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Combinatorial optimization problems (COPs) play an important role in both industrial production and theoretical research. As a classical class of optimization algorithms, evolutionary algorithms have shown good performance in COPs. However, as the size of the problem increases, it becomes difficult for evolutionary algorithms to produce satisfactory solutions within the specified time. In recent years, neural network algorithms have been applied to solve COPs and have shown superior effects. Inspired by this, this paper proposes a new algorithm based on graph structure for solving COPs. Through relaxation, COPs are transformed into their corresponding continuous optimization problems. The output state of our algorithm is converted to a discrete solution by approximation method. Compared with previous algorithms, our algorithm can not only solve COPs with general form such as binary quadratic programming (BQP), but also solve COPs with special form, which greatly expands the application scope of neural network algorithms. In addition, our algorithm can be embedded into other problem frameworks such as solving multi-objective combinatorial optimization problem (MOCOP) through scalarization method. A new penalty function is added to the objective function, which enables our algorithm to overcome the inconsistency between the optimal solution of the original problem and the corresponding continuous problem, so that our algorithm can perform well on various problems. Several numerical simulations are used to illustrate the effectiveness of our algorithm in solving COPs. Experimental results show that our algorithm can effectively solve COPs with general form and has a shorter solving time compared to traditional evolutionary algorithms. [ABSTRACT FROM AUTHOR]