Four new topological indices based on the molecular path code

The sequence of all paths pi of lengths i) 1 to the maximum possible length in a hydrogen-depleted molecular graph ( which sequence is also called the molecular path code) contains significant information on the molecular topology, and as such it is a reasonable choice to be selected as the basis of topological indices ( TIs). Four new ( or five partly new) TIs with progressively improved performance ( judged by correctly reflecting branching, centricity, and cyclicity of graphs, ordering of alkanes, and low degeneracy) have been explored. ( i) By summing the squares of all numbers in the sequence one obtains Sigma(i)p(i)(2), and by dividing this sum by one plus the cyclomatic number, a Quadratic TI is obtained: Q=Sigma(i)p(i)(2)/(mu+ 1). ( ii) On summing the Square roots of all numbers in the sequence one obtains Sigma(i)p(i)(1/2), and by dividing this sum by one plus the cyclomatic number, the TI denoted by S is obtained: S= Sigma(i)p(i)(1/2)/(mu+ 1). ( iii) On dividing terms in this sum by the corresponding topological distances, one obtains the Distance-reduced index D=Sigma(i){p(i)(1/2)/[(i)(mu+ 1)]}. Two similar formulas define the next two indices, the first one with no square roots: (iv) distance-Attenuated index: A=Sigma i{p(i)/[i( mu + 1)]}; and (v) the last TI with two square roots: Path-count index: P= Sigma i{p(i)(1/2)/ [ i(1/2)( mu + 1)]}. These five TIs are compared for their degeneracy, ordering of alkanes, and performance in QSPR ( for all alkanes with 3-12 carbon atoms and for all possible chemical cyclic or acyclic graphs with 4-6 carbon atoms) in correlations with six physical properties and one chemical property.