On Vertex-Edge-Degree Topological Descriptors for Certain Crystal Networks

Due to the combinatorial nature of graphs they are used easily in pure sciences and social sciences. The dynamical arrangement of vertices and their associated edges make them flexible (like liquid) to attain the shape of any physical structure or phenomenon easily. In the field of ICT they are used to reflect distributed component and communication among them. Mathematical chemistry is another interesting domain of applied mathematics that endeavors to display the structure of compounds that are formed in result of chemical reactions. This area attracts the researchers due to its applications in theoretical and organic chemistry. It also inspires the mathematicians due to involvement of mathematical structures. Regular or irregular bonding ability of molecules and their formation of chemical compounds can be analyzed using atomic valences (vertex degrees). Pictorial representation of these compounds helps in identifying their properties by computing different graph invariants that is really considered as an application of graph theory. This paper reflects the work on topological indices such as ev-degree Zagreb index, the first ve-degree Zagreb index, the first ve-degree Zagreb index, the second ve-degree Zagreb index, ve-degree Randic index, the ev-degree Randic index, the ve-degree atom-bond connectivity index, the ve-degree geometric-arithmetic index, the ve-degree harmonic index and the ve-degree sum-connectivity index for crystal structural networks namely, bismuth tri-iodide and lead chloride. In this article we have determine the exact values of ve-degree and ev-degree based topological descriptors for crystal networks.


Introduction
Computation of topological indices for large chemical structures becomes very challenging but still useful in depicting the structure and physico-chemical properties that are extremely important in reticular chemistry. Recently reticular Metal-organic frameworks MOFs are evolved as porous conductive solids with great applicability in fuel cells, batteries, capacitors, sensors and electronics. In MOFs covalent fibers of carbon atoms form mesh like crystals [1,2]. In reticular chemistry, the numerically representation of structural characteristics of molecules, are the topological indices, which are obtained by using the graphical methods. These indices play an important role in the area of mathematical chemistry and control theory, mainly in QSAR/QSPR investigations [3,4].
The networks that are topologically equivalent, although they exibit different labelings of distinct atoms but due to topological indices they are invariant. These indices describe the connections among the atoms and in this way they are basic invariants that show a relationship with biological activity and chemical reactivity. Topological study of a MOF means transforming the connectivity of any structure into a unique number representing an index of the metal-organic framework under consideration.

Preliminaries
Let G be a simple connected graph with vertex sets V ðGÞ and edge sets EðGÞ. The degree of a vertex e, denoted by dðeÞ, is the number of edges that are incident to the e. The open neighborhood of e is defined as N ðeÞ ¼ fe 2 V ðGÞ : e 2 EðGÞg and closed neighborhood N½e ¼ NðeÞ [ feg [23]. The ve-degree, denoted by d ve ðeÞ, of any vertex e 2 V is the number of different edges that are incident to any vertex from the N ½e. In [24] defined the ev-degree of the edge e ¼ e 2 E; denoted by d ev ðeÞ; the number of vertices of the union of the closed neighborhoods of and e: For details see [25][26][27][28][29][30].
The ve-degree and ev-degree topological indices are defines as: ve ) index, ve-degree Randic (R ve ) index, the ev-degree Randic (R ev ) index, the ve-degree atom-bond connectivity (ABC ve ) index, the ve-degree geometric-arithmetic (GA ve ) index, the ve-degree harmonic (H ve ) index and the ve-degree sum-connectivity (v ve ) index, respectively.

Crystal Structures
The physical structure of solid materials is significant for engineering applications. It depends on the arrangements of the atoms, ions, or molecules that becomes the reason for strength of solid materials. The connectivity pattern of ions or atoms in a solid and repetitive patterns in three dimensions is known as crystal structure and material is called crystalline solid or crystalline material. Due to different crystalline structure of a materials their performance and characteristics varies. The unit cell is the basic structure that explains the crystal structure and repetition of this unit cell forms the whole crystal. Some of the examples of crystalline materials are alloys, metals, and some ceramic materials. In this paper topological indices for the bismuth tri-iodide and lead chloride are determined by mapping their crystalline structures in the form of graphs.

