Revan Indices and Revan Polynomials of Silicon Carbide Graphs

Topological polynomials are algebraic expressions which are related to the topology of graphs up to graph isomorphism. They are used to indicate the invariants of graphs of chemical structures. In a chemical graph, vertices and edges correspond to atoms and bonds respectively. The quantitative structure property relationship (QSPR) depicts a connection between the structure and the properties of molecules. There are numerous types of topological indices among them degree based are more popular. In this paper we compute Revan first index, Revan second index, Revan third index and their polynomials for list of silicon carbide graphs. Comparison between Revan first index, Revan second index and Revan third index is also part of our study. Keywords— Chemical graph theory; chemical graph; atoms; bonds; silicon carbide; Revan vertex degree; first Revan index; second Revan index, third Revan index, Revan polynomials.


I. INTRODUCTION
Chemical graph theory is a topology branch of mathematical chemistry which deals with applications of graph theory to mathematical structure of chemical phenomena. Molecular graphs are the basic models of chemical graph theory in which vertices corresponds to vertices and edges correspond to bonds. The quantitative structure property relationship (QSPR) depicts a connection between the structure and the properties of molecules [1], [2]. Various numerical graph invariants and their related polynomials have been defined and used for correlation analysis in theoretical chemistry, pharmacology, toxicology and environmental chemistry. A topological index is a type of molecular descriptor which is calculated for a molecular graph of chemical phenomena. Topological indices are used to study the topology and graph invariants of molecular graphs. For a simple, connected and undirected graph = ( ( ), ( )) with vertex set ( ) and edge set ( ), is an edge with end vertices and , denotes the ordinary degree of vertex in , ∆( ) denotes the maximum degree of and ( ) is the minimum degree of . For a simple and connected graph , V. R. Kulli [5] introduced Revan first index, Revan second index and third Revan index as:

II. SILICON CARBIDE GRAPHS
We consider a family of silicon carbide graphs, and calculate necessary computations in this section. This family includes eight silicon carbide graphs with the names and notation as: 2  Description and computations about these graphs are as follows: 1. The 2 structure of silicon carbide graph ≅ 2 3 − [ , ], , ≥ 1, is shown in Figure 1  This structure consists of cells (in a row) and connected rows. Left figure is a unit cell and right figure is of 2 3 − [4,3]. Order of this graph is 10 and size is 15 − 2 − 3 . Edge partition and cardinalities of partite sets are given as: 3,2 3   This structure consists of cells (in a row) and connected rows. Order of this graph is 10 and size is 15 − 3 − 3 . Left picture is a unit cell and right is of 2 3 − [ 3,3]. Edge set is partitioned into following sets:

4.
The 2 structure of ≅ 2 3 − [ , ], , ≥ 1, is shown in Figure 3. Left picture is a unit cell and right is of 2 3 − [4,3] This structure consists of cells (in a row) and connected rows. Order of this graph is 10 and size is 15 − 2 − 3 . Edge set is partitioned into following sets:  Left picture is a unit cell and right is of 3 − [ 4,3]. Order of this graph is 8 and size is 12 − 2 − 3 . Edge set is partitioned inteo following sets:  Figure 5. This structure consists of cells (in a row) and connected rows. Order of this graph is 8 and size is 12 − 2 − 2 . Left picture is a unit cell and right is of 3 − [ 4,3].

Figure 5
Edge set is partitioned into following sets  This structure consists of cells (in a row) and connected rows. Order of this graph is 8 and size is 12 − 3 − 2 . Left picture is a unit cell and right is of 3 − [ 5,4]. Edge set is partitioned into following sets:  Figure 7. This structure consists of cells (in a row) and connected rows. Order of this graph is 10 and size is 15 − 4 − 2 + 1. Left picture is a unit cell and right is of 4 − [ 4,3]. Partition of edge set is 3,2 3,1 And cardinalities of partite sets are | 3,2   This structure consists of cells (in a row) and connected rows. Order of this graph is 10 and size is 15 − 4 − 2 . First picture is a unit cell and second is of 4 − [ 3,3]. Edge set is partitioned into following sets:  Adding entries of last column, we get 3 ( ) for > 1, ≥ 1.   • Comparison of Revan first index 1 ( ), Revan second index 2 ( ) and Revan third index 3 ( ) of

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Comparison of Revan first index 1 ( ), Revan second index 2 ( ) and Revan third index 3 ( ) of 3 − [ , ]. Yellow color sheet represents Revan first index, blue color sheet represents Revan second index and red color sheet represents Revan third index.

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Comparison of Revan first index 1 ( ), Revan second index 2 ( ) and Revan third index 3 ( ) of 3 − [ , ]. White color sheet represents Revan first index, sea green color sheet represents Revan second index and yellow color sheet represents Revan third index.