Jason Grout

UNIVERSALLY OPTIMAL MATRICES AND FIELD INDEPENDENCE OF THE MINIMUM RANK OF A GRAPH

The minimum rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose (i, j)th entry (for ij) isnonzero whenever {i, j} isan edge in G and is zero otherwise. A universally optimal matrix is defined to be an integer matrix A such that every off-diagonal entry of A is0, 1, or −1, and for all fields F , the rank of A isthe minimum rank over F of its graph. Universally optimal matrices are used to establish field independence of minimum rank for numerousgraphs . Examplesare als o provided verifying lack of field independence for other graphs.

Using variants of zero forcing to bound the inertia set of a graph

Zero forcing is a combinatorial game played on a graph with a goal of changing the color of every vertex at minimal cost. This leads to a parameter known as the zero forcing number that can be used to give an upper bound for the maximum nullity of a matrix associated with the graph. A variation on the zero forcing game is introduced that can be used to give an upper bound for the maximum nullity of such a matrix when it is constrained to have exactly q negative eigenvalues. This constrains the possible inertias that a matrix associated with a graph can achieve and gives a method to construct lower bounds on the inertia set of a graph (which is the set of all possible pairs (p,q) where p is the number of positive eigenvalues and q is the number of negative eigenvalues).

Minimum rank of powers of trees

The minimum rank of a simple graph G over a field F is the smallest possible rank among all real symmetric matrices, over F, whose (i, j)-entry (for i 6= j) is nonzero whenever ij is an edge in G and is zero otherwise. In this paper, the problem of minimum rank of (strict) powers of trees is studied.