Zagreb first

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 Sect. 3 some classical results of enumeration of polyhexes are reported. They include Klamer numbers , Harary-Read numbers and Lunnon numbers , which were so beautifully (but not infallibly) sorted out for the first time by the Dusseldorf-Zagreb group [ 1 Reference to the Dusseldorf-Zagreb numbers is also contained in Sect. 3. Furthermore, original supplements to the different tables are found therein. 

The planar (non-helicenic) circulenes have also been considered separately, first by the Diisseldorf-Zagreb group [16,17, 35] see Table 5. He and He [33] depicted the forms of these systems for h <, 9 from these figures we have extracted separate numbers for the catacondensed and pericondensed systems up to this h value. For the catacondensed planar monocirculenes with h < 11 the numbers may be deduced from a work of Brunvoll et al. [54], who enumerated the benzenoid systems composed of coronene with catacondensed appendages up to h = 12. On... 

A list of numbers of catacondensed benzenoids was produced for the first time in 1983 by Knop and Trinajstic with collaborators (the Diisseldorf-Zagreb group) [5, 42-44]. These investigators generated and enumerated all benzenoids up to h = 10. Somewhat later also Cu [45] and C12 [31, 46] were computed cf. Table 3 and 4, respectively. Table 4 also includes the very recent values of C13 and C14 [47],... 

The two Zagreb indices are strictly related to zero-order °x and first-order - connectivity indices, respectively. The 1 Zagreb index M (also called Gutman index) is also related to the - Platt number F and the - connection number Na by the relationship ... 

First Zagreb index by valence vertex degrees Second Zagreb index M2... 

The Balaban centric index is calculated from the pruning partition in analogy with the first Zagreb index Mi as... 

The number of connections N2 of a molecular graph is related to the —> first Zagreb index Mi and the qiuadratic index Q by the following relationships ... 

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

Gutman index = first Zagreb index Zagreb indices... 

Moreover, the first order overall Zagreb index coincides with the second Zagreb index Mj-... 

The first variable Zagreb index is then defined as... 

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... 

Gutman, I. and Das, K.C. (2004) The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem., 50, 83-92. 

Let us describe some frequently encountered polycyclic structures. Bicyclo[2.2.1] heptane, also known as norbomane, is the basic structure of a series of natural products. The tricyclic compound adamantane (tricyclo[3.3.1.L ]decane) is in principle the smallest structural fragment of diamond and its structure appears on the list of pharmaceutically important products. Adamantane was synthesized for the first time in 1941 at the University of Zagreb by Vladimir Prelog. 

Contrary to this positive practice and recommendations, student surveys at Faculty of Civil Engineering in Zagreb, at first accepted enthusiastically by students, proved not to be productive. The main reason for this is that survey results, except for statistics at the level of the whole institution, have been kept under the lid, so students have no information at all on their evaluations of teachers. They are, thus, less and less motivated to fill out the questionnaires. Another problem is that university and faculty managements do not take any productive steps toward poorly graded teachers, even though, formally, student survey results should affect their promotion. 

Summation of the diagonal elements gives the first Zagreb index. This index is discussed in a number of reports (Gutman and Trinajstid, 1972, Gutman et al., 1975,... 

We will consider now adamantane, for which we have already described the assignment of the canonical labels. We have selected adamantane for the following reasons (i) It requires about half a dozen trials to find canonical labels one labeling among 10 = 3,628,800 (ii) it is a beautiful molecule, and (iii) it was, for the first time, synthesized by Prelog and Seiwerth [63] in our neighborhood, Zagreb, Croatia, about 80 years ago. Admittedly, at the time, some of us were not yet around. 

FIGURE 13.4 The dual graphs of the four-color maps of the first exon of 3-globin gene of the nine species listed in Table 13.2. (Reproduced from On map representation of DNA, Croat. Chem. Acta (2013) Croatian Chemical Society, Zagreb, Croatia.)... 

At this point, it should be noted that the concept of extensive structural modification is imbedded in the invention of the first aza-macrolide, the antibiotic azithromycin, discovered in 1982 by PLIVA chemists in Zagreb, Croatia [21, 22]. Azithromycin became a multibillion drug on the world market, and represents an early example of therapeutic success with a non-natural natural compound , available via extensive structural modification. 

---