Compact Graph Structures and Quantum-Theoretic Concepts

Compact graphs are defined as graphs whose fractional automorphism polytope contains no genuinely fractional vertices. This essay introduces a quantum analogue of this concept by examining the fundamental magic unitary associated with the quantum automorphism group acting on graph states. We construct a closed convex set of doubly stochastic matrices that lies between the classical automorphism polytope and the full fractional automorphism polytope. Our principal result establishes that the natural quantum analogue of compactness is, in fact, classical; namely, every quantum compact graph is necessarily classically compact. By investigating the relationship between this convex set and the quantum orbital algebra, we further derive a hierarchy connecting classical and quantum notions of compactness and their corresponding pseudo-compact variants. Moreover, this framework extends several well-known consequences of classical compactness through commutant techniques while providing quantum counterparts to concepts such as generous transitivity and distance transitivity. Finally, the study resolves several open problems, presents illustrative examples of quantum symmetries, and demonstrates that the proposed theory constitutes a strict refinement of the classical theory of graph compactness.

کلیدواژه ها:

Regular graph ، Perfect state transfer ، Continuous-time quantum walk