Computing Topological Invariants of Triangular Chandelier Lattice

A numerical parameter mathematically derived from the graph structure is a topological index. The topological index is the first actual choice in QSAR research and these indices are used to build the correlation model between the chemical structures of various chemicals compounds. Here, we investigate some old degree-based topological indices like Randic index, sum connectivity index, ABC index, GA index, 1st and 2nd Zagreb indices, modified second Zagreb index, redefined version of 1st, 2nd and 3rd Zagreb indices, hyper and augmented Zagreb indices, forgotten index and symmetric division degree index, and some new degree-based indices like SK index, SK1 index, SK2 index, and AG1 index of triangular chandelier-lattice (TCL). The results are generalized by using edge partition and closed formulas for topological indices of triangular chandelier-lattice are analysed.


Introduction
In the previous two decades topological indices of graphs are being widely applied in analyzing the topology of theoretical and computer-aided models of different physical and chemical phenomena and have found extensive usage in such different areas of scientific research as theoretical physics, chemistry, pharmacology and pharmaceutical chemistry, toxicology, engineering, computer science, sociology, geography, architecture and linguistics. A topological index is a unique mathematical quantity associated with a graph or network which is associated with different properties of the network like connectivity, stability, stress and many others. In the case of chemical graphs, topological indices predict different physiochemical properties like boiling point, biological reactivity, stress, kovat's constant. The study of topological indices is now one of the most vital research field in the chemical graph theory. Topological indices define the structure of molecules mathematically and are used in the growth of qualitative structure-activity relationships (QSARs). Most frequently known invariants of such kinds are degree-based topological indices. In this work, all molecular graphs are considered to be connected, finite and deprived of parallel edges. Let ς be a graph with n vertices and m edges. The degree of a vertex is the number of vertices adjacent to r and is signified as r ∆ . By these expressions, certain topological indices are well-defined as follows. A much-investigated degree-based topological index is Randic index and is indicated as ( ) χ ς and proposed by Randic [Randic (1975)] while analyzing the boiling point of paraffin.
This index is certainly the most extensively applied in chemistry and pharmacology. Definition 1.1 Consider a molecular graph ς , then the Randic index is defined as The sum connectivity index [Zhou and Trinajstic (2009)] is a variation of the Randic connectivity index.
Definition 1.2 Consider a molecular graph ς , then the sum connectivity index is defined as Estrada et al. [Estrada, Torres and Rodriguez (1998) For simple connected graphs and trees Zhong [Zhong (2012)] computed the minimum and maximum values of harmonic index. Definition 1.6 The first and second Zagreb invariants [Gutman and Trinajstic (1972)] are the oldest vertex degree-based topological indices. These indices are defined as Definition 1.7 Nikolic et al. [Nikolic, Kovacevic, Milcevic et al. (2003)] introduced modified Zagreb indices. The second modified Zagreb index is defined as After this, Hao [Hao (2011)] proposed the connection between the Zagreb and modified Zagreb indices and Hao [Hao (2012) Gao et al. [Gao, Siddiqui, Naeem et al. (2017)] computed the closed formula of hyper Zagreb index, first and second multiple Zagreb indices and first and second Zagreb polynomials of carbon graphite and crystal structure of cubic carbon. Definition 2.0 Furtula et al. [Furtula, Graovac, and Vukicevic (2010)] proposed a topological index named as augmented Zagreb index and defined as Ali et al. [Ali, Bhatti and Raza (2017) Mohanappriya et al. [Mohanappriya and Vijayalakshmi (2018)] obtained the general expression for SDD index and inverse sum index of the transformation networks.

Definition 2.3
Shigehalli et al. [Shigehalli and Kanabur (2016)] proposed new degree-based topological indices like The objective of this article is to analyze the degree-based topological indices of triangular Chandelier lattice. Computing topological indices are one of the central problems in chemical graph theory and are being extensively used quantitative structure-activity relations and network topologies. For further interesting results see: Khalid et al. [Khalid and Idrees (2018); Idrees, Said, Rauf et al. (2017); Idrees, Hussain and Sadiq (2018); Qiong and Li (2019)].

Main results for triangular chandelier-lattice ( ) TCL
Lattices are important mathematical objects which have been successfully employed to study different models in biology, chemistry, and physics. For instance, many researchers have employed the Ising and Potts models [Akin (2018); Uguz and Akin (2010)] in conjunction with the Triangular chandelier lattice (Cayley tree-like lattice).
Chandelier lattices can be realized as simple connected undirected graphs : ( , ) V E ς = , where V denotes the set of nodes and E denotes the relation between the nodes. Let k + nearest neighbors with V as the set of vertices and the set of edges . It is clear that the root vertex (0) x has k the nearest neighbors (see Fig. 1). For k =3, chandelier lattice is termed as triangular chandelier lattice, similarly for k =4,5,…, we have a terminology of rectangular lattice, pentagonal lattice and so forth. Our object of interest in the present study is triangular chandelier lattice (TCL). We compute several well-established degree-based topological indices of triangular chandelier-lattice (TCL).

Figure 1: Triangular chandelier-lattice 4
We use edge partition method to compute the topological indices of TCL. A TCL is said to be 1-chandelier if it comprises of a root vertex along with three neighboring vertices. A TCL is obtained from 1-chandelier by successively attaching 1-chandeliers with all vertices except the root vertex. Let denotes the number of 1-chandeliers in a TCL. The graph of TCL is denoted by c T has 4c+1 number of total vertices or nodes and 6c edges.
Degree of root vertex is three and all the vertices in middle layers of TCL have degree 6 except the boundary layer vertices which again have degree three. So all edges of TCL has degrees of end vertices as (3,3), (3,6) and (6,6).We denote each type of edges as 3,3 E , 3,6 E and 6,6 E , respectively. The number of these types of edges is analyzed by considering different values of . These results aresummarised in Tab. 1 below. ix.
The forgotten index of TCL is F( ) 270 522 x.