This paper presents a novel constraint-based approach for generating regular graphs, addressing two critical challenges: avoiding redundant isomorphic graphs and ensuring the connectivity of the generated graphs. We introduce a reverse lexicographic ordering (RevLex) for adjacency matrices, which significantly reduces computational overhead and minimizes redundant graph generation, especially for larger instances. Experimental results demonstrate that using the RevLex ordering outperforms the state-of-the-art Lex method on regular graphs in \( K_n(d) \) with degree \( d < n/2 \) . Furthermore, we propose an optimized connectivity constraint that seamlessly integrates with both methods. These findings establish a robust framework for large-scale graph generation and open new avenues to explore the performance disparities between Lex and RevLex. Moreover, the proposed method serves as a valuable tool for real-world applications in fields such as network design, chemistry and combinatorics.