With labeled vertices

LisV69 Liskovec, V. A. On a recurrence method of counting graphs with labelled vertices. Dokl. Akad. Nauk. SSSR 184 (1969) No. 6 1284-1287. Translated in Soviet Math. Dokl. 10 (1969) 242-246. 

We may immediately construct the Huckel Hamiltonian-matrix for butadiene it is going to be 4 x 4. The Coulomb integral, a, will occur along the diagonal since all atoms in the conjugated system are carbon and the only resonance integrals, p, which are non-zero are those between centres which are bonded. Thus the (1-2)-, (2-3)-, and (3-4)-elements are p, and so also, of course, are the (2-1)-, (3-2)-, and (4-3)-elements—the matrix must clearly be symmetrical since, if atom i is joined to atom j, then certainly atom j is joined to atom i Thus we have, with the vertex labelling adopted in Fig. 2-6,... 

Start with the vertex having the smallest label and move to the vertex having the next smallest label. 

Label La is updated with label Lg of the previous pixel, Lg. Label Lg is updated with the minimum label in the neighborhood of the previous pixel, if a merger pattern was detected at the previous position (MergerPat ) and the left label of this merger pattern was identified to be the minimum label. Otherwise, Lg is updated with L. Label Lc is updated with its parent, M[L[C]], to identify the label of the root vertex and Ld is updated with the pixel assigned to the previous pixel. 

The vertex associated with label in S is made the child of the minimum label Lmins which propagates the root back to the leaves, effectively flattening the forest structure in M to a height of one. 

The integer label associated with a vertex v,- represents the time (or equivalently the cycle) with respect to the beginning of the schedule (operation modeled by may begin execution, i.e. r(v,) is the start... 

Figure 6.6 Photograph from the vertex of the rotunda of Saint Catherine in Znojmo (Czech Republic). The places of sampling are labelled. Samples with numbers 6, 7 and 9 were not analysed because they were parts of bricks and stone. (Courtesy of Dr Jaromfr Kovarnfk)...

Figure 5. A DNA molecule whose helix axes have the connectivity of a cube. The molecule shown consists of six cyclic strands that have been catenated together in this particular arrangement. They are labeled by the first letters of their positional designations, Up, Down, Front, Back, Left, and Right. Each edge contains 20 nucleotide pairs of DNA, so we expect that their lengths will be approximately 68 A. All of the twisting has been shown in the middle of the edges for clarity, but the DNA is base-paired from vertex to vertex. From model building, the axis-to-axis distance across a square face seems to be approximately 100 A, with a volume (in a cubic configuration) of approximately 1760 nm3 when the cube is folded as shown.

To deal with the problem in a rigorous fashion, we couch it within the branch of mathematics called graph theory. A graph G = (V,E) is a finite collection V of vertices and a finite collection E of edges. Each edge (v,w") consists of an unordered pair of distinct vertices. Each edge and each vertex may in addition have a label specifying certain information... 

We define a total ordering with vertex labels by the following rules. 

Figure 3.16. A few interactions contributing to the Raman-Stokes process (with creation of one phonon). The propagators are labeled with the notation of the text. The process a is of the first order b could be included by renormalization of the created exciton c would require the renormalization of the interaction (or the vertex).

Annihilate a graph Suppose we consider a certain graph G with N vertices each of which is denoted by the letter Ki, with the index i labelling the vertices. Now let us delete one of the vertices together with the edges incident at it. This way we get a new graph that has one vertex less ... 

Following Wales scheme we can track the theoretical movement of every vertex in the polyhedron as the isomerisation proceeds. By labelling vertices we can do the same thing experimentally and so compare theory with experiment. 

Figure 13. Hypergraphs with the symbolic vertex and edge labels representing allylic and cyclopropenyl complexes given in Figure 12...

In order to obtain a vertex numbering similar to the Morgan algorithm, the convention to associate label 1 to the vertex with the largest coefficient and label A (the number of atoms) to that with the smallest one was established. 

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