Investigation graph isomorphism problem via entanglement entropy in strongly regular graphs

TL;DR

This paper explores the entanglement entropy in strongly regular graphs within quantum networks, analyzing their properties, calculating entanglement measures analytically, and applying these insights to distinguish graph isomorphism.

Contribution

It provides an analytical approach to compute entanglement entropy in SRGs and introduces a method to distinguish non-isomorphic SRGs using adjacency matrix elements.

Findings

Entanglement entropy relates to boundary-to-system size ratio.

Area-law behavior observed with no entanglement entropy at maximum system size.

Adjacency matrix elements can distinguish non-isomorphic SRGs.

Abstract

We investigate the quantum networks that their nodes are considered as quantum harmonic oscillators. The entanglement of the ground state can be used to quantify the amount of information one part of a network shares with the other part of the system. The networks which we studied in this paper, are called strongly regular graphs (SRG). These kinds of graphs have some special properties like they have three strata in the stratification basis. The Schur complement method is used to calculate the Schmidt number and entanglement entropy between two parts of graph. We could obtain analytically, all blocks of adjacency matrix in several important kinds of strongly regular graphs. Also the entanglement entropy in the large coupling limit is considered in these graphs and the relationship between Entanglement entropy and the ratio of size of boundary to size of the system is found. Then, area-law is studied to show that there are no entanglement entropy for the highest size of system. Then, the graph isomorphism problem is considered in SRGs by using the elements of blocks of adjacency matrices. Two SRGs with the same parameters: $(\eta,\kappa,\lambda,\nu)$ are isomorphic if they can be made identical by relabeling their vertices. So the adjacency matrices of two isomorphic SRGs become identical by replacing of rows and columns. The nonisomirph SRGs could be distinguished by using the elements of blocks of adja- cency matrices in the stratification basis, numerically.