Detecting non-isomorphic strongly regular graphs via entanglement entropy in spinless free fermion model

In this research, the graph isomorphism problem is investigated in the non-isomorphic strongly regular graphs by using the entanglement entropy. Strongly regular graphs are a class of regular graphs with high symmetry. Accordingly, classical and quantum algorithms often fail to detect all non-isomorphic strongly regular graphs. Nevertheless, entanglement entropy has an efficient power in the graph isomorphism problem for strongly regular graphs. The entanglement entropy is surveyed in the ground state of the spinless free fermion Hamiltonian, where its hopping matrix is given by the adjacency matrix of the graph. After that, calculating the entanglement entropy for all kinds of partitions leads to entanglement entropy spectra. We show that all chosen sets of non-isomorphic strongly regular graphs distinguish from each other by using entanglement entropy spectra.