Revan Indices and Revan Polynomials of Silicon Carbide Graphs

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Revan Indices and Revan Polynomials of Silicon Carbide Graphs

Rehana Ashraf

Department of Mathematics

Lahore College for Women University, Jhang Campus Pakistan

Sadia Akhter

Department of Mathematics

Lahore College for Women University, Jhang Campus Pakistan

Abstract 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

Where

2() = ( )

()

3() = | |

()

is Revan degree of a vertex defined by Kulli as

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.

  1. 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:

    1() = ( + )

    ()

    = () + () .

    In 2018, V. R. Kulli [6] introduced first, second and third Revan polynomials of a simple connected graph as:

    1(, ) = (+)

    ()

    2(, ) = ( )

    ()

    3(, ) = ||

    ()

  2. 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: 23 [, ], 23 [, ], 23

    [, ], 3 [, ] , 3 [, ], 3

    [, ], 4 [, ], and 4 [, ] for arbitrary , 1. Here we list some graph terms and properties which is a common possession of all these eight graphs:

    1. In all graphs of silicon carbides, silicon atoms are colored blue and carbon atoms are colored red, green color edges are used to connect two cells in a colums and red color edges are used to connect two cells in a row.

    2. In all graphs, vertices have ordinary degree 1, 2 or 3 and since () + () = 4, so vertices have Revan degree 3, 2 or 1 respectively.

    3. An edge has a type {, } if vertex has Revan degree and vertex has Revan degree .

    4. Edge sets of all eight silicon carbide graphs are portioned with respect to type defined above and cardinalities of partite sets are computed.

      Description and computations about these graphs are as follows:

      1. The 2 structure of silicon carbide graph

    23 [, ], , 1, is shown in Figure 1. 2.

    Figure 1

    This structure consists of cells (in a row) and connected rows. Left figure is a unit cell and right figure is of 23

    [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 = { (23 [, ])| = 3, = 2},

    3,1 = { (23 [, ])| = 3, = 1},

    2,2 = { (23 [, ])| = 2, = 2},

    = { ( [, ])| = 2, = 1},

    and cardinalities of partite sets are:|3,2| = 2, |3,1| = 1,

    |2,2| = 2 + 2, |2,1| = 8 + 8 14, |1,1| = 15 13 13 + 11

    4. The 2 structure of 23 [, ],

    , 1, is shown in Figure 3. Left picture is a unit cell and right is of 23 [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.

    Figure 3

    Edge set is partitioned into following sets:

    3,1 = { (23 [, ])| = 3, = 1},

    2,2 = { (23 [, ])| = 2, = 2},

    2,1 = { (23 [, ])| = 2, = 1},

    1,1 = { (23 [, ])| = 1, = 1},

    And cardinalities of partite sets are |3,1| = 1, |2,2| = 2 + 2, |2,1| = 8 + 8 12, |1,1| = 15

    10 13 + 8

    2,1 2 3

    1,1 = { (23 [, ])| = 1, = 1},

    And cardinalities are |3,2| = 1, |3,1| = 1, |2,2| =

    + 2, |2,1| = 6 + 8 9, |1,1| = 15 9 13 + 7.

    3. The 2 structure of 23 [, ], , 1, is shown in Figure 2.

    Figure 2

    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

    4. The 2 structure of 3 [, ], , 1, is shown in Figure 4. This structure consists of cells (in a row) and connected rows.

    Figure 4

    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:

    3,2 = { (3 [, ])| = 3, = 2}

    2 3 3,1 = { (3 [, ])| = 3, = 1}, = 1

    [3, 3]. Edge set is partitioned into following sets:

