@vcu.edu Research
I have many mathematical interests, but am currently working on problems in C*-algebras, matrix analysis, and graph theory. I am always looking for new and interesting questions and collaborations. My publications include the following:
- On the Minimum Rank Among Positive Semidefinite Matrices With A Given Graph to appear in SIAM Journal on Matrix Analysis and Applications.
- Unitary Matrix Digraphs and Minimum Semidefinite Rank in Linear Algebra and its Applications (2008).
For an undirected simple graph $G$, the minimum rank among all positive semidefinite matrices with graph $G$ is called the minimum semidefinite rank (msr) of $G$. In this paper, we show that the msr of a given graph may be determined from the msr of a related bipartite graph. Finding the msr of a given bipartite graph is then shown to be equivalent to determining which digraphs encode the zero/nonzero pattern of a unitary matrix. We provide an algorithm to construct unitary matrices with a certain pattern, and use previous results to give a lower bound for the msr of certain bipartite graphs.
- Simplicity of C*-Algebras Using Unique Eigenstates. Ph.D. Dissertation, University of Kansas (2007).
- Maximal Total Absolute Displacement of a Permutation in Discrete Mathematics (2004).
- A Characterization of Tree Type in Rose-Hulman Undergraduate Mathematics Journal (2003).
The entries of an $n$ by $n$ Hermitian matrix $A=(a_{ij})$ over the complex numbers naturally determine an undirected graph $G(A)$ with vertices $v_i$ and edge set \[E=\{(v_i,v_j) \colon a_{ij} \neq 0, i \geq j \} \; .\] The minimum positive semidefinite rank (msr) of a graph $G$ is the minimum rank among all positive semidefinite matrices with graph $G$. In this paper, we show that the minimum semidefinite rank of all chordal graphs (graphs that do not contain an induced cycle on four or more vertices) is equal to the (edge) clique-cover number.
We consider a one-parameter family of operators \[ X_q = S_1\left(1 - \left(1-q\right)S_2S_2^*\right) + \left( 1 - \left(1-q\right)S_1S_1^* \right) S_2^* \] constructed from a pair of isometries $S_1$ and $S_2$ on Hilbert space with complementary orthogonal ranges that is, \[ S_1S_1^*+ S_2S_2^* = 1 = S_1^*S_1 = S_2^*S_2 \; . \] For special values of $q$, the operator $X_q$ plays a role in the representation theory of free groups and in free probability theory. For each value of $q$, we identify the irreducible *-representations of the pair of isometries in which the operator has an eigenvalue. This yields a new technique for showing that certain subalgebras of the Cuntz algebra $\mathcal{O}_2$, including the C*-algebra generated by the operator, are simple.