Higher Order Connectivity Indices Of Hexagonal Systems

Let G=(V, E) be a simple and connected graph with the vertex set V andthe edge set E. Then the Randi(?) index (also called connectivity index) of a graphG introduced by Milan Randi(?) in 1975 is: x(G)=sum from uv∈E(G) (d(u)d(v))^-1／2where d(u) denotes the degree of the vertex u in G.Connectivity index is one of the most important topological indices inChemical Graph Theory. There is a good correlation between it and several physic-ochemical properties of alkanes: boiling points, surface areas, energy levels, etc.Connectivity index has been widely investigated and applied in mathematics andchemistry. Afterward, Randi(?), Kier, Hall considered the higher order connectivityindices of a general graph G as ^hx(G)=sum from u₁u₂…u_h+1 1／(d_u1…d_uh+1)^1／2where the summation is taken over all possible paths of length h of G. They approvedthat higher order connectivity indices have widely practice meaning in physics andchemistry.For Benzenoid systems and Phenylenes, J.Rada, O.Aranjo, I. Gutman haveworked out a computational formula, and established a relation between the connec-tivity indices of a phenylene and the corresponding hexagonal squeeze; Afterward,J. Rada has worked out a computational formula of Benzenoid systems. In this pa-per, we go on to investigate the second order and third order connectivity indices ofhexagonal systems, and get some results below: (â…°) Second order connectivity indices of phenylenes and its correspondingheptagonal squeezes, include a relation between them;(â…±) Third order connectivity index of phenylenes;(â…²) The hexagonal chains with the extremal third order connectivity index...