    3,2 = { (23 [, ])| = 3, = 2},

    2,2

    = { (3

    [, ])|

    = 2,

    = 2},

    = { ( [, ])| = 3,

    = 1},

    2,1 = { (3 [, ])| = 2, = 1},

    3,1

    2 3

    = { ( [, ])| = 1, = 1},

    2,2 = { (23 [, ])| = 2, = 2},

    1,1

    3

    2,1 = { (23 [, ])| = 2, = 1},

    And cardinalities are |3,2| = 2, |3,1|

    1,1 = { (23 [, ])| = 1, = 1},

    |2,2

    | = { 3 1 for = 1, 1 2 + 2 3 for > 1, 1

    |2,1

    | = { 6 4 for = 1, 1 4 + 8 8 for > 1, 1

    And cardinalities of partite sets are |3,2| = 2,|3,1| = 1,

    |2,2| = 3 + 2 3,|2,1| = 6 + 4 8,|1,1| = 12 12 8 + 8.

    |1,1| =

    12 2 12 + 2 for = 1, 1

    {

    12 8 13 + 8 for > 1, 1

    1. The 2structure of 4

      [, ], , 1, is

      5. The 2 structure of 3 [, ], , 1, is shown in 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

      3,1 = { (3 [, ])| = 3, = 1},

      2,2 = { (3 [, ])| = 2, = 2}

      2,1 = { (3 [, ])| = 2, = 1},

      1,1 = { (3 [, ])| = 1, = 1},

      And cardinalities of partite sets are computed as |3,1| = 2, |2,2| = 2 + 1,|2,1| = 4 + 8 10,|1,1| =

      12 8 10 + 7.

      6. The 2 structure of 3 [, ], , 1, is shown in Figure 6.

      Figure 6

      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:

      3,2 = { (3 [, ])| = 3, = 2},

      3,1 = { (3 [, ])| = 3, = 1},

      2,2 = { (3 [, ])| = 2, = 2},

      2,1 = { (3 [, ])| = 2, = 1},

      1,1 = { (3 [, ])| = 1, = 1},

      shown in 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 = { (4 [, ])| = 3, = 2},

      3,1 = { (4 [, ])| = 3, = 1},

      2,2 = { (4 [, ])| = 2, = 2},

      2,1 = { (4 [, ])| = 2, = 1},

      1,1 = { (4 [, ])| = 1, = 1},

      And cardinalities of partite sets are |3,2| = 2, |3,1| = 3 2,|2,2| = + 2 2, |2,1| = 2 + 4 2,

      |1,1| = 14 10 8 + 5

      Figure 7

      8. The 2 structure of 4 [, ], , 1, is shown in Figure 8.

      Figure 8

      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:

      3,2 = { (4 [, ])| = 3, = 2},

      2,2 = { (4 [, ])| = 2, = 2},

      2,1 = { (4 [, ])| = 2, = 1},

      1,1 = { (4 [, ])| = 1, = 1}, And cardinalities of partite sets are |3,2| = 2

      |2,2| =

      5 + 2 for = 1, 1

      {

      2 + 2 for > 1, 1

      |2,1

      | = { 6 6 for = 1, 1 12 + 8 14 for > 1, 1

      Revan third polynomial is

      (, ) = || = 2 + (6 + 8 8) +

      3 ()

      |1,1

      | = { 15 15 2 + 2 for = 1, 1 15 18 10 + 10 for > 1, 1

      (15 8 11 + 7)

      Theorem 3.2. Let

      [, ], , 1 be the

      2 3

      silicon carbide graph, then

  3. MAIN RESULTS

    In this section, we prove results about family of silicon carbide graphs listed in previous section. Here we present results of Revan indices and Revan polynomials for our list of silicon carbide graphs.

    Theorem 3.1. Let 23 [, ], , 1 be the silicon carbide graph, then

    1. 1() = 30 + 4 + 6 4.

    2. 2() = 15 + 7 + 11 2.

    3. 3() = 6 + 8 6.

    Proof. Adding entries of last column, we get 1().

    1. 1() = 30 + 6 + 6 6.

    2. 2() = 15 + 11 + 11 2.

    3. 3() = 8 + 8 10.

