We investigate the graph isomorphism (GI) in some cospectral networks. Two graph are isomorphic when they are related to each other by a relabeling of the graph vertices. We want to investigate the GI in two scalable (n + 2)-regular graphs G4(n; n + 2) and G5(n; n + 2), analytically by using the multiparticle quantum walk. These two graphs are a pair of non-isomorphic connected cospectral regular graphs for any positive integer n. In order to investigation GI in these two graphs, we rewrite the adjacency matrices of graphs in the antisymmetric fermionic basis and show that they are different for thesepairs of graphs. So the multiparticle quantum walk is able to distinguish pairs of non- 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 using multiparticle quantum walk. Then we study GI by using the entanglement entropy. To this aim, we calculate entanglement entropy between two parts of network. In our 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 can distinguish pairs of non-isomorphic cospectral graphs too.
The graph isomorphism problem asks whether two graphs are identical up to vertex relabeling. While the exact problem admits quasi-polynomial-time classical algorithms, many applications in molecular comparison, noisy network analysis, and pattern recognition require a flexible notion of structural similarity. We study the quantum query complexity of approximate graph isomorphism testing, where two graphs on
n vertices drawn from the Erdős--Rényi distribution
G(n,1/2) are considered approximately isomorphic if they can be made isomorphic by at most
k edge edits.
We present a quantum algorithm based on MNRS quantum walk search over the product graph
Γ(G,H) of the two input graphs. When the graphs are approximately isomorphic, the quantum walk search detects vertex pairs belonging to a dense near isomorphic matching set; candidate pairings are then reconstructed via local consistency propagation and verified via a Grover-accelerated consistency check. We prove that this approach achieves query complexity
O(n3/2logn/ε), where
ε parameterizes the approximation threshold. We complement this with an
Ω(n2) classical lower bound for constant approximation, establishing a genuine polynomial quantum speedup in the query model.
We extend the framework to spectral similarity measures based on graph Laplacian eigenvalues, as well as weighted and attributed graphs. Small-scale simulation results on quantum simulators for graphs with up to twenty vertices demonstrate compatibility with near-term quantum devices.
Studies on graph isomorphism play an important role in graph research, and graph isomorphism algorithms have a wide range of applications in image matching, pattern recognition, computer vision, biochemistry and other fields. Previous research demonstrated that involving discrete-time quantum walk in the graph isomorphism algorithm could achieve complexity O ( N 7 ) for general graphs, since quantum walk could be utilized as a new toolbox for solving graph problems. We develop an enhanced classical approach to graph isomorphism using continuous-time quantum walk, which has lower complexity O ( N 5 ) and can effectively distinguish the graphs that are generally considered difficult. In addition, we define a graph similarity measure based on the proposed algorithm, which can be used for graph isomorphism and graph clustering. In the experiment, we test a wide variety of classes of graphs; the results show that the algorithm has a wide range of applications rather than being limited to a specific type of graph.