Frequent connected subgraph (FCS) mining is an interesting problem with wide applications in real life. Most of the FCS mining algorithms have been focused on detecting duplicate candidates using canonical form tests. Canonical form tests have high computational complexity, and therefore, they affect the efficiency of graph miners. In this paper, we introduce novel properties to reduce the number of canonical form tests in FCS mining. Based on these properties, a new algorithm for FCS mining called gRed is presented. The experimentation on real world datasets shows the impact of the proposed properties on the efficiency of gRed reducing the number of canonical form tests regarding gSpan. Besides, the performance of our algorithm is compared against gSpan and other state-of-the-art algorithms.