The Vertex-Connectivity Matrix

The vertex-connectivity matrix, denoted by x, was introduced by Randid (1992). This matrix is also named the (connectivity) % matrix (Todeschini and Consonni, 2000, 2009) and the product-connectivity matrix (Zhou and Trinajstid, 2010a Lucid et al., 2014). It can be regarded as an edge-weighted matrix of a graph that is defined as 

The summation of elements in the upper (or lower) matrix-triangle gives the vertex-connectivity index of G, (Gutman, 2002 Hu et al 2003). 

The vertex-connectivity matrix has also been used in computing the connectivity identification (ID) number (Randid, 1984a Szymanski et al., 1985). The connectivity ID number was successfully tested in QSAR (Randid, 1984b Carter et al., 1987). 

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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 ... 

One can generate the distance-sum-connectivity matrix, denoted by if one substitutes vertex-degrees with the distance-sums (Szymanski et al., 1986) in the formula for the vertex-connectivity matrix, presented in Section 2.13 ... 

An analogous quantity, called the bond vertex degree 6f, is calculated from the - atom connectivity matrix C as ... 

While the vertex - adjacency matrix A and -> edge adjacency matrix E contain information only about vertex-edge connectivity in the graph, weighted adjacency matrices allow the distinguishing of different bonds and atoms in a molecule. 

The bond vertex degree 5 is another local invariant that accounts for atom connectedness and also for bond multiplicity. It is calculated from the atom connectivity matrix C as the sum of row entries [Kier and Hall, 1986] ... 

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) ... 

Lukovits (2000, 2002, 2004) and Lukovits and Gutman (2002) offered an approach by which the vertex-adjacency matrix of an acyclic structure can be replaced by a single number, called the compressed (vertex-) adjacency matrix code, denoted by CAM. Here we present, besides the CAM code, the N-tuple code of trees that induces the unique labeling of trees (Aringhieri et al 1999). A graph is acyclic if it does not contain cycles. A tree is a connected acyclic graph. 

The sum-vertex-connectivity matrix, denoted by S, was introduced independently by Zhou and Trinajstid (2009, 2010a) and Randid et al. (2010). Randid et al. (2010) named this matrix the distance-weighted adjacency matrix. It is defined as follows ... 

The sum-edge-connectivity matrix, denoted by S, of a graph G is the sum-vertex-connectivity matrix of the corresponding line graph L(G) (Ludid et al., 2014) ... 

By substituting [d(i) + d(j)V with [d i) + dO)l in (2.55), oue obtains the sum-vertex-counectivity matrix (Zhou and Trinajstid, 2009, 2010a Randid et al., 2010) see Section 2.15. If one substitutes [d(i) + d(j)f with either [d(i) + d(jj or [d(i) + dO)] the sum-edge-Zagreb matrix or modified-sum-edge-Zagreb matrix is obtained, respectively. Thus, it appears that all these matrices can be traced to the Randid connectivity matrix proposed years ago (Randid, 1992). 

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... 

Walks can be generated from powers of the vertex-adjacency matrix A (see Section 2.1), and this may be viewed as an identification of the distribution for equipoise random walks. Similarly, the distribution for simple random walks can be generated by powers of a Markov matrix. The random-walk Markov matrix, denoted by MM, of a vertex-labeled connected graph G is a real unsymmetrical V x V matrix whose elements are probabilities for the associated individual steps (Klein etal.,2004) ... 

The most basic element in the molecular structure is the existence of a connection or a chemical bond between a pair of adjacent atoms. The whole set of connections can be represented in a matrix form called the connectivity matrix [249-253]. Once all the information is written in the matrix form, relevant information can be extracted. The number of connected atoms to a skeletal atom in a molecule, called the vertex degree or valence, is equal to the number of a bonds involving that atom, after hydrogen bonds have been suppressed. 

The Zagreb group was the first to propose indices (Mj and M2) that were based directly on the graph adjacency matrix (Hall and Kier, 2001). Mj and M2 are defined as the sum of the squared vertex degrees (i.e., the number of edges with which it is connected, a,), and the sum of vertex degrees products (a ) over all pairs of adjacent vertices, respectively (Gutman et al., 1975) ... 

From vertex to vertex. In the graph theory the matrix elements Aij can be interpreted as follows an Atj is the number of unitary walks between the vertices i and /. Then the product of two elements of the matrix A, ArjAj is equal to 1 if the vertex r is connected with the vertex /, and the latter in its turn is connected with s, i.e. between r and s there is a walk of length 2 passing through /. If there is no such a walk, Aj.jAjs = 0. 

The formerly proposed and the most important of this series of topological indices is the Baiaban distance connectivity index J (also called distance connectivity index or average distance sum connectivity). It is one of the most discriminating - molecular descriptors and its values do not increase substantially with molecule size or number of rings it is defined in terms of sums over each ith row of the - distance matrix D, i.e. the vertex distance degree o [Baiaban, 1982 Baiaban, 1983a]. It is defined as ... 

A first square unsymmetric A xA Cluj matrix is derived from the - distance matrix D and is called the Cluj-distance matrix CJDu. The off-diagonal entry i-j of the matrix is the count A, p. of all vertices lying closer to the focused vertex /, but out of the shortest path pij connecting vertices v, and j, i.e. the count of the external paths on the side of V, including the path i-j. The focused vertex v, is included in the set Nij,... In the event of more than one shortest path i-j being encountered, the maximal cardinality of the sets of vertices is taken. Therefore, the off-diagonal matrix elements are ... 

The sum of the positive eigenvalues X of the - / matrix, based on the path connectivities calculated by the - valence vertex degree 6 of the atoms in the path ... 

A layer matrix whose entry i-k is defined as the sum of the conventional bond order ji of the bonds connecting the vertices situated in the kth layer with the vertices of the (k - l)th layer with respect to the focused ith vertex [Hu and Xu, 1996]. 

Another important connectivity matrix is obtained weighting each bond i-j by the -+ edge connectivity (6,- bj) b being the - vertex degree of the atoms. This matrix is known as the Kier-Hall connectivity matrix or sparse x matrix ... 

The X matrix [Hall, 1990] is the most popular weighted distance matrix, based on the path contributions arising in constructing -> connectivity indices. Using the vertex degree 8 of the atoms, each path pij between the vertices v, and vj is weighted by the path connectivity defined as ... 

The procedure for counting S3 and D3 3 is based on the extended vertex-adjacency matrix and the adjacency bonding array. The extended vertex-adjacency matrix of a Kekule structure contains elements 1 or 2 depending on the single or double bond between carbon atoms in the Kekule structure and it is called extended because of the added column and row in which elements are set to 1 or 0 depending whether the carbon atom in the Kekule structure is connected to the hydrogen bond or not. For example, the extended vertex-adjacency matrix for the Kekule structure of benzene (see Figure 15) is ... 

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