    Proof. Adding entries of last column, we get 1()

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {3,1}

    1

    4

    4

    {2,2}

    2 + 2

    4

    8 + 8

    {2,1}

    8 + 8 14

    3

    24 + 24 42

    {1,1}

    15 13

    13 + 11

    2

    30 26

    26 + 22

    Revan first polynomial is

    {, }

    = |, |

    +

    ( + )

    {3,2}

    1

    5

    5

    {3,1}

    1

    4

    4

    {2,2}

    + 2

    4

    4 + 8

    {2,1}

    6 + 8 9

    3

    18 + 24 27

    {1,1}

    15 9

    13 + 7

    2

    30 18

    26 + 14

    {, }

    = |, |

    +

    ( + )

    {3,2}

    1

    5

    5

    {3,1}

    1

    4

    4

    {2,2}

    + 2

    4

    4 + 8

    {2,1}

    6 + 8 9

    3

    18 + 24 27

    {1,1}

    15 9

    13 + 7

    2

    30 18

    26 + 14

    (, ) = (+) = 25 + (2 + 2 + 1)4

    1 ()

    {, }

    = |, |

    ( )

    {3,2}

    2

    6

    12

    {3,1}

    1

    3

    3

    {2,2}

    2 + 2

    4

    8 + 8

    {2,1}

    8 + 8 14

    2

    16 + 16 28

    {1,1}

    15 13

    13 + 11

    1

    15 13

    13 + 11

    {, }

    = |, |

    ( )

    {3,2}

    2

    6

    12

    {3,1}

    1

    3

    3

    {2,2}

    2 + 2

    4

    8 + 8

    {2,1}

    8 + 8 14

    2

    16 + 16 28

    {1,1}

    15 13

    13 + 11

    1

    15 13

    13 + 11

    +(8 + 8 14)3+(15 13 13 + 11)2. Adding entries of last column, we get 2()

    Revan first polynomial is

    1(, ) = () (+) = 5 + ( + 2 + 1)4

    {, }

    = |, |

    ( )

    {3,2}

    1

    6

    6

    {3,1}

    1

    3

    3

    {2,2}

    + 2

    4

    4 + 8

    {2,1}

    6 + 8 9

    2

    12 + 16 18

    {1,1}

    15 9

    13 + 7

    1

    15 9

    13 + 7

    6

    {, }

    = |, |

    ( )

    {3,2}

    1

    6

    {3,1}

    1

    3

    3

    {2,2}

    + 2

    4

    4 + 8

    {2,1}

    6 + 8 9

    2

    12 + 16 18

    {1,1}

    15 9

    13 + 7

    1

    15 9

    13 + 7

    +(6 + 8 9)3+(15 9 13 + 7)2. Adding entries of last column, we get 2()

    Revan second polynomial is

    2(, ) = () () = 6 + ( + 2)4 + 3 + (6 + 8 9)2 + (15 9 13 + 7)

    Adding entries of last column, we get 3()

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    1

    1

    1

    {3,1}

    1

    2

    2

    {2,2}

    + 2

    0

    0

    {2,1}

    6 + 8 9

    1

    6 + 8 9

    {1,1}

    15 9

    13 + 7

    0

    0

    Revan second polynomial is 2(, ) = () () = 26 + (2 + 2)4 + 3 +

    (8 + 8 14)2 + (15 13 13 + 11)

    Adding entries of last column, we get 3()

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {3,1}

    1

    2

    2

    {2,2}

    2 + 2

    0

    0

    {2,1}

    8 + 8

    14

    1

    8 + 8 14

    {1,1}

    15 13

    13 + 11

    0

    0

    Revan third polynomial is 3(, ) = () || =

    2 + (8 + 8 12) + (15 11 11 + 11).

    Theorem 3.3. Let 23 [, ], , 1 be the silicon carbide graph, then

    1. 1() = 30 + 4 + 6 4.

