For a graph G we define S(G) to be the set of all real symmetric n by n matrices whose off-diagonal zero/nonzero pattern is described by G. We show how to compute the minimum rank of all matrices in S(G) for a class of graphs called outerplanar graphs. In addition, we obtain results on the possible eigenvalues and possible inertias of matrices in S(G) for certain classes of graph G. We also obtain results concerning the relationship between two graph parameters, the zero forcing number and the path cover number, related to the minimum rank problem.
College and Department
Physical and Mathematical Sciences; Mathematics
BYU ScholarsArchive Citation
Kempton, Mark Condie, "The Minimum Rank, Inverse Inertia, and Inverse Eigenvalue Problems for Graphs" (2010). Theses and Dissertations. 2182.
matrix, graph, minimum rank, inertia, zero forcing, eigenvalue