Vertex sequence

Those sequences correspond to vertex sequences, i.e. a path in the (r, )-polycycle P. Locally, there is no obstruction to the coherence of the definition of boundary vertex there is no problem, and around an interior vertex, the condition of having degree q provides coherence. Also, if we consider flags around a face /, we do not encounter coherence problems. The simple connectedness of P allows us to modify the path P" into the path P by changing around faces and vertices. So, no ambiguity will appear. ... 

A walk from the vertex C to vertex a finite sequence of edges of the... 

The structure of LiTa02F2, as reported by Vlasse et al. [218], is similar to a ReC>3 type structure and consists of triple layers of octahedrons linked together through their vertexes. The layers are perpendicular to the c axis, and each layer is shifted, relative to the layer below, by half a cell in the direction (110). Lithium atoms are situated in the centers of the tetragonal pyramids (coordination number = 5). The other lithium atoms are statistically distributed along with tantalum atoms (coordination number = 6) at a ratio of 1 3. The sequence of the metal atoms in alternating layers is (Ta-Li) - Ta - (Ta-Li). Positions of oxygen and fluorine atoms were not determined. The main interatomic distances are (in A) Ta-(0, F) - 1.845-2.114 Li-(0, F) - 2.087-2.048 (O, F)-(0,F) - 2.717-2.844. 

There is a series of metallaboranes of the type [SBgHjoM], [S2B7H7M], etc., where M seems to contribute two cluster electrons giving a nido-electron count. Nevertheless, these metallaboranes must be designated as arachno-dusters. Though the principal structure of the 10-vertex nido- and arachno-skeleton is the same, they differ structurally by the position of the bridging H atoms (which are the gunwale positions 5-10 and 7-8 in the arachno-case) and, moreover, in their physical properties, particularly in the sequence of the 11B NMR shift values. We will discuss the ar-10 clusters with formal ni-10 electron count later. 

A directed path is a sequence of adjacent edges where each step goes in direction of an edge. A vertex A is reachable by a vertex B, if there exists an oriented path from B to A. 

The most outstanding examples of heteronuclear gold clusters are the series of Au-Ag supraclusters whose metal frameworks are based on vertex-sharing. In the structure of these compounds, the basic building block is the 13-metal atom (Au7Ag6) icosahedra. These high nuclearity clusters have been termed clusters of clusters and they follow a well-defined growth sequence by successive additions of... 

The code can be written more concise as an ordered sequence, omitting the vertex degrees since they are uniquely defined ... 

Take an elementary (5,3)-polycycle A3 (see Figure 7.2) and remove its central vertex. The result is a (5,3)gen-polycycle with two boundary sequences. It turns out that those boundary sequences are identical, namely, (322)3. Hence, we can fill both those boundaries by the same structure, which is again bRj. So, we obtain a larger ( 5, 8], 3)-sphere, which is bRj. This operation can, obviously, be repeated and we obtain larger ( 5, b, 3)-spheres, which are bRj. By creating a chain of such spheres, we get an infinity of them. ... 

Since every b-gon is adjacent to exactly two 5-gons, in the (5,3)-polycycle, the runs of 3 (if. sequence of 3 bounded by 2) of the boundary sequence have length at most one, i.e. every 3-valent vertex is bounded by two 2-valent vertices. This implies V3 < vz and so, p < 0, by Theorem 11.0.3. Hence, (i) holds. 

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