    2. 2() = 15 + 6 + 11 2.

    3. 3() = 8 + 8 8.

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {3,1}

    1

    4

    4

    {2,2}

    3 1

    4

    12 4

    {2,1}

    6 4

    3

    18 12

    {1,1}

    12 2

    12 + 2

    2

    24 4

    24 + 4

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {3,1}

    1

    4

    4

    {2,2}

    3 1

    4

    12 4

    {2,1}

    6 4

    3

    18 12

    {1,1}

    12 2

    12 + 2

    2

    24 4

    24 + 4

    Proof. Adding entries of last column, we get 1().

    {, }

    = |, |

    +

    ( + )

    {3,1}

    2

    4

    8

    {2,2}

    2 + 2

    4

    8 + 8

    {2,1}

    8 + 8 12

    3

    24 + 24 36

    {1,1}

    15 10

    13 + 8

    2

    30 20

    26 + 16

    Revan first polynomial is

    1(, ) = () (+) = (2 + 4)4+ (8 + 8 12)3+ (15 10 13 + 8)2.

    Adding entries of last column, we get 2()

    Adding entries of last column, we get 1() for > 1,

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {3,1}

    1

    4

    4

    {2,2}

    2 + 2 3

    4

    8 + 8 12

    {2,1}

    4 + 8 8

    3

    12 + 24 24

    {1,1}

    12 8

    13 + 8

    2

    24 16

    26 + 16

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {3,1}

    1

    4

    4

    {2,2}

    2 + 2 3

    4

    8 + 8 12

    {2,1}

    4 + 8 8

    3

    12 + 24 24

    {1,1}

    12 8

    13 + 8

    2

    24 16

    26 + 16

    1.

    Revan second polynomial is

    ( )

    Revan first polynomial is 1(, ) for = 1, 1

    {, }

    = |, |

    ( )

    {3,1}

    2

    3

    6

    {2,2}

    2 + 2

    4

    8 + 8

    {2,1}

    8 + 8 12

    2

    16 + 16 24

    {1,1}

    15 10

    13 + 8

    1

    15 10

    13 + 8

    {, }

    = |, |

    ( )

    {3,1}

    2

    3

    6

    {2,2}

    2 + 2

    4

    8 + 8

    {2,1}

    8 + 8 12

    2

    16 + 16 24

    {1,1}

    15 10

    13 + 8

    1

    15 10

    13 + 8

    () (+) = 25 + 34+ (6 4)3+ (12

    4 3 2 12 + 2)2.

    2(, ) = ()

    = (2 + 4)

    + 2 +

    (8 + 8 12)2 + (15 10 13 + 8)

    Adding entries of last column, we get 3()

    {, }

    = |, |

    |

    |

    | |

    {3,1}

    2

    2

    4

    {2,2}

    2 + 2

    0

    0

    {2,1}

    8 + 8

    12

    1

    8 + 8 12

    {1,1}

    15 10

    13 + 8

    0

    0

    Revan third polynomial is 3(, ) = () || = 22 + (8 + 8 12) + (15 10 11 + 10).

    Theorem 3.4. Let 3 [, ], , 1 be the silicon carbide graph, then

    1. 1() =

    24 4 + 6 + 2 for = 1, 1

    {

    24 + 4 + 6 8 for > 1, 1

    2. 2() =

    12 2 + 12 + 5 for = 1, 1

    {

    12 + 8 + 11 8 for > 1, 1

    3. () = {6 for = 1, 1

    3 4 + 8 4 for > 1, 1

    Proof. Adding entries of last column, we get

    1() for for = 1, 1.

    And Revan first polynomial for > 1, 1

    1(, ) = 25 + (2 + 2 2)4+ (4 + 8 8)3+ (12 8 13 + 8)2.

