On the minimum rank of certain graphs with path cover number 2

The minimum rank problem is an interesting and ongoing problem in spectral graph theory which seeks to answer the question "Given a simple graph G what is the minimum rank of a matrix whose off-diagonal zero/nonzero pattern is described by G?" In recent years, the minimum rank of trees, unicyclic graphs, and cases of extreme minimum rank have been completely characterized. However, little is known about other families of graphs. Recent work in zero-forcing parameters, minimum semidefinite rank, and ranks of outerplanar graphs have given more ways to calculate upper and lower bounds for the minimum rank of a graph. We define a family of graphs with path cover number two and consider restrictions on the structure and minimum rank of these types of graphs. We consider a sub-family of these graphs and calculate the zero-forcing number and the positive semidefinite minimum rank. We also conjecture toward the minimum rank of these graphs.

Grantor dc:publisher

University of Tennessee at Chattanooga

Author dc:creator

• Corley, Christopher M.

Contributors dc:contributor

• Barioli, Francesco
• Van der Merwe, Lucas; Smith, Ron; Ledoan, Andrew
• College of Arts and Sciences

• Graph theory (Mathematics)
• Combinatorial analysis

Licence

• https://rightsstatements.org/page/InC/1.0/?language=en

Language dc:language

English, eng

Repository record dc:identifier

https://scholar.utc.edu/theses/166

OAI identifier oai:identifier

oai:scholar.utc.edu:theses-1307