Quantum-Theoretic Perspectives on Compact Graph Structures

A compact graph can be understood as a graph whose fractional automorphism polytope possesses only integral extreme points, thereby excluding genuinely fractional vertices. This study develops a quantum-theoretic extension of compactness by analyzing the fundamental magic unitary associated with the quantum automorphism group acting on graph states. On this basis, we define a closed convex collection of doubly stochastic matrices situated between the classical automorphism polytope and the complete fractional automorphism polytope. The main finding reveals that the proposed quantum formulation of compactness does not yield a fundamentally broader class: every quantum compact graph is necessarily compact in the classical sense. We further investigate the interaction between this convex structure and the quantum orbital algebra, establishing a systematic hierarchy relating classical compactness, quantum compactness, and their respective pseudo-compact formulations. Through commutant methods, the framework also extends several established implications of classical compactness and develops quantum-theoretic counterparts of structural properties such as generous transitivity and distance transitivity. In addition, the study addresses a number of previously open questions, presents representative examples illustrating quantum symmetries, and demonstrates that the resulting framework provides a more refined perspective on the classical theory of graph compactness.

کلیدواژه ها:

Graph Automorphisms ، Quantum Automorphism Groups ، Compact Graphs ، Fractional Automorphism Polytopes ، Magic Unitaries ، Quantum Symmetry ، Quantum Walks ، Perfect State Transfer