MUHAMMAD Shakeel，ABAID Ur Rehman Virk

(1.Department of Mathematics，Minhaj University，Lahore，Pakistan 54000；2.Department of Mathematics，University of Management and Technology，Lahore，Pakistan 54000)

Abstract: In order to further study the derivative network of cellular network Type 1, we studied its reversed degree-based topological indices. Meanwhile, we computed the first and second reversed Zagreb indices, reversed modified second Zagreb index, reversed symmetric division index, reversed Randic and inverse Randic index, reverse inverse sum index and reversed augmented Zagreb index.

Key words: type 1；cellular network；degree；reversed index

1 Background Knowledge

In mathematical chemistry, mathematical tools such as polynomials and numbers predict properties of compounds without using quantum mechanics. These tools, in combination, capture information hidden in the symmetry of molecular graphs. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. They describe the structure of molecules numerically and are used in the development of qualitative structure activity relationships (QSARs). Most commonly known invariants of such kinds are degree-based topological indices. These are actually the numerical values that correlate the structure with various physical properties, chemical reactivity and biological activities. It is an established fact that many properties such as heat of formation, boiling point, strain energy, rigidity and fracture toughness of a molecule are strongly connected to its graphical structure and this fact plays a synergic role in chemical graph theory.

Throughout this report,Gis a connected graph,V(G) andE(G) are the vertex set and the edge set respectively anddvdenotes the degree of a vertexv.

The first topological index was introduced by Wiener[1]and it was named path number, which is now known as Wiener index. In chemical graph theory, this is the most studied molecular topological index due to its wide applications, see for details[2-3]. Randic index[4], denoted byR-1/2(G) and introduced by Milan Randic in 1975 is also one of the oldest topological index. The Randic index is defined as:

In 1998, Bollobas and Erdos[5]proposed the generalized Randic index and has been studied extensively by both chemist and mathematicians.

The general Randic index is defined as:

The reverse First Zagreb index is defined as:

The reverse Second Zagreb index is defined as:

The reverse Second Modified Zagreb index is defined as:

The reverse Symmetric division index is defined as:

The reverse Harmonic index is defined as:

The reverse Inverse Sum-Index is defined as:

The reverse Augmented Zagreb Index is defined as:

In this paper, we aim to study the above mentioned reversed degree-based indices for the Honey comb derived network of type 1.

2 Main Results

The Honey comb derived network of type 1 is shown in Figure 1 and is denoted byHCN1.

Figure 1 Honey Comb Derived Network of Type 1 for n=3

The edge set ofHCN1(n)has following five subclasses:

E1(HCN1(n)1)={uv∈E(HCN1(n)1):du=dv=3} ;

E2(HCN1(n)1)={uv∈E(HCN1(n)1):du=3,dv= 5};

E3(HCN1(n)1)={uv∈E(HCN1(n)1):du=3,dv=6};

E4(HCN1(n)1)={uv∈E(HCN1(n)1):du=5,dv=6};

E5(HCN1(n)1)={uv∈E(HCN1(n)1):du=dv=6}.

The maximum degree ofHCN1is 6. Hence the reverse edge partition is:

CE1(HCN1(n)1)={uv∈E(HCN1(n)1):cu=cv=4};

CE2(HCN1(n)1)={uv∈E(HCN1(n)1):cu=4,cv=2};

CE3(HCN1(n)1)={uv∈E(HCN1(n)1):cu=4,cv=1};

CE4(HCN1(n)1)={uv∈E(HCN1(n)1):cu= 2,cv=1};

CE5(HCN1(n)1)={uv∈E(HCN1(n)1):cu=cv=1}.

Now,

|E1(HCN1(n)1)|=|CE1(HCN1(n)1)|=6;

|E2(HCN1(n)1)|=|CE2(HCN1(n)1)|=12(n-1);

|E3(HCN1)|=|CE3(HCN1)|=6n；

|E4(HCN1)|=|CE4(HCN1)|=18(n-1)；

|E5(HCN1)|=|CE5(HCN1)|=27n2-57n+30.

For this edge partition, one can compute the following results．

Theorem1LetHCN1be the honey comb derived network. Then

CM1(HCN1)= 54n2+12n-18.

ProofFrom the reverse edge partition given above, we have

=|CE1(HCN1)|(4+4)+|CE2(HCN1)|(4+2)+|CE3(HCN1)|(4+1)

+|CE4(HCN1)|(2+1)+|CE5(HCN1)|(1+1)

=(6×8)+12(n-1)×6+(6n×5)+18(n-1)3+(27n2-57n+30)2

=54n2+12n-18.

Theorem2LetHCN1be the honey comb derived network. Then

CM2(HCN1)=27n2+99n-6.

ProofFrom the reverse edge partition given above, we have

=|CE1(HCN1)|(4×4)+|CE2(HCN1)|(4×2)+|CE3(HCN1)|(4×1)

+|CE4(HCN1)|(2×1)+|CE5(HCN1)|(1×1)

=(6×16)+12(n-1)×8+(6n×4)+18(n-1)2+(27n2-57n+30)1

=27n2+99n-6.

Theorem3LetHCN1be the honey comb derived network. Then

ProofFrom the reverse edge partition given above, we have

Theorem4LetHCN1be the honey comb derived network. Then

ProofFrom the reverse edge partition given above, we have

Theorem5LetHCN1be the honey comb derived network. Then

ProofFrom the reverse edge partition given above, we have

Theorem6LetHCN1be the honey comb derived network. Then

ProofFrom the reverse edge partition given above, we have

Theorem7LetHCN1be the honey comb derived network. Then

ProofFrom the reverse edge partition given above, we have

Theorem8LetHCN1be the Honey comb derived network. Then

CRRα(HCN1)= (27×1α)n2+ (12×8α+6×4α+18×2α-57×1α)n

+ (6×16α-12×8α+30×1α-18×2α).

ProofFrom the reverse edge partition given above, we have

=|CE1(HCN1)|(4×4)α+|CE2(HCN1)|(4×2)α+|CE3(HCN1)|(4×1)α

+|CE4(HCN1)|(2×1)α+|CE5(HCN1)|(1×1)α

=(6×16α)+12(n-1)×8α+(6n×4α)+18(n-1)2α+(27n2-57n+30)1α

=(27×1α)n2+(12×8α+6×4α+18×2α-57×1α)n

+(6×16α-12×8α+30×1α-18×2α).

3 Conclusions

Topological indices are helpful to understand the topology of networks. In this paper, we have studied nine reversed degree-based indices for the Honey comb derived network of type 1. To study the reversed indices for the honey comb derived networks of type 2, type 3 and type 4 is an interesting problem.