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Using the shuffle structure of the graphs, we introduce a new kind of theHopf algebraic structure for tagged graphs with, or without loops. Like aquantum group structure, its product is non-commutative. With the help of theHopf algebraic structure, after taking account symmetry of the tagged graphs,we reconstruct the topological recursion on spectral curves proposed by B.Eynard and N. Orantin, which includes the one-loop equations of various matrixintegrals as special cases.

Hopf Algebraic Structure for Tagged Graphs and Topological Recursion