Tenth Spectral Moment for Molecular Graphs of Phenylenes

Abstract

Our investigations are motivated by recent papers concerning the spectral moments of the edge-adjacency matrix, which were successfully employed in QSAR and QSPR studies of different classes of compounds [Estrada, E. J. Chem. Inf. Comput. Sci. 1996, 36, 844−849; 1997, 37, 320−328; 1998, 38, 23−27]. In this work, the evaluation of the 10th spectral moment of the vertex-adjacency matrix for phenylenes is considered. This is achieved by using the method of Hall [Hall, G. G. Theor. Chim. Acta 1986, 70, 323−332], which was originally applied to benzenoid systems and then adopted for molecular graphs of phenylenes [Marković, S.; Stajković, A. Theor. Chem. Acc. 1997, 96, 256−260]. It is found that the 10th spectral moment can be expressed in terms of 6 mutually independent invariants that can be easily deduced from the molecular graph.

The anti-Kekulé number of graphs

“…Adding an edge e i to G − A will increase the neighbors of S by one (at most). Hence Combining inequalities (23), (24) and (25), we obtain…”

“…Zhang et al [47] considered the problem of finding the anti-Kekulé number of a special class of conjugated hydrocarbons-phenylenes [24]. These compounds are composed of six-and four-membered rings, where the six-membered rings (hexagons) are adjacent only to four-membered rings and every four-membered ring is adjacent to a pair of hexagons.…”

Application of spectral graph theory on the enthalpy change of formation of acyclic saturated ketones

“…Explicit formulas, in terms of counts of simple structural properties, were designed for the first few spectral moments of various classes of molecular graphs. [7][8][9]12,17,18,22,24,26,27 A procedure based on energy partitioning via spectral moments was proposed for investigating the aromaticity of conjugated systems. 15,16 In a series of works, moments were used to estimate the HMO total π-electron energy and examine its dependence on molecular structure.…”

Topology of naphthylenic tori

“…Phenylenic systems 1 are cata-condensed species with alternating aromatic C 6 and antiaromatic C 4 rings (i.e., C 6 , C 4 , C 6 , C 4 ,...,C 6 sequences) and exhibit unique physico-chemical properties. 2 This explains the increasing interest of both synthetic [3][4][5][6] and theoretical [7][8][9][10] chemists, during the last decade. The phenylenic structures synthesized or theoretically studied so far can be drawn as plane (molecular) graphs.…”