Block structures of graphs and quantum isomorphism

TL;DR

This paper explores the relationship between quantum isomorphism of graphs and their block structures, establishing isomorphism conditions for block trees and graphs, and deriving new necessary conditions for quantum isomorphic but non-isomorphic graphs.

Contribution

It proves that quantum isomorphic graphs have isomorphic block trees and graphs with blocks that are also quantum isomorphic, preserving 2-connectedness and providing new criteria for quantum isomorphism.

Findings

Block trees and block graphs of quantum isomorphic graphs are isomorphic.

Quantum isomorphism preserves 2-connectedness in graphs.

Derived necessary conditions for quantum isomorphic, non-isomorphic graphs.

Abstract

We prove that for every pair of quantum isomorphic graphs, their block trees and their block graphs are isomorphic, and that such an isomorphism can be chosen so that the corresponding blocks are quantum isomorphic -- in particular, 2-connectedness is preserved under quantum isomorphism. We conclude with some corollaries, including obtaining some necessary conditions on a pair of quantum isomorphic, not isomorphic graphs with a minimal number of vertices.