Vertex labels

While the rows and columns of A obviously depend on a particular choice of vertex labels, the generic structural j)roperties of G must remain invariant under a permutation of rows and columns. Much of this structural information can in fact be extracted from the spectrum of G the spectrum of a graph G,... 

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

The vertex and edges alphabets of hypergraph H2 representing cyclopropenyl complex look like A V H2)) = CH, M and A E H2)) = [El, E2,E3,E4. The vertex labels are presented by symbols of functional chemical groups. It is easy to see that the vertex alphabets are the different for these hypergraphs. 

Figure 14. Hypergraphs with the numerical vertex labels representing allylic and cyclopropenyl complexes given in Figure 12...

These counterintuitive properties of racemization paths of three-dimensional labeled and unlabeled chiral tetrahedra, noted by Mislow, are referred to as Mislow s label paradox. More recently, it has been shown that Mislow s label paradox is general for n-chiral simplices in all finite dimensions n, and sufficient and necessary partial vertex labeling conditions have been given for chirality-preserving interconversion paths of mirror images of chiral -dimensional simplexes. 

The regular orbit of a point symmetry group is the set of positions for which the only operation that leaves each vertex label invariant is the identity all other operations permute... 

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,... 

A corollary of this is that every vertex label on the perimeter appears at least twice. Two of them (at the comers ) appear three times. 

Figure 2. The Kj 3 bipartite graph as a topological representation of the degenerate planar isomerization of a tetrahedron (Tj) to its enantiomer through a square planar intermediate (O4/,). The isomers corresponding to the vertices of the K22 bipartite graph are depicted next to the vertex labels.

In other words, after a symmetry has been applied to an object there will be no change in the visual presentation of the object. Consider the rectangle in Figure 4.3 with each comer or vertex labeled a, b, c, and d. A reflective symmetry is applied to it by reflecting... 

We define a map ip . F(Ind(L )) —> C by the following rule. The simplices that contain the vertex labeled 3 get mapped to C3 the simplices that do not contain the vertex labeled 3, but contain the vertex labeled 6 get mapped to cg the simplices that do not contain the vertices labeled 3 and 6, but contain the vertex labeled 9 get mapped to cg and so on. Finally, the simplices that contain none of the vertices labeled 3, 6,. .., 3k all get mapped to [Pg.193]

The vertex-adjacency matrix or binary matrix, denoted by A, of a vertex-labeled connected simple graph G with Vvertices is a square VxVmatrix, which is determined by the adjacencies of vertices in G (Harary, 1971) ... 

The term vertex-adjacency matrix was first used in chemical graph theory by Mallion in his interesting paper on graph-theoretical aspects of the ring current theory (Mallion, 1975). Below we give the vertex-adjacency matrix of the vertex-labeled graph Gi (see structure A in Figure 2.1). 

FIGURE 2.1 A vertex-labeled (A) and edge-labeled (B) graph G. ... 

FIGURE 2.5 A branched tree representing the carbon skeleton of 2,2,3-trimethylhexane and the vertex-labels induced by the N-tuple code. 

The vertex-adjacency matrix of a vertex-labeled multiple graph G is a square V x V matrix defined as... 

FIGURE 2.9 The vertex-labeled multiple graph Gj representing the earbon skeleton of one Kekule structure of styrene. 

FIGURE 2.15 A vertex-labeled vertex-weighted graph representing the carbon skeleton... 

The vertex-connectivity matrix of Gj (using the vertex-labels presented in structure A in Figure 2,1 and vertex-degrees from Figure 2.17) is given below ... 

FIGURE 2.19 Vertex-labels and vertex-degrees of branched tree T2 representing the carbon skeleton of 2,3-dimethylhexane. 

The Laplacian matrix is sometimes also called the Kirchhojf matrix (Mohar, 1989 Kunz, 1992, 1993, 1995) due to its role in the matrix-tree theorem (Cvetkovic et al., 1995), implicit (Moon, 1970) in the electrical network work of Kirchhoff (1847 in his paper Kirchhoff also introduced the concept of the spanning tree, though he did not use this term). It is also known as the admittance matrix (Cvetkovic et al., 1995). However, the name Laplacian matrix appears to be more appropriate since this matrix is just the matrix of a discrete Laplacian operator, which is one of the basic differential operators in quantum chanistry and beyond. Below we give the Laplacian matrix of the vertex-labeled graph Gj (see structured in Figure 2.1) ... 

FIGURE 3.1 Edge-labels and edge-degrees of the branched tree representing the carhon skeleton of 2,3-diniethylhexane. Vertex-labels are given in Figure 2.19. 

The standard distance matrix or the vertex-distance matrix (or the minimum path matrix) of a vertex-labeled connected graph G (Harary, 1971 Gutman and Polansky, 1986 Buckley and Harary, 1990 Trinajstid, 1992 Mihalic et al., 1992 Todeschini and Consonni, 2000,2009 Consonni and Todeschini, 2012), denoted by D, is a real symmetric VxV matrix whose elements are defined as... 

The standard distance matrix of the vertex-labeled graph Gj (see structure A in Figure 2.1) is as follows ... 

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