    Adding entries of last column, we get

    2() for = 1, 1

    {, }

    = |, |

    ( )

    {3,2}

    2

    6

    12

    {3,1}

    1

    3

    3

    {2,2}

    3 1

    4

    12 4

    {2,1}

    6 4

    2

    12 8

    {1,1}

    12 2

    12 + 2

    1

    12 2

    12 + 2

    Adding entries of last column, we get 2() for > 1,

    1.

    {, }

    = |, |

    ( )

    {3,2}

    2

    6

    12

    {3,1}

    1

    3

    3

    {2,2}

    2 + 2 3

    4

    8 + 8 12

    {2,1}

    4 + 8 8

    2

    8 + 8 16

    {1,1}

    12 8

    13 + 8

    1

    12 8

    13 + 8

    Revan second polynomial 2(, ) = 1, 1

    () () = 26 + (3 1)4 + 3 +

    +(6 4)2 + (12 2 12 + 2).

    And Revan second polynomial > 1, 1 is

    2(, ) = 26 + (2 + 2 3)4 + 3 +

    +(4 + 8 8)2 + (12 8 13 + 8).

    Adding entries of last column, we get

    3() for = 1, 1.

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {3,1}

    1

    2

    2

    {2,2}

    3 1

    0

    0

    {2,1}

    6 4

    1

    6 4

    {1,1}

    12 2

    12 + 2

    0

    0

    Adding entries of last column, we get 3() for > 1,

    1.

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {3,1}

    1

    2

    2

    {2,2}

    2 + 2 3

    0

    0

    {2,1}

    4 + 8 8

    1

    4 + 8 8

    {1,1}

    12 8

    13 + 8

    0

    0

    Revan third polynomial is 3(, ) for = 1, 1

    = () || = 2 + (6 2) + (12 2 9 + 1).

    And Revan third polynomial for > 1, 1

    3(, ) = 2 + (4 + 8 6) + (12 6 11 + 5).

    Theorem 3.5. Let 3 [, ], , 1 be the silicon carbide graph, then

    1. 1() = 24 + 4 + 4 8.

    2. 2() = 12 + 8 + 6 6.

    3. 3() = 4 + 8 6.

    Proof. Adding entries of last column, we get 1().

    {, }

    = |, |

    +

    ( + )

    {3,1}

    1

    4

    4

    {2,2}

    2 + 1

    4

    8 + 4

    {2,1}

    4 + 8 10

    3

    12 + 24 30

    {1,1}

    12 8

    10 + 7

    2

    24 8

    10 + 7

    Revan first polynomial is 1(, ) = () (+) = (2 + 1)4+ (4 + 8 10)3+ (12 8 10 + 7)2.

    {, }

    = |, |

    ( )

    {3,1}

    1

    3

    3

    {2,2}

    2 + 1

    4

    8 + 4

    {2,1}

    4 + 8 10

    2

    8 + 16 20

    {1,1}

    12 8

    10 + 7

    1

    12 8

    10 + 7

    {, }

    = |, |

    ( )

    {3,1}

    1

    3

    3

    {2,2}

    2 + 1

    4

    8 + 4

    {2,1}

    4 + 8 10

    2

    8 + 16 20

    {1,1}

    12 8

    10 + 7

    1

    12 8

    10 + 7

    Adding entries of last column, we get 2()

    Revan second polynomial is 2(, ) = () () = (2 + 1)4 + 3 + (4 + 8 10)2 + (12 8 10 + 7)

    Adding entries of last column, we get 3()

    {, }

    = |, |

    |

    |

    | |

    {3,1}

    2

    2

    4

    {2,2}

    2 + 1

    0

    0

    {2,1}

    4 + 8

    10

    1

    4 + 8 10

    {1,1}

    12 8

    10 + 7

    0

    0

    Revan third polynomial is 3(, ) = () || = 22 + 4 + 8 10 + (12 6 10 + 8).

    Theorem 3.6. Let 3 [, ], , 1 be the silicon carbide graph, then

    1. 1() = 24 + 6 + 4 6.

