Graph theory adjacency matrix

Although the investigation of complex cognitive maps cannot be very easy, the graph theory from matrix algebra provides a suitable analytical framework. Empirical studies in social sciences are referred to social network analysis (SNA), which is considered a key technique in organizational studies. The first step is to put the map into a matrix form. The variables are listed both on the vertical axis and on the horizontal one forming a matrix, technically called the adjacency matrix [47]. 

In graph theory, the conversion of the adjacency matrix into the distance matrix is known as the "all pairs shortest path problem",... 

In Chapter 1 we have stated that the classical structural theory is the only way to "visualise" the synthesis of a more or less complex organic compound. However, all or most of the information given by a structural formula can also be expressed.by a matrix (see also Appendix A-1). There are different kinds of matrices for example, the adjacency matrix J, which originates in graph theory and indicates only which atoms are bonded, or the connectivity matrix C, whose off-diagonal entries are the formal covalent bond orders. For instance, the corresponding matrices of hydrogen cyanide are ... 

The ideas of chemical graph theory have a special meaning for benzenoid hydrocarbons (BHs). As the structures of BHs do not allow for the cis/trans isomers, there is a rigorous correspondence between the adjacency matrices of BHs and their properties. In other words, any property JP(G) of the benzenoid molecule G is a function of only its adjacency matrix A(G)... 

Much of the material we present in this paragraph is discussed exhaustively in two classical textbooks on the chemical graph theory [7, 8] and therefore we review it only very briefly. Let A(G) be the adjacency matrix pertinent to a molecular graph... 

The correspondence between mathematical graph theory and classical chemical structure theory is manifested in Table I. A widely used mathematical representation for graphs is the adjacency matrix (A). The rank of this matrix equals the number of the vertices (atoms), and its entries a,y are equal to either 0 or 1 ... 

However, Group Theory can be employed in the following manner. There are N permutations of the graph vertices II, some of them leaving the adjacency matrix invariant, i. e. ... 

Another rather interesting relation between the resonance theory and elementary MO theory was established by Ham and Ruedenberg 107>. Let Pp be the Pauling bond-order matrix known from resonance theory 179>. If the adjacency matrix of the -th Kekute graph is Aj, then... 

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). 

Norbomane matrices adjacency matrix and six additional matrices, which arise in various applications of chanical graph theory... 

In Table 2.2, we have shown, in addition to the adjacency matrix A, the distance matrix D, also the valence matrix V, the Laplacian matrix L, and the powers and A of the adjacency matrix, all for norbomane. Distance matrix D of a graph, another useful matrix in chemical graph theory, was introduced by F. Harary [118], The element [D] , of the D matrix are defined as... 

We end with a brief comment on getting the characteristic polynomial of chanical graphs nsing the symmetric function theory. R. Barakat in his paper has shown that the Frame s method is nothing but symmetric functions and Newton identities [52], In the view that a reference is made to Newton, this paper deserves to be included here if even at the very end of this section. Let c be the coefficients of the characteristic polynomial. They are the elementary symmetric fnnctions from the eigenvalues of the adjacency matrix (see, for example, Weyl [53] or other books on higher algebra). Thus,... 

When the row sums as entries in construction of the connectivity-type indices are applied to the adjacency matrix of molecular graphs, one obtains the connectivity index % of Randid [10,11]. When they are applied to the graph distance matrix, one obtains Balaban s index J [12]. Both indices, x and J, have been very nseful in QSAR, which is an incentive to explore properties of additional structural indices of this type. At our disposal are several novel sparse matrices that have been recently introduced in chemical graph theory, which, in the view that their row sums cover a wider range of values, can be expected to produce novel highly discriminatory topological indices. 

The two matrix representations of the protein segments, the amino acid adjacency matrix and the decagonal isometries matrix, are derived from the sequence information alone. As has been demonstrated, mathematical descriptors, dependent on the sequence information alone, have successfully revealed the underlying characteristics and patterns of given sequences. Their numerical nature also makes them easier to incorporate into a mathematical model. In addition, as has been well illustrated in chemical graph theory, when considering characterization of molecules, one can... 

Branched topologies as generated by the conditional Monte Carlo methods described in this section are most conveniently represented in matrix forms from graph theory [33, 53]. We name two of them the adjacency matrix A and the incidence matrix C (see Figure 9.22). They both describe connectivity. Note that in... 

As discussed in Ref. [20], some matrix invariants within the context of chemical graph theory may occasionally be identical despite being derived from different molecular graphs. A known example is that of the characteristic polynomial of 1,4-divinylbenzene and that of 2-phenylbutadiene which are identical - lOx H- 33x - 44x" H- 24x - 4). This problem is extremely unlikely when the molecules are coded not by topological connectivity matrices consisting of ones and zeroes but rather by their respective LDMs (or electron density-weighted adjacency/connectivity matrices, discussed below) since these matrices cannot contain elements that are all of identical magnitudes. 

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