David E. Roberson

Graph Homomorphisms for Quantum Players

A homomorphism from a graph X to a graph Y is an adjacency preserving mapping f:V(X) -> V(Y). We consider a nonlocal game in which Alice and Bob are trying to convince a verifier with certainty that a graph X admits a homomorphism to Y. This is a generalization of the well-studied graph coloring game. Via systematic study of quantum homomorphisms we prove new results for graph coloring. Most importantly, we show that the Lovasz theta number of the complement lower bounds the quantum chromatic number, which itself is not known to be computable. We also show that other quantum graph parameters, such as quantum independence number, can differ from their classical counterparts. Finally, we show that quantum homomorphisms closely relate to zero-error channel capacity. In particular, we use quantum homomorphisms to construct graphs for which entanglement-assistance increases their one-shot zero-error capacity.

Bounds on Entanglement Assisted Source-channel Coding Via the Lovász Theta Number and Its Variants

We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs G and H. Such vectors exist if and only if theta(G) <= theta(H) where theta represents the Lovasz number. We also obtain similar inequalities for the related Schrijver theta^- and Szegedy theta^+ numbers. These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: alpha^*(G) <= theta^-(G). Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lovasz number. Beigi introduced a quantity beta as an upper bound on alpha^* and posed the question of whether beta(G) = \lfloor theta(G) \rfloor. We answer this in the affirmative and show that a related quantity is equal to \lceil theta(G) \rceil. We show that a quantity chi_{vect}(G) recently introduced in the context of Tsirelsons conjecture is equal to \lceil theta^+(G) \rceil.

Homomorphisms of Strongly Regular Graphs

We prove that if

Fractional zero forcing via three-color forcing games

G

Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions

and

Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings

H

A new property of the Lovasz number and duality relations between graph parameters

are primitive strongly regular graphs with the same parameters and

Bounds on Entanglement-Assisted Source-Channel Coding via the Lovász \(\vartheta \) Number and Its Variants

\varphi

Quantum and non-signalling graph isomorphisms

is a homomorphism from

Note on von Neumann and Rényi entropies of a graph

G