The aim of this paper is to explore the use of backtrackless walks and prime cycles for characterizing both labeled and unlabeled graphs. The reason for using backtrackless walks and prime cycles is that they avoid tottering, and can increase the discriminative power of the resulting graph representation. However, the use of such methods is limited in practice because of their computational cost. In this paper, we present efficient methods for computing graph kernels, which are based on backtrackless walks in a labeled graph and whose worst case running time is the same as that of kernels based on random walks. For clustering unlabeled graphs, we construct feature vectors using Ihara coefficients, since these coefficients are related to the frequencies of prime cycles in the graph. To efficiently compute the low order coefficients, we present an O(|V|(3)) algorithm which is better than the O(|V|(6)) worst case running time of previously known algorithms. In the experimental evaluation, we apply the proposed method to clustering both labeled and unlabeled graphs. The results show that using backtrackless walks and prime cycles instead of random walks can increase the accuracy of recognition.