Zagreb Matrices

Zagreb matrices are a generalization of the x matrix in terms of a variable exponent X as... 

Zagreb matrices —> weighted matrices (0 weighted adjacency matrices)... 

The Zagreb matrices can also be considered the vertex- and edge-weighted matrices related to the vertex- and edge-connectivity matrices discussed above. They can be formulated in terms of the vertex- or edge-degrees. 

Variable Zagreb Matrices in Terms of the Vertex-Degrees AND Edge-Degrees... 

Variable Zagreb matrices in terms of the vertex-degrees VZM and the edge-degrees WZM are given by (Milicevid and Nikolid, 2004 Milicevid et al., 2004 Nikolid etal.,2004) ... 

The variable Zagreb matrices are used to generate the variable Zagreb indices, which found application in structure-boiling point modeling and structure-molar volume modeling for aliphatic organic molecules (Milicevid and Nikolid, 2004 Nikolid et al., 2004). 

Finally, the sum-edge-Zagreb matrix ZM and modified sum-edge-Zagreb matrix ZM can be collected as the variable sum-Zagreb matrices WZM ... 

Examples of molecular descriptors derived from generalized matrices are distance distribution moments and W% indices from the distance matrix and molecular profiles from the geometry matrix. Moreover, vertex Zagreb matrix (X = 2) and modified vertex Zagreb matrix (X = —2) are a generalization of the vertex degree matrix, and the generalized... 

Particular realizations of the generalized vertex degree matrix are the vertex Zagreb matrix, denoted by ZM, and defined as [Janezic, Milicevic et al., 2007]... 

The sum of the diagonal elements of the vertex Zagreb matrix results into the first Zagreb index, while the sum of the diagonal elements of the modified vertex Zagreb matrix results into... 

For X = —1/2, the Zagreb matrix obviously reduces to the edge-% matrix. 

Edge-Zagreb matrix (ZM ) and modified edge-Zagreb matrix (" ZM ) of 5-methyl-1,3.4-oxathiazol-2-one VS, is the matrix row sum. 

The edge-weighted Zagreb matrix "ZM for Gj (see structure A in Figure 2.1) is presented below ... 

The modified edge-Zagreb matrix, denoted by is defined as... 

Finally, the path-Zagreb matrix, denoted by eZM, has been introduced by Vukidevic and his coworkers (Vukidevid et al., 2009). The path-Zagreb matrix for trees pZM is defined as follows ... 

Below we give the path-Zagreb matrix for a branched tree T2 representing the carbon skeleton of 2,3-dimethylhexane. In Figure 2.19, we give the labeled tree Tj and the corresponding vertex-degrees. 

Another way to modify the edge-Zagreb matrix ZM is to sum up the degrees of vertices i and j, making up bond i-j. The novel Zagreb matrix we caU the sum-edge-Zagreb matrix and denote it by ZM. It is defined as... 

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). 

D. Vukiievic, S. Nikolic, and N. Trinajstic, On the path-Zagreb matrix, J. Math. Chem. 45 (2009) 538-543. 

Our studies also included IR spectroscopic investigation of the observed ions (Fig. 6.2). John Evans, who was at the time a spectroscopist at the Midland Dow laboratories, offered his cooperation and was able to obtain and analyze the vibrational spectra of our alkyl cations. It is rewarding that, some 30 years later, FT-IR spectra obtained by Denis Sunko and his colleagues in Zagreb with low-temperature matrix-deposition techniques and Schleyer s calculations of the spectra showed good agreement with our early work, considering that our work was... 

The first numerical topological indices, based on the adjacency matrix, were introduced by the Zagreb group 3) ... 

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) ... 

In Equation 5.9, the sum in the brackets equals the vertex degree products for half of the adjacency matrix. It is multiplied by two in order to obtain summation over all pairs of adjacent vertices. Note that by definition M2 is not equal to 21 IM,. Although it is difficult to derive bond contributions for the index based only on the vertex degrees (M,), the formal bond distribution of the Zagreb index M2 (Figure 5.9) shows that the terminal bonds are again underestimated, although by a different amount. 

Calculation of internal and external first Zagreb index. The molecular fragment consists of vertices 6, 7, and 8, and A is the adjacency matrix. 

Variable connectivity index, variable Balaban index, and variable Zagreb indices for 2-pentanol. A(x, y) and D(x, y) are the variable augmented adjacency matrix and the variable augmented distance matrix, respectively. VS, indicates the matrix row sums x and y are the variable parameters for carbon and oxygen atom, respectively. 

Chemical adjacency matrices based on relative atomic masses were used to calculate the —> atomic molecular connectivity index, Zagreb topochemical indices, and the superadjacency topochemical index, all defined in terms of the Madan chemical degree 5 ", which is the row sum of the atomic weight-weighted adjacency matrix. 

The Zagreb topochemical indices [Bajaj, Sambi et al, 2005] are modifications of the original Zagreb indices, which account both for the presence and relative position of heteroatoms in a H -depleted molecular graph. They are calculated from the chemical adjacency matrix based on relative atomic weights as... 

Sununation of the off-diagonal elements in the upper (or lower) matrix-triangle produces the second Zagreb index. This molecular descriptor is also discussed... 

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