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The robustness of an integrated power grid and natural gas network can be transformed into a graph theory problem theoretically. With the concept of tree-coritivity, a novel assessment to analyze the robustness of the interdependent network is proposed in this paper. This method contains a two-stage optimization method and the calculation framework to reduce the computation burden and examine the results, respectively. Compared with existing robustness indices, the proposed tree-coritivity shows the advanced performance in terms of the connected and unconnected network. Furthermore, this index is able to measure the vulnerability of graphs with the same coritivity. Case studies on the standard and modified IEEE New England 39-bus systems integrated with the natural gas network show the effectiveness of the proposed algorithm and index.