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 that has been published.  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 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. 

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, 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 June 24, 2018