Early Graduate Research (EGR)
Math 610: Early Graduate Research (EGR) has now
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).
EGR is modeled on the successful Combinatorial Matrix Theory (CMT)
research group that has involved students in publishable
research since 2000.
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.
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.
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. This group
also spun off two subgroups, each of which published a paper.
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 2017, the research problems were
aspects of the inverse eigenvalue of
a graph and related parameters. Groups worked on grapsh with
maximum nullity equal to zero forcing number, the mninimum number of
eogenvalues of a graph, and throttling for positive semidefintite zero
forcing. Each group submitted a paper; one is accepted and two
are under review.
Throttling positive semidefinite zero forcing propagation time on graphs
J. Carlson, L. Hogben, J. Kritschgau, K. Lorenzen, M.S. Ross, V. Valle Martinez. [PDF preprint] (2017 EGR)
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. To appear in Special Matrices [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)
Minimum rank of graphs with loops.
C. Bozeman, AV. Ellsworth, L. Hogben, J.C.-H. Lin, G. Maurer, K. Nowak, A. Rodriguez, J.
Strickland. Electron. J. Linear Algebra 27 (2014): 907 – 934. [PDF] (2014 EGR)
Computing positive semidefinite minimum rank for small graphs. S. Osborne, N. Warnberg. To appear in Involve: a journal of mathematics. (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)