Generalized Second Neighborhood Zagreb Index: Mathematical Inequalities and Chemical Applicability of PAHs

Topological indices are graphical invariants that relate a numeric number to a graph, which is structurally invariant and predicts the chemical, biological and physical features of the molecular graphs. In this work, mathematical inequalities of generalized second neighborhood Zagreb index are obtained. Further, generalized second neighborhood Zagreb indices for some particular values are computed for some basic polycyclic aromatic hydrocarbons, and the QSPR analysis are also obtained.


Introduction
In this paper we are concerned with simple graphs that is graphs without multiple, directed, or weighted edges, and without self-loops. Let ( , ) be such a graph with vertex set ( ) and edge set ( ). Let | ( )| = and | ( )| = . The degree ( ) of a vertex v is the number of vertices adjacent to v. The set of all vertices which are adjacent to a vertex v is called open neighborhood of v and denoted by ( ). The closed neighborhood set of a vertex v is the set ( )= ( ) ∪ { }. For graph-theoretical terminology and notation not defined here we follow [12].
Chemical graph theory is a branch of mathematical chemistry deals with the chemical graph obtained by considering molecules or atoms as vertices and chemical bonds as edge. Topological indices are numeric quantities that transform chemical structure to real number, which are used in QSAR/QSPR studies to correlate the bioactivity and physiochemical properties of molecule. For their history, applications and mathematical properties, see [2,11,14,25,26] and the references cited therein.
For any real number , the generalized second neighborhood Zagreb index

Mathematical Inequalities
In this section, we obtain some mathematical inequalities of 2 ( ) ( ) in terms of order, size, minimum/maximum degree, minimum/maximum neighborhood degree sum and generalized Randic index of a graph G. For more details, we refer [1,3,7,8,9,15,16].
Let G be a non-trivial (p, q) -graph with > 0.

Chemical Applicability
The properties and activities of chemicals are strongly related to their molecular structures, which are capable to predict the higher correlation factor has greater importance in quantitative structure-property relationships (QSPR).

Polycyclic Aromatic Hydrocarbons (PAHs)
Polycyclic aromatic hydrocarbons (PAHs) are the primary source of environmental pollution and most are carcinogenic and mutagenic. The accumulation and influence of PAHs on the environment and human health depends on their physico-chemical properties. Therefore, the QSPR study of physical and chemical properties helps to manage these PAHs. Recently the QSPR analysis of PAHs was studied in [5,6]. In this paper, we examined the chemical graph of certain fundamental PAHs.

Table-1: Bond partition for molecular graphs of PAHs
The generalized second neighbor index of Naphthalene for = 1 is calculated as follows: Similarly, we have         From the Table -8    The study of the table -10 reveals that the index for R greater than 0.9.   ) .

Conclusion
In this article, some mathematical inequalities for generalized second neighborhood indices are obtained. We computed generalized second neighborhood Zagreb indices for α = {1, 2, 1/2, −1/2} for PAHs and the QSPR analysis is performed for the Physico-chemical properties of the PAHs. It was found, the correlation between these indices with different properties of PAHs are often strong and hence these indices are suitable for QSPR analysis.