Four New Topological Indices Based on the Molecular Path Code

Abstract

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 Sigmaipi(2), and by dividing this sum by one plus the cyclomatic number, a Quadratic TI is obtained: Q = Sigmaipi(2)/(mu+1). (ii) On summing the Square roots of all numbers in the sequence one obtains Sigmaipi(1/2), and by dividing this sum by one plus the cyclomatic number, the TI denoted by S is obtained: S = Sigmaipi(1/2)/(mu+1). (iii) On dividing terms in this sum by the corresponding topological distances, one obtains the Distance-reduced index D = Sigmai{pi(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 = Sigmai{pi/[i(mu + 1)]}; and (v) the last TI with two square roots: Path-count index: P = Sigmai{pi(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.

Novel graph distance matrix

“…There are number of topological indices of high discriminatory power 55–62. Hence, of indices of high discriminatory power, there is a lack of such indices having straightforward structural interpretation, because most of such descriptors involve various arbitrary constructional elements, and in addition, some involve conceptual or computational complexity.…”

Study of peptide fingerprints of parasite proteins and drug–DNA interactions with Markov-Mean-Energy invariants of biopolymer molecular-dynamic lattice networks

“…In another paper, Liao proposed a 2D graphical representation of a DNA sequence [124]. One special class of indices used in QSAR are the TIs of molecular graphs; which indicates the presence of vertices or nodes (atoms) and connections or edges between nodes (chemical bonds) [136][137][138][139]. Among all above-mentioned, Liao, Randic, Basak, Vackro, Nandy and Wang [108,118] associated a DNA sequence having n bases with n  n non-negative real symmetric matrix A with elements a ij and use its leading eigenvalue to characterize the DNA sequence in phylogenetic studies.…”

TI2BioP — Topological Indices to BioPolymers. A Graphical– Numerical Approach for Bioinformatics

“…They translate the connectivity/adjacency relations between nodes or edges in the graph to a matrix arrangement [23]. Later, several algorithms can be applied on the adjacency matrix to provide different TI types such as the Winner index (W) [24], first defined in a chemical context; and others like Randić invariant (χ) [25], Balaban index (J) [26], Broto-Moreau autocorrelation (ATSd) [27] and the spectral moments introduced by Estrada [28]. The spectral moments were defined as the sum of main diagonal entries of the different powers of the bond adjacency matrix [29].…”