    2. 2() = 12 + 12 + 8 5.

    3. 3() = 6 + 4 4.

    Proof. Adding entries of last column, we get 1().

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {3,1}

    1

    4

    4

    {2,2}

    3 + 2 3

    4

    12 + 8 12

    {2,1}

    6 + 4 8

    3

    18 + 12 24

    {1,1}

    12 12

    8 + 8

    2

    24 24

    16 + 16

    Revan first polynomial is 1(, ) = () (+) = 25 + (3 + 2 3)4

    +(6 + 4 8)3+(12 12 8 + 8)2. Adding entries of last column, we get 2()

    {, }

    = |, |

    ( )

    {3,2}

    2

    6

    12

    {3,1}

    1

    3

    3

    {2,2}

    3 + 2 3

    4

    12 + 8 12

    {2,1}

    6 + 4 8

    2

    12 + 8 16

    {1,1}

    12 12

    8 + 8

    1

    12 12

    8 + 8

    Revan second polynomial is 2(, ) = () () = 26 + (3 + 2 3)4 + 3 + +(6 + 4 8)2 + (12 12 8 + 8)

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {3,1}

    1

    2

    2

    {2,2}

    3 + 2 3

    0

    0

    {2,1}

    6 + 4 8

    1

    6 + 4 8

    {1,1}

    12 12

    8 + 8

    0

    0

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {3,1}

    1

    2

    2

    {2,2}

    3 + 2 3

    0

    0

    {2,1}

    6 + 4 8

    1

    6 + 4 8

    {1,1}

    12 12

    8 + 8

    0

    0

    Adding entries of last column, we get 3()

    Revan third polynomial is 3(, ) = () || =

    2 + (6 + 4 6) + (12 9 6 + 5).

    Revan third polynomial is 3(, ) = () || = (3 2)2 + (2 + 4) + (15 9 6 + 3).

    Theorem 3.8. Let 4 [, ], , 1 be the silicon carbide graph, then

    1. 1() =

    { 30 + 8 4 + 4 for = 1, 1 30 + 8 + 4 4 for > 1, 1

    2. 2() =

    15 + 17 2 + 10 for = 1, 1

    {15 14 + 6 + 2 for > 1, 1

    3. () = { 6 4 for = 1, 1

    Theorem 3.7. Let 4

    carbide graph, then

    [, ], , 1 be the silicon

    3

    Proof.

    12 + 8 12 for > 1, 1

    1. 1() = 30 + 2 + 4 2.

    2. 2() = 15 + 7 + 8 1.

    3. 3() = 8 + 4 4.

    Proof. Adding entries of last column, we get 1().

    Proof. Adding entries of last column, we get

    1() for = 1, 1.

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {2,2}

    5 + 2

    4

    20 + 4

    {2,1}

    6 6

    3

    18 18

    {1,1}

    15 15

    2 + 2

    2

    30 30

    4 + 4

    Adding entries of last column, we get 1() for > 1,

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {3,1}

    3 2

    4

    12 8

    {2,2}

    + 2 2

    4

    4 + 8 8

    {2,1}

    2 + 4 2

    3

    6 + 12 6

    {1,1}

    15 10

    8 + 5

    2

    30 20

    16 + 10

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {3,1}

    3 2

    4

    12 8

    {2,2}

    + 2 2

    4

    4 + 8 8

    {2,1}

    2 + 4 2

    3

    6 + 12 6

    {1,1}

    15 10

    8 + 5

    2

    30 20

    16 + 10

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {2,2}

    2 + 2

    4

    8 + 8

    {2,1}

    12 + 8

    14

    3

    36 + 24 42

    {1,1}

    15 18

    10 + 10

    2

    30 36

    20 + 20

    {, }

    = |, |

    +

    ( + )

    {3,2}

    2

    5

    10

    {2,2}

    2 + 2

    4

    8 + 8

    {2,1}

    12 + 8

    14

    3

    36 + 24 42

    {1,1}

    15 18

    10 + 10

    2

    30 36

    20 + 20

    Revan first polynomial is (, ) = (+) =

    1.