Graph of Bismuth Tri-Iodide
Bismuth tri-iodide (BiI 3 ) is an inorganic compound. It is the result of the response of bismuth and iodine, which is important in qualitative inorganic analysis. Layered BiI 3 crystal is considered to be a three-layered stacking structure, where bismuth atom planes are lying between iodide atom planes, which form the sequence I-Bi-I planes. The rhombohedral BiI 3 crystal with R-3 symmetry is formed by the periodic stacking of three layers [31,32]. In 1995, Nason et al. [33] synthesized a unit crystal of BiI 3 . The Fig. 1 shows one unit of bismuth tri-iodide.
The graph of a single unit of bismuth tri-iodide contains six 4-cycles of which two are at the bottom, two on the top and two in the middle. The unit cells of bismuth tri-iodide can be arranged either linearly or in a sheet form. A linear arrangement with q unit cells is called q-bismuth chain, p Â q bismuth sheet is obtained by arrangements of pq unit cells into p rows and q columns. A p Â q bismuth sheet throughout onward represented by BiI 3 . A sheet of BiI 3 contains 11pq þ 10p þ 7q þ 2 vertices and 18pq þ 12p þ 6q edges, which are shown in Tab We partitioned the edges of BiI 3 , based on ev-degree in Tab. 4.
In Tab. 5, we partitioned the vertices of BiI 3 , based on ev-degree.
We partitioned the edge of BiI3 with respect to ve-degrees.

ev-Degree Zagreb Index
By using ev-degree of BiI 3 from Tab. 4, we compute the ev-degree based Zagreb index: The First ve-degree Zagreb a Index Using Tab. 5 we compute the first ve-degree Zagreb a index: The First ve-degree Zagreb b Index Using Tab. 6 we compute the first ve-degree Zagreb b index: The second ve-degree Zagreb index Using Tab. 6 we compute the second ve-degree Zagreb index:

ve-degree Randic Index
Using Tab. 6 we compute the ve-degree Randic index:

The ve-degree Atom-bond Connectivity Index
Using Tab. 6 we compute the ve-degree atom-bond connectivity index: 4.10 The ve-degree Sum-connectivity Index Using Tab. 6 we compute the ve-degree sum-connectivity index:

The Graph of Lead Chloride
Lead chloride is a precious halide stone that usually occurs in mineral cotunnite. The structure of lead chloride is orthorhombic dipyramidal. The diagram of a solitary unit of lead chloride is obtained from bismuth tri-iodide by joining an extra vertex to only one 2-degree vertex of every one of the 4-cycles. Fig. 2 shows one unit of lead chloride.
The unit cells of lead chloride can be arranged either linearly or in a sheet form. A linear arrangement with q unit cells is called q-lead chloride chain, p Â q lead chloride sheet is obtained by arrangements of pq unit cells into p rows and q columns. A p Â q lead chloride sheet throughout onward represented by LC. A sheet of LC contains 12pq þ 10p þ 7q þ 2 vertices and 24pq þ 12p þ 6q edges, which are shown in Tab. 7. The number of vertices corresponding to their degrees of LC are shown in Tab. 8 and the edge partition based on degree of end vertices of each edge is shown in the Tab. 9.
We partitioned the edge of LC with respect to ve-degrees.

The First ve-degree Zagreb b Index
Using Tab. 12 we compute the first ve-degree Zagreb b index:

Conclusion
For numerical representaion of topologies for graphs or netwroks, topological descriptors are most useful invariants. That is why such investigations are widely used in computer applications and mathematical chemistry. Calculated results in this paper for ev-degree and ve-degree based topological indices for the crystal networks are shown pictorial in form of line chart from Figs. 3-6. In all the line  Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.