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This paper investigates a minimum spanning tree (MST) problem with fuzzy costs and quadratic cost structure, which we call the fuzzy quadratic minimum spanning tree problem (FQMST). After formulating the FQMST problem as a chance-constrained programming model based on a credibility measure, the deterministic equivalent is proposed when the fuzzy direct costs and fuzzy interactive costs are characterized by trapezoidal fuzzy numbers. Then, a genetic algorithm is designed for solving FQMST problems. Finally, a numerical example is provided for illustrating the effectiveness of the genetic algorithm.