Journal of Linear Algebra 18 (2009): 403-419. (2008 EGR)


 
Leslie Hogben's Homepage October 24, 2018