Graph-Theoretical Approach to Clusters – Analysis of Clusters with 5–11 Vertices –

In this study, we systematically analyzed all isomers of 5–11 atom clusters using a graph-theoretical approach and evaluated their maximum Hückel energy (HE) within the Hückel approximation. The maximum HE increased with the number of bonds, reached a maximum at an intermediate value, and then decreased again, with the peak located near the average of the minimum and maximum possible edge numbers. This relationship was well reproduced by a simple linear expression. The isomers with maximum HE were found to have nearly uniform degree distributions, and classification by graph radius (Gr) and maximum degree (Ma) revealed that they possess compact and homogeneous structural characteristics. By extrapolating these trends, structural features of maximum-HE isomers were predicted for clusters with 12–14 atoms. These results demonstrate the effectiveness of a graph-theoretical description for analyzing cluster stability and provide a foundation for extending the analysis to larger clusters and for comparison with first-principles calculations.