Table 9 Magic topological formulas for BCC and FCC clusters
From: Magic Mathematical Relationships for Nanoclusters—Errata and Addendum
Index | Formula |
|---|---|
bcc cube | |
Wiener | \(\frac {76}{35}n^{7} + \frac {38}{5}n^{6} + \frac {74}{5}n^{5} + 18n^{4} + \frac {206}{15}n^{3}+\frac {32}{5}n^{2}+\frac {136}{105}n\) |
Reverse Wiener | \(\frac {64}{35}n^{7} + \frac {22}{5}n^{6} + \frac {31}{5}n^{5} + 2n^{4} - \frac {26}{15}n^{3}-\frac {17}{5}n^{2}-\frac {136}{105}n\) |
HyperWiener | \(\frac {47}{35}n^{8}+\frac {226}{35}n^{7} + \frac {469}{30}n^{6} + \frac {241}{10}n^{5} + \frac {119}{5}n^{4} + \frac {149}{10}n^{3}+\frac {1097}{210}n^{2}+\frac {19}{35}n\) |
Szeged | NA |
bcc rhombic dodecahedron | |
Wiener | \(\frac {76}{7}n^{7} + 38n^{6} + \frac {302}{5}n^{5} + 56n^{4} + \frac {94}{3}n^{3}+10n^{2}+\frac {148}{105}n\) |
Reverse Wiener | \(\frac {148}{7}n^{7} + 58n^{6} + \frac {378}{5}n^{5} + 48n^{4} + \frac {38}{3}n^{3}-2n^{2}-\frac {148}{105}n\) |
HyperWiener | \(\frac {359}{42}n^{8}+\frac {832}{21}n^{7} + \frac {1217}{15}n^{6} + \frac {1454}{15}n^{5} + 73n^{4} + \frac {103}{3}n^{3}+\frac {1957}{210}n^{2}+\frac {39}{35}n\) |
Szeged | \(\frac {4637}{105}n^{9}+\frac {15655}{84}n^{8} + \frac {7661}{21}n^{7}+\frac {2615}{6}n^{6} + \frac {5194}{15}n^{5} + \frac {2245}{12}n^{4} + \frac {1412}{21}n^{3}+\frac {103}{7}n^{2}+\frac {32}{21}n\) |
bcc truncated cube | |
Wiener | \(\frac {76}{35}n^{7} + \frac {38}{5}n^{6} + \frac {74}{5}n^{5} - 6n^{4} - \frac {394}{15}n^{3}-\frac {168}{5}n^{2}+\frac {4336}{105}n\) |
Reverse Wiener | \(\frac {64}{35}n^{7} + \frac {22}{5}n^{6} + \frac {31}{5}n^{5} - 6n^{4} - \frac {146}{15}n^{3}-\frac {57}{5}n^{2}+\frac {1544}{105}n\) |
HyperWiener | \(\frac {47}{35}n^{8}+\frac {226}{35}n^{7} + \frac {469}{30}n^{6} + \frac {49}{10}n^{5} - \frac {121}{5}n^{4} - \frac {1313}{30}n^{3}+\frac {4457}{210}n^{2}+\frac {1933}{105}n\) |
Szeged | NA |
fcc truncated 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 | \(\frac {1564}{105}n^{7} + \frac {602}{15}n^{6} + \frac {1343}{30}n^{5} + \frac {70}{3}n^{4} + \frac {43}{15}n^{3}-\frac {59}{30}n^{2}-\frac {36}{355}n\) |
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 | \(\frac {14822}{945}n^{9}+\frac {2099}{35}n^{8} + \frac {30781}{315}n^{7}+\frac {941}{10}n^{6} + \frac {1073}{18}n^{5} + \frac {251}{10}n^{4} + \frac {12629}{1890}n^{3}+\frac {29}{35}n^{2}+\frac {32}{105}n\) |