Most of the Frequent Connected Subgraph Mining (FCSM) algorithms have been focused on detecting duplicate candidates using canonical form (CF) tests. CF tests have high computational complexity, which affects the efficiency of graph miners. In this paper, we introduce novel properties of the canonical adjacency matrices for reducing the number of CF tests in FCSM. Based on these properties, a new algorithm for frequent connected subgraph mining called grCAM is proposed. The experiments on real world datasets show the impact of the proposed properties in FCSM. Besides, the performance of our algorithm is compared against some other reported algorithms.