    1

    25 + ( + 2 2)4

    ()

    {, }

    = |, |

    ( )

    {3,2}

    2

    6

    12

    {3,1}

    3 2

    3

    9 6

    {2,2}

    + 2 2

    4

    4 + 8 8

    {2,1}

    2 + 4 2

    2

    4 + 8 4

    {1,1}

    15 10

    8 + 5

    1

    15 10

    8 + 5

    {, }

    = |, |

    ( )

    {3,2}

    2

    6

    12

    {3,1}

    3 2

    3

    9 6

    {2,2}

    + 2 2

    4

    4 + 8 8

    {2,1}

    2 + 4 2

    2

    4 + 8 4

    {1,1}

    15 10

    8 + 5

    1

    15 10

    8 + 5

    +(2 + 4 2)3+(15 10 8 + 5)2. Adding entries of last column, we get 2()

    Revan second polynomial is

    2(, ) = () () = 26 + ( + 2 2)4 + (3 2)3 + +(2 + 4 2)2 + (15 10

    8 + 5)

    Adding entries of last column, we get 3()

    Revan first polynomial 1(, ) for = 1, 1 is

    () (+) = 25 + (5 + 2)4+ (6 6)3+ (15 15 2 + 2)2.

    Revan first polynomial for > 1, 1 is

    1(, ) = 25 + (2 + 2)4+ (12 + 8 14)3+ (15 18 10 + 10)2.

    Adding entries of last column, we get

    2() for = 1, 1

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {3,1}

    3 2

    2

    6 4

    {2,2}

    + 2 2

    0

    0

    {2,1}

    2 + 4 2

    1

    2 + 4 2

    {1,1}

    15 10

    8 + 5

    0

    0

    {, }

    = |, |

    ( )

    {3,2}

    2

    6

    12

    {2,2}

    5 + 2

    4

    20 + 8

    {2,1}

    6 6

    2

    12 12

    {1,1}

    15 15

    2 + 2

    1

    15 15

    2 + 2

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {3,1}

    3 2

    2

    6 4

    {2,2}

    + 2 2

    0

    0

    {2,1}

    2 + 4 2

    1

    2 + 4 2

    {1,1}

    15 10

    8 + 5

    0

    0

    {, }

    = |, |

    ( )

    {3,2}

    2

    6

    12

    {2,2}

    5 + 2

    4

    20 + 8

    {2,1}

    6 6

    2

    12 12

    {1,1}

    15 15

    2 + 2

    1

    15 15

    2 + 2

    Adding entries of last column, we get 2() for > 1,

    1.

    {, }

    = |, |

    ( )

    {3,2}

    2

    6

    12

    {2,2}

    2 + 2

    4

    8 + 8

    {2,1}

    12 + 8

    14

    2

    24 + 16 28

    {1,1}

    15 18

    10 + 10

    1

    15 18

    10 + 10

    Revan second polynomial 2(, ) = 1, 1

    ( ) 6 4

      • Comparison of Revan first index 1(), Revan

    ()

    = 2

    + (2 + 2)

    + (12 + 8

    second index () and Revan third index () of

    14)2 + (15 18 10 + 10). 2 3

    Revan second polynomial > 1, 1 is

    2(, ) = 26 + (5 + 2)4 + (6 6)2 + (15 15 2 + 2).

    Adding entries of last column, we get

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {2,2}

    5 + 2

    0

    0

    {2,1}

    6 6

    1

    6 6

    {1,1}

    15 15

    2 + 2

    0

    0

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {2,2}

    5 + 2

    0

    0

    {2,1}

    6 6

    1

    6 6

    {1,1}

    15 15

    2 + 2

    0

    0

    3() for = 1, 1.

