Investigation graph isomorphism problem via entanglement entropy in strongly regular graphs

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 * E-mail:jafarizadeh@tabrizu.ac.ir † E-mail:F.Egbali@tabrizu.ac.ir ‡ E-mail:S.Nami@tabrizu.ac.ir 1 Entanglement entropy 2 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:(n, κ, λ, ν) 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 adjacency matrices in the stratification basis, numerically.

Entanglement entropy in the spinless free fermion model and its application to the graph isomorphism problem

“…So, the correlation matrix is obtained as Γ = E * B XE * B and the entanglement entropy is calculated from the eigenvalues of the correlation matrix. The adjacency matrix of an SRG can be written as [42]…”

“…. We investigate entanglement entropy in three sets of SRGs in which the blocks A 11 , A 12 and A 22 are completely identified [42].…”

“…The graph isomorphism problem has many applications in other fields of science and engineering, such as mathematical chemistry [37] (for identifying a chemical compound within a chemical database), biochemical tools [38], electronic design [39] and computer science [40,41]. The graph isomorphism problem in strongly regular graphs has been investigated by using bosonic entanglement entropy [42].…”

“…For the association scheme graphs [45], we use the the Bose-Mesner algebra [46] to construct the projection operators from the idempotent operators. Various kinds of association schemes are investigated in this paper such as SRGs [42,47], distance-regular graphs [48] and graphs of three-class association schemes [49]. In these graphs the projection operator to a subset B of graph (P B ), can be obtained from the basis of the Terwilliger algebra [50] and by using stratification techniques [51].…”

“…We show that there are some non-isomorphic SRGs that can be distinguished from each other by using the fermionic entanglement entropy. Some of these pairs could not be distinguished by bosonic entanglement entropy [42].…”