The Polarity Number

Wiener has also formulated 28 the so-called polarity number, p. This is the number of pairs of vertices, separated by three edges, or otherwise, half of the number of distances of lengths three, d3, in D(G), or half of the sum of all entries in the B3 matrix as  

One can also see that the Gordon-Scantlebury N2 index 19) can similarly be expressed as half the number of distances of length two, but for acyclic graphs only  

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It is seen that the Smolenskii term X2 is identical to the N2 index, while the term X3 is identical (in the case of acyclic graphs only) to the polarity number p. 

The polarity number p (or Wiener polarity number) was also defined by Wiener in 1947 [Wiener, 1947c] as the number of pairs of graph vertices which are separated by three edges. It is usually assumed that the polarity number accounts for the flexibility of acyclic structures, p being equal to the number of bonds around which free rotations can take place. Moreover, it relates to the steric properties of molecules. 

The polarity number is usually calculated on the distance matrix as the number of pairs of vertices at a topological distance equal to three, i.e. 

For example, in cyclopropane, the polarity number pis equal to zero because there are no pairs of vertices that are separated by three edges, while the polarity number is equal to 3 because there are three paths of length 3 pii = 1,2, 3,1, P22 = 2, 3,1, 2, p33 = 3,1,2, 3. The polarity number p4 is 1.8 x 3 = 5.4, which is rounded to 5, because the contribution of the acyclic part is zero and there are three ways the path of length 3 can be laid upon the cyclic part of the graph (in this case this number coincides with P3). 

R is the topological radius, D the topological diameter, T the eccentricity, f the average atom eccentricity, At the eccentric, p2 the polarity number, and MS D the mean square distance index. 

Apart from the use of the carbon number index, the first use of gri h invariants for the correlation of the measured properties of molecules with their structural features was made in 1947. In that yeax, Wiener [121,122] introduced two parameters designed for this purpose. The first of these was termed the path number and was defined as the "sum of the distances between any two carbon atoms in the molecule, in terms of carbon-carbon bonds. A simple algorithm was given for the calculation of this number and it was shown that its value for normal alkanes assumes the form - n). The second parameter was called the polarity number and was defined as "the number of pairs of carbcm atthree carbon-carbon bonds it took the general value n-3 f< normal alkanes. Wiener proposed that the variation of any physical property for an isomeric structure as compared to a normal alkane would be ven by the linear expression ... 

The charges on the ions when a compound is written in its hypothetical ionic form are called the polar numbers or oxidation numbers of the elements concerned. Thus the polar number of carbon in CCI4 is +4 and of chlorine -1. Polar number is the original term and somewhat more expressive than oxidation number but the latter is now used almost universally. 

Those based on the topological distance matrix, including the Wiener index, the polarity number, the distance sum, the Altenburg polynomial, the mean square distance, the Hosoya index, and the distance polynomial ... 

Now, one has to sum over all crystal-field levels of the ground state and over all crystal-field levels of the excited state. This is a sum over all M and M of the ground and final states, as well as a sum over the polarization numbers p=-l, 0 and -i-l. 

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