A Recently, Ghorbani et. al. introduced the eccentric versions of first and second Zagreb indices called third and fourth Zagreb indices defined asM3 (G) = Σuv∊E(G) (ε (u) + ε (ν)) and M4 (G) = Σν∊V(G)ε (ν)2, respectively, where ε (ν)is the eccentricity of the vertex ν. In this paper, we compute the closed formula for third Zagreb index of Polycyclic Aromatic Hydrocarbons (PAHk).
A topological index of a graph G is a numeric quantity related to G which is invariant under automorphisms of G. The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = Σe=uv∈E(G) [nu + nv], where nu is the number of edges of G lying closer to v than u, analogously nv. In this paper, we compute the vertex PI index of Titania carbon Nanotubes TiO2[m,n].
Let G be a simple connected graph. The geometric-arithmetic index of G is defined as $\begin{array}{}
G{A_1}\left( G \right) = {\sum\nolimits _{u\nu \in E(G)}}\frac{{2\sqrt {d(u)d(\nu)} }}{{d(u) + d(\nu)}}
\end{array}$, where d(u) represents the degree of the vertex u in the graph G. Recently, Graovac defined the fifth version of geometric-arithmetic index of a graph G as $\begin{array}{}
G{A_5}\left( G \right) = {\sum\nolimits _{u\nu \in E(G)}}\frac{{2\sqrt {{S_\nu}{S_u}} }}{{{S_\nu} + {S_u}}}
\end{array}$, where Su is the sum of degrees of all neighbors of vertex u in the graph G. In this paper, we compute the fifth geometric arithmetic index of Polycyclic Aromatic Hydrocarbons (PAHk).
A novel topological index, the face index ( F I ), is proposed in this paper. For a molecular graph G, face index is defined as F I ( G ) = ∑ f ∈ F ( G ) d ( f ) = ∑ v ∼ f , f ∈ F ( G ) d ( v ) , where d ( v ) is the degree of the vertex v. The index is very easy to calculate and improved the previously discussed correlation models for π - e l e c t r o n energy and boiling point of benzenoid hydrocarbons. The study shows that the multiple linear regression involving the novel topological index can predict the π -electron energy and boiling points of the benzenoid hydrocarbons with correlation coefficient r > 0.99 . Moreover, the face indices of some planar molecular structures such as 2-dimensional graphene, triangular benzenoid, circumcoronene series of benzenoid are also investigated. The results suggest that the proposed index with good correlation ability and structural selectivity promised to be a useful parameter in QSPR/QSAR.
Irregularity indices are usually used for quantitative characterization of the topological structures of non-regular graphs. In numerous problems and applications, especially in the fields of chemistry and material engineering, it is useful to be aware of the irregularity of a molecular structure. Furthermore, the evaluation of the irregularity of graphs is valuable not only for quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies but also for various physical and chemical properties, including entropy, enthalpy of vaporization, melting and boiling points, resistance, and toxicity. In this paper, we will restrict our attention to the computation and comparison of the irregularity measures of different classes of dendrimers. The four irregularity indices which we are going to investigate are σ irregularity index, the irregularity index by Albertson, the variance of vertex degrees, and the total irregularity index.
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