    Adding entries of last column, we get 3() for > 1,

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {2,2}

    2 + 2

    0

    0

    {2,1}

    12 + 8

    14

    1

    12 + 8 14

    {1,1}

    15 18

    10 + 10

    0

    0

    {, }

    = |, |

    |

    |

    | |

    {3,2}

    2

    1

    2

    {2,2}

    2 + 2

    0

    0

    {2,1}

    12 + 8

    14

    1

    12 + 8 14

    {1,1}

    15 18

    10 + 10

    0

    0

    1.

    Revan third polynomial is 3(, ) for = 1, 1

    () || = (6 4) + (15 10 2 + 4).

    Revan third polynomial for > 1, 1 is

    3(, ) = (12 + 8 12) + (15 16 10 + 12).

    23 [, ]. Black color sheet represents Revan first index, yellow color sheet represents Revan second index and red color sheet represents Revan third index.

    • Comparison of Revan first index 1(), Revan second index 2() and Revan third index 3() of

      23 [, ]. Sky color sheet represents Revan first index, yellow color sheet represents Revan second index and purple color sheet represents Revan third index.

    • 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.

  4. ANALYSIS

  • Comparison of Revan first index 1(), Revan second index 2() and Revan third index

    3() for 23 [, ]. Blue color sheet represents Revan first index, purple color sheet represents Revan second index and green color sheet represents Revan third index.

  • 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.

  • Comparison of Revan first index 1(), Revan second index 2() and Revan third index 3() of 3

    [, ]. Green color sheet represents Revan first index, sky blue color sheet represents Revan second index and yellow color sheet represents Revan third index.

  • Comparison of Revan first index 1(), Revan second index 2() and Revan third index 3() of 4

    [, ]. Yellow color sheet represents Revan first index, blue color sheet represents Revan second index and sky blue color sheet represents Revan third index.

  • Comparison of Revan first index 1(), Revan second index 2() and Revan third index 3() of 4

[, ]. Red color sheet represents Revan first index, sky blue color sheet represents Revan second index and purple color sheet represents Revan third index.

IV. REFERENCES

  1. J. Devillers, A. T. Balaban, Topological indices and related descriptors in QSAR and QSPR. Gordon and Breach, Amsterdam, 1999.

  2. M. Karelson, Molecular descriptors in QSAR/QSPR. Wiley-Inter science, New York, 2000.

  3. M. Kamran, M. K. Siddiqui, M. Naeem, M. A. Iqbal, On topological properties of symmetric chemical structures. Symmetry 2018, 10(5), 173 April (2018).

  4. V. R. Kulli, 2012. College Graph heory, Vishwa International Publications,Gulbarga, India.

  5. V. R. Kulli, Revan indices of oxide and honeycomb networks, International Journal of Mathematics and its Applications, 5(4-E) (2017) 663-667.

  6. V. R. Kulli. Revan indices and their polynomials of certain Rhombus networks. ReserachGate article, International Journal of Current Research in Life Sciences. Vol.07, No. 05, pp. 2110-2116, May, 2018.

  7. V. R. Kulli. Hyper-Revan indices and their polynomials of silicate networks. International Journal of Current Research in Science and Technology.Volume 4, Issue3 (2018),17-21.

  8. V. R. Kulli, On K edge index of some nanostructures, Journal of Computer and Mathematical Sciences, 7(7), (2016) 373-378.

  9. V. R. Kulli, On the sum connectivity Gourava index, International Journal of Mathematical Archive, 8(7)(2017) 211-217.

  10. Y. C. Kwun, A. R. Virk, W. Nazeer, M. A. Rehman, S. M. Kang. On the multiplicative degree based topological indices of silicon- carbon 23 [, ] and 23 [, ]. Symmetry 2018, 10, 320.

  11. H. Wiener, Structural determination of paraffin boiling points. J. Amer. Chem. Soc. 69 (1947), 1720.

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