Skeleton of a polyhedron

We will consider only 3-dimensional polytopes they are called polyhedra. Their 0-dimensional faces are called vertices and the 2-dimensional faces are called just faces. Two vertices are called adjacent if there exist an edge, i.e. a 1-dimensional face containing both of them. The skeleton of a polyhedron is the graph formed by all its vertices with two vertices being adjacent if they share an edge. This graph is 3-connected and admits a plane embedding. 

The overlap of the n unique internal orbitals to form an n-center core bond may be hard to visualize since its topology corresponds to that of the complete graph K which for n > 5 is non-planar by Kuratowski s theorem and thus cannot correspond to the 1-skeleton of a polyhedron realizable in three-dimensional space. However, the overlap of these unique internal orbitals does not occur along the edges of the deltahedron or any other three-dimensional polyhedron. For this reason, the topology of the overlap of the unique internal orbitals in the core bonding of a deltahedral cluster need not correspond to a graph... 

The topology of a polyhedron can be described by a graph, called the 1-skeleton of the polyhedron. The vertices and edges of the 1-skeleton correspond to the vertices and edges, respectively, of the underlying polyhedron. Of fundamental importance are relationships between possible numbers and... 

We annotate that Eqs. (7.2) and (7.3) correspond geometrically in cutting off a vertex as shown in Fig. 7.3, bottom. The graphs shown in this figure are Schlegel graphs or else addressed as the skeleton of the polyhedron. 

Since the first reported example of such a species, [Rli6C(CO)i5] ,f the encapsulation of a p-block element within a rhodium-containing polyhedron is now quite a familiar phenomenon. For rhodium-containing clusters there are several examples of interstitial p-block elements (B, C, N) within an octahedral and/or trigonal prismatic skeleton and these are often interconvertable, sometimes quantitatively (Eq. For hexanuclear clusters, however, only Ose polyhedra have so far been found to accommodate the more sterically demanding phosphorus. " ... 

The best (and historically the first) understood are boron clusters, which class comprises boranes (poly-nuclear boron hydrides), carboranes (boranes with one or two B atoms replaces with C), and metallo-carboranes with a metal atom incorporated into the cluster. Their structures have been rationalized by Wade in a set of rules [238]. The structure of a boron cluster depends on the numbers of its skeleton atoms (u) and of skeleton electron pairs (SEPs)(p) available for bonding these atoms. If p = a- -l,the cluster is a closed deltahedron, i.e. a polyhedron having only triangular faces (closo structure). If p = a + 2, it adopts the form of a higher-order polyhedron with one... 

Tridentate ligands can span either two coplanar edges or two edges of the same face of a coordination polyhedron, as sketched in Scheme 8. In the case of an octahedral center, these two options are tantamount to the mer and fac conformations, respectively. Ligands with a relatively flexible skeleton may adopt different hinge angles around... 

The bonds that form the skeleton have a pronounced covalent character due to the polarization interactions, while the alkaline ions are connected by the weaker ionic bond, upon the vertices of the polyhedron of KOjq coordination. The formation of the tridimensional skeleton is found to a reduced number of combinations and only for compounds able to accomplish the tridimensional concatenation of the basic structural (C, Si) units. 

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