On topological descriptors of ceria oxide and their applications

: A topological descriptor is a mathematical illustration of a molecular construction that relates particular physicochemical properties of primary molecular structure as well its mathematical depiction. Topological co-indices are usually applied for quantitative structure actions relationships (QSAR) and quantitative structures property relationships (QSPR). Topological co-indices are topological descriptors which are considered the noncontiguous vertex set. We study the accompanying some renowned topological co-indices: first and second Zagreb co-indices, first and second multiplicative Zagreb co-indices and the F-coindex, and some other degree-based indices of the co-indices of ceria oxide.


Introduction
In graph theory, a vertex describes an atom and edges bonds between atoms. A topological descriptor is a mathematical entity which is handled methodically on the atomic structure. Which is linked with copied composition displaying for relationships among the substances of the structure through several physical, chemical characteristics. Topological descriptors of some nanostar dendrimers studied and plotted in Imran et al. (2016) and Siddiqui et al. (2016)." Topological index is a mathematical parameter of a structure which depict the topology and generally structure invariants. In QSAR/QSPR research, physicochemical characteristics, and topological co-indices such as Randic, atom bond connectivity and geometric arithmetic coindices are utilized to forecast bio activity of molecular structure. Graph theory has observed a substantial usage in this part of study." Many molecular compounds are vital for the existence of alive things. Many molecular structures are important for the existence of alive thing. Such as carbon, oxygen, hydrogen, and nitrogen help in construction of living cells. Carbon is a vital element for alive cells. Carbon is valuable in creation of protein, carbohydrate, and nucleic acid. It is vibrant for development of plant in forms of carbon dioxide. The carbon atom link composed in numerous states, named allotrope of carbons. The renowned form is graphite and diamonds. Lately, numerous innovative forms have been revealed together with nanotubes, buck minster fullerenes, and sheet, crystals cubic structures." Let G be a chemical graph, this graph is simple connected with set of atoms V(G) and set of bonds E(G). Degree of an atom is the amount of neighboring atoms to A simple connected structure is called r regular, if each atom of G has exactly r-neighbour. Complement of a structure G that is represented as is a structure with the same set of atoms V(G), and any two atoms if and only if . Zagreb and Randic indices are amongst the eldest and utmost completely researched structure-based topological descriptor along with various application. The first Zagreb index and second Zagreb index was defined by Gutman and Trinajstic (Doslic, 2008) and give a relation with the total π-electron energy (ε) of a structure, in which estimated formulations of the total π-electron energy (ε). These co-indices deliver a quantity of the separating of the carbon-atoms skeletons. These indices are defined as follows: (1) ( 2) Recently, multiplicative Zagreb co-indices are studied by Xu et al. (2013), these indices are counted as the multi plicative version of Zagreb co-indices and studied few characteristics and upper bound and lower bound of a structure, and defined as: ( 3) (4) De et al. (2016) presented the F-co-index of a structure, symbolized by ; later attesting that correlation coefficient amongst the logarithm of octanol-water divider number and the equivalent F-co-index value of octane isomer is learnt to be approximately 0.996 that show F-co-index can also forecast the value with great precision. F co-index is formulated as: (5) Atom bond connective index is renowned connectivity topological descriptor defined by Estrada et al. (1998): (6) Second very important topological index is geometric arithmetic, defined by Vukicevic and Furtula (2009): First topological index constructed on degree of a structure is Randic index (Randic, 1975) presented by Milan Randic. The general Randic co-index was defined as:

Lemma 1
For a simple graph G order n with where n i are amount vertices with degrees i and m ij are the amount of edges joining the vertex of degree i with the vertex of degree j (Berhe and Wang, 2019). Then, for: In this paper, we study the result for ceria oxide CeO 2 and computed analytically close results of first, second Zagreb and multiplicative Zagreb, forgotten, atom bond connectivity, geometric-arithmetic, and general Randic co-indices for different values of α, for result for ceria oxide CeO 2 .

Result for ceria oxide CeO 2
Let be the structure of CeO 2 with p × q unit cell and t layers. We create the structure by choosing first p × q units in the pq−plane after that joining them up in t layers see Figure 1. The cardinality of vertex and edge set of CeO 2 [p, q, t] are 12pqt + 2pq + 2pt + 2qt + p + q + t + 1 and 22pqt + 4pq + 4pt + 4qt + 2p + 2q + 2t + 4, respectively. As, three types of vertices are in CeO 2 [p, q, t] namely, the vertices of degree 4, 6, and 8. The partitions of the vertex set of CeO 2 [p, q, t] are given in Table 1. Also, the edge partitions of CeO 2 [p, q, t] constructed on degree of end vertices of edges are given in Table 2.
We will use Lemma 1 to calculate the equations below. For ;   For i = 4, j = 6; For i = 4, j = 8; For i = j = 6;

Conclusion
In this study, we computed the degree based topological coindices such as first, second Zagreb, and multiplicative Zagreb coindices, forgotten atom-bond connectivity (ABC), geometric arithmetic, and general Randic coindices of silicate network and triangular chain cactus graphs for different values of α, and shown their numerical comparisons."