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Table 23 Magic topological formulas for clusters

From: Magic Mathematical Relationships for Nanoclusters

Simple cubic  
 Wiener 64n7−16n5
 Reverse Wiener 128n7−96n6+16n5−24n4+12n3
 HyperWiener \(\frac {224}{3}n^{8} + 32n^{7} - \frac {88}{3}n^{6} - 8n^{5} + \frac {8}{3}n^{4}\)
 Szeged 256n9−64n7
fcc cube  
 Wiener \(\frac {956}{105}n^{7} + \frac {478}{15}n^{6} + \frac {1357}{30}n^{5} + \frac {110}{3}n^{4} + \frac {589}{30}n^{3}+\frac {97}{15}n^{2}+\frac {36}{35}n\)
 Reverse Wiener \(\frac {1564}{105}n^{7} + \frac {602}{15}n^{6} + \frac {1343}{30}n^{5} + \frac {70}{3}n^{4} + \frac {43}{15}n^{3}-\frac {59}{30}n^{2}-\frac {36}{35}n\)
 HyperWiener \(\frac {59}{10}n^{8}+\frac {2956}{105}n^{7} + \frac {1089}{20}n^{6} + \frac {701}{12}n^{5} + \frac {817}{20}n^{4} + \frac {1153}{60}n^{3}+\frac {53}{10}n^{2}+\frac {5}{7}n\)
 Szeged \(\frac {14822}{945}n^{9}+\frac {2099}{35}n^{8} + \frac {30781}{315}n^{7}+\frac {941}{10}n^{6} + \frac {1073}{18}n^{5} + \frac {251}{10}n^{4} + \frac {12629}{1890}n^{3}+\frac {29}{35}n^{2}+\frac {32}{105}n\)
bcc cube  
 Wiener \(\frac {12}{7}n^{7} + 6n^{6} + \frac {59}{5}n^{5} + \frac {29}{2}n^{4} + \frac {34}{3}n^{3}+\frac {11}{2}n^{2}+\frac {121}{105}n\)
 Reverse Wiener \(\frac {16}{7}n^{7} + 6n^{6} + \frac {46}{5}n^{5} + \frac {11}{2}n^{4} + \frac {2}{3}n^{3}-\frac {5}{2}n^{2}-\frac {121}{105}n\)
 HyperWiener \(\frac {71}{84}n^{8}+\frac {89}{21}n^{7} + \frac {53}{5}n^{6} + \frac {253}{15}n^{5} + \frac {421}{24}n^{4} + \frac {143}{12}n^{3}+\frac {4211}{840}n^{2}+\frac {137}{140}n\)
 Szeged NA
fcc octahedron  
 Wiener \(\frac {59}{420}n^{7} + \frac {59}{60}n^{6} + \frac {179}{60}n^{5} + \frac {61}{12}n^{4} + \frac {77}{15}n^{3}+\frac {44}{15}n^{2}+\frac {26}{35}n\)
 Reverse Wiener \(\frac {383}{1260}n^{7} + \frac {101}{60}n^{6} + \frac {743}{180}n^{5} + \frac {59}{12}n^{4} + \frac {104}{45}n^{3}-\frac {3}{5}n^{2}-\frac {26}{35}n\)
 HyperWiener \(\frac {173}{3360}n^{8}+\frac {27}{56}n^{7} + \frac {463}{240}n^{6} + \frac {87}{20}n^{5} + \frac {2891}{480}n^{4} + \frac {41}{8}n^{3}+\frac {699}{280}n^{2}+\frac {19}{35}n\)
 Szeged \(\frac {397}{5040}n^{9}+\frac {397}{560}n^{8} + \frac {347}{120}n^{7}+\frac {841}{120}n^{6} + \frac {891}{80}n^{5} + \frac {2897}{240}n^{4} + \frac {2801}{315}n^{3}+\frac {1769}{420}n^{2}+1n\)
fcc cuboctahedron  
 Wiener \(\frac {204}{35}n^{7} + \frac {102}{5}n^{6} + \frac {168}{5}n^{5} + 33n^{4} + \frac {98}{5}n^{3}+\frac {33}{5}n^{2}+\frac {34}{35}n\)
 Reverse Wiener \(\frac {1664}{315}n^{7} + \frac {194}{15}n^{6} + \frac {713}{45}n^{5} + 7n^{4} - \frac {52}{45}n^{3}-\frac {44}{15}n^{2}-\frac {34}{35}n\)
 HyperWiener \(\frac {487}{140}n^{8}+\frac {589}{35}n^{7} + \frac {433}{12}n^{6} + \frac {183}{4}n^{5} + \frac {548}{15}n^{4} + \frac {357}{20}n^{3}+\frac {103}{21}n^{2}+\frac {4}{7}n\)
 Szeged \(\frac {68867}{7560}n^{9}+\frac {12589}{336}n^{8} + \frac {3269}{45}n^{7}+\frac {10403}{120}n^{6} + \frac {23759}{360}n^{5} + \frac {1475}{48}n^{4} + \frac {30929}{3780}n^{3}+\frac {467}{420}n^{2}+\frac {1}{15}n\)