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We investigate the graph isomorphism (GI) in some cospectral networks. Twograph are isomorphic when they are related to each other by a relabeling of thegraph vertices. We want to investigate the GI in two scalable (n + 2)-regulargraphs G4(n; n + 2) and G5(n; n + 2), analytically by using the multiparticlequantum walk. These two graphs are a pair of non-isomorphic connectedcospectral regular graphs for any positive integer n. In order to investigationGI in these two graphs, we rewrite the adjacency matrices of graphs in theantisymmetric fermionic basis and show that they are different for thesepairsof graphs. So the multiparticle quantum walk is able to distinguish pairs ofnon- isomorph graphs. Also we construct two new graphs T4(n; n + 2) and T5(n; n+ 2) and repeat the same process of G4 and G5 to study the GI problem by usingmultiparticle quantum walk. Then we study GI by using the entanglement entropy.To this aim, we calculate entanglement entropy between two parts of network. Inour model the nodes are considered as identical quantum harmonic oscillators.The entanglement entropy between two special parts of G4(n; n+2) and G5(n; n+2)are calculated analytically. It is shown that the entanglement entropy candistinguish pairs of non-isomorphic cospectral graphs too.

Investigating graph isomorphism in cospectral graphs via multiparticle quantum walk in fermionic basis and entanglement entropy