The Reverse Cuthill-McKee (RCM) algorithm is a graph-based method for reordering sparse matrices, renowned for its effectiveness in minimizing matrix bandwidth and profile. This reordering enhances the efficiency of matrix operations, making RCM pivotal among reordering algorithms. In the context of executing the RCM algorithm, it is often necessary to select a starting node from the graph representation of the matrix. This selection allows the execution of BFS (Breadth-First Search) to construct the level structure. The choice of this starting node significantly impacts the algorithm's performance, necessitating a heuristic approach to identify an optimal starting node, commonly referred to as the RCM starting node problem. Techniques such as the minimum degree method and George-Liu (GL) algorithm are popular solutions. This paper introduces a novel algorithm addressing the RCM starting node problem by considering both the eccentricity and the width of the node during the run. Integrating this algorithm with the RCM algorithm, we introduce RCM++. Experimental results demonstrate that RCM++ outperforms existing RCM methods in major software libraries, achieving higher quality results with comparable computation time. This advancement fosters the further application and development of the RCM algorithm.The code related to this research has been made available at https://github.com/SStan1/RCM\_PP.git.

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