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A chain of quantum subgroups of the quantum automorphism group of finitegraphs has been introduced. It generalizes the construction of J. Bichon (see[3]) in a sense. A better bound of the non zero eigenvalues of the graphLaplacian has been obtained using the chain of quantum subgroups.

Improved bound on the non zero eigenvalues of the graph Laplacian coming from quantum symmetry of vertex transitive graphs