Traditional Genetic Algorithm (GA), first proposed by John Holland in 1975, demonstrates advantages in versatility, robustness, and convergence speed. This paper proposes a genetic algorithm based on critical edge guidance and adaptive mutation. By introducing a dynamic adjustment mechanism for mutation rate and a mutation and repair strategy guided by critical edges, the algorithm significantly improves the ability to escape local optima while enhancing the search efficiency for the shortest path in a layer-two network topology. Experimental results show that compared to traditional GA and Spanning Tree Protocol (STP), CEGA-AM performs better in terms of convergence speed, solution quality, and computational efficiency, especially in high-dimensional optimization tasks.