Due to the nonlinearity of artificial neural networks, designing topologies for deep convolutional neural networks (CNN) is a challenging task and often only heuristic approach, such as trial and error, can be applied. An evolutionary algorithm can solve optimization problems where the fitness landscape is unknown. However, evolutionary algorithms are computing resource intensive, which makes it difficult for problems when deep CNNs are involved. In this paper, we propose an evolutionary strategy to find better topologies for deep CNNs. Incorporating the concept of knowledge inheritance and knowledge learning, our evolutionary algorithm can be executed with limited computing resources. We applied the proposed algorithm in finding effective topologies of deep CNNs for the image classification task using CIFAR-10 dataset. After the evolution, we analyzed the topologies that performed well for this task. Our studies verify the techniques that have been commonly used in human designed deep CNNs. We also discovered that some of the graph properties greatly affect the system performance. We applied the guidelines learned from the evolution and designed new network topologies that outperform Residual Net with less layers on CIFAR-10, CIFAR-100, and SVHN dataset.