Graph Theory of Capping Golden Clusters

Skeletal numbers and their valences have been extremely useful in analyzing and categorizing clusters especially boranes, carbonyls, and Zintl ions. This approach is being extended to the analysis and categorization of golden clusters. The newly introduced concept of graphing will also be applied to the clusters. The capping symbol Kp = CC[Mx] which has been restricted to post-closo clusters will be adapted for pre-closo cluster series. The concept of the existence of black holes in the nuclei of capping golden clusters will be introduced and explained.


Results and Discussion
In order to be able to compare and contrast the structures of gold as compared to other clusters, let us compare and contrast the structures of two hypothetical clusters, namely C 6 H 10 and Au 6 H 10 .The K value of C 6 H 10 : K =6[2]-5 =7; K(n) = 7(6),S =4n+10.In the case of Au 6 H 10 : K = 6[3.5]-5= 16; K(n) = 16(6), S =4n-8, Kp = C 5 C[M1].The calculation of skeletal linkages of clusters has been discussed extensively (Kiremire, 2017d).Let us analyze these cases one at a time.
Ex-2: Au 6 H 10 : K = 6[3.5]-5= 16, K(n) = 16(6), S = 4n-8, Kp = C 5 C[M1].This symbol means that the cluster is capped in such a way that 5 skeletal elements are surrounding 1 central skeletal atom at the nucleus.In addition, the skeletal framework of 6 golden elements is bound by 16 linkages.Since each capping skeletal element uses 3 linkages, the 5 capping elements means utilizing 15 of the skeletal linkages out of the 16.Since the cluster has one skeletal element {[M1]; n =1}, then the element obeys the CLOSO series formula S= 4n+2 and K =2n-1 =2(1)-1 = 1.Thus, the skeletal element in the nucleus utilizes 1 skeletal element and the 5 capping atoms consume 15 skeletal elements giving us a total of 15+1 = 16.For categorization, we can use S = 4n+q for main group and transition elements.However for calculating the number of valence electrons we have to use Ve = 14n+q equation.In this case, the number of valence electrons will be given by Ve = 14n-8 = 14(6)-8 = 76.We can verify this from the cluster formula F = Au 6 H 10 ; VF = 10( 6) 13( 7) SC-1 Ex-1: C6H10

Without Hydrogen Ligands
Is K(n) = 10(6) below [M7]?We can test this by a simple check on the flow of K(n) numbers.
According to the series method, K(n) = 16(6), Kp = C 5 C[M1].This means it belongs to the group or a clan of clusters which have got 1 skeletal element in the nucleus.Thus, K(n)= 16(6) parameter will be found to be a member of [M1]based numbers.This can be seen to be the case as indicated in the [M1] series.
Is K(n) = 13(6) a member of [M4] series of numbers?.Let us check it out.

SC-5
The rest of the examples we can just demonstrate the categorization and the flow of K(n) number series.

Data Interpretation
A number of golden clusters have been studied by UV-Vis-NID method (Wang, et al, 2016).For the purposes of illustrating how the series method analysis of clusters could be utilized, these findings are given in Table 3.For ease of differentiation of the clusters and grouping them, let us refer to the clusters with the same closo nucleus [Mx] as being in the same CLUSTER GROUP but different series S = 4n+q(q varies), while those which belong to the same series S = 4n+q ( q is the same) as being in the same CLUSTER PERIOD.This approach is similar to what we have in the ordinary periodic table of elements was introduced.This is easily illustrated by the scheme SC-7.

Conclusion
A collection of known golden clusters have been analyzed and categorized using skeletal numbers.A large percentage of them are centered around [M0], [M1] and [M2] closo-based axes.The [Mx] may be regarded as similar to groups of elements in the periodic table.The clusters which lie on the closo lines [Mx], x≤ 0, have a tendency of possessing metallic character.In this connection, the golden clusters are closer to the metallic region of metallic clusters than that of non-metallic.A concept of a black-hole nucleus has been introduced as well as that of numerical categorization of clusters which lie lower than the closo base line.The two dimensional ideal shapes of clusters have been sketched using graph theory derived from the 4n series method.

Table 1
. The clusters were regrouped according to[Mx]series and are presented in Table2.In order to emphasize the idea of grouping the clusters according to [Mx] series, cluster group trees of selected clusters were constructed F-35 to F-37 for selected clusters of [M1], [M2] and [M6].A proposed scheme for broad grouping of clusters is shown GR-1.

Table 2 .
Grouping of Golden Clusters

Table 5 .
Some of Rudolph Type of Series and Corresponding Boran