Hyper Zagreb Index and Reduced Zagreb Index of Four New Operations of Graphs

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Hyper Zagreb Index and Reduced Zagreb Index of Four New Operations of Graphs

J. Buragohain*1, A. Mahanta2 and A. Bharali3 Department of Mathematics,

Dibrugarh University, Assam-786004, INDIA.

email:j.bgohain75@yahoo.com1, am02dib@gmail.com2, a.bharali@dibru.ac.in3

(Received on: September 17, 2018) ABSTRACT

In 2016 a new graph operation has been proposed by Indulal and Balakrishnan called ‘Indu-Bala product’. Based on this product four sums were defined in the paper entitled “Four new operations of graphs and their Zagreb indices”. In this communication we propose explicit formulas for Reduced first Zagreb index and Hyper Zagreb index of these four new graph operations.

AMS classification (2010): 05C76, 05C07.

Keywords: Topological index, Zagreb indices, Reduced Zagreb indices, Hyper Zagreb index.

1. INTRODUCTION

Topological index of a chemical compound is calculated based on its molecular graph.

A molecular graph of a chemical compound is obtained by considering the atoms as vertices and the chemical bonds as its edges. These molecular structure descriptors are utilized in various purpose of chemical graph theory, molecular biology, mathematical chemistry etc and are generally graph invariants.

In this paper we have taken only connected, simple finite graphs. Let G be a simple finite graph. Then, by V(G) and E(G) we mean the set of vertices and set of edges of G respectively. The degree of a vertex u in G is the number of direct connections it has with the other vertices of G and this is denoted as 𝑑𝐺(𝑢) or simply by d(u) if there is no scope for misunderstanding. We adopt the notation uv ∈ E(G) to represent that there is an edge between u and v in G.

The First Zagreb index is defined as

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𝑀1(𝐺) = ∑𝑢∈𝑉(𝐺)𝑑𝐺2(𝑢)= ∑𝑢∈𝑉(𝐺)(𝑑𝐺(𝑢) + 𝑑𝐺(𝑣)).

The Second Zagreb index is defined as 𝑀2(𝐺) = ∑𝑢𝑣∈ 𝐸(𝐺)𝑑𝐺(𝑢)𝑑𝐺(𝑣).

Historically Zagreb index is considered to be the first degree based molecular structure descriptor which was invented in 19709. Several other topological indices based on vertex degree and distance is defined and is used in modeling information of chemical compounds.

Various works and research papers have surfaced since its recognition as topological indices.

Some of these works may be found in2,4,6,8,13.

Another version of Zagreb index called ‘forgotten topological index’ which was reintroduced by Furtula and Gutman in7 is given as

𝑀3(𝐺) = ∑𝑢∈ 𝑉(𝐺)𝑑𝐺3(𝑢)= ∑𝑢𝑣∈ 𝐸(𝐺)(𝑑𝐺2(𝑢) + 𝑑𝐺2(𝑣)).

The reduced first Zagreb index is defined as 𝑅𝑀1(𝐺) = ∑𝑢∈𝑉(𝐺)(𝑑𝐺(𝑢) − 1)2.

The Hyper Zagreb index12 of a simple graph G can be defined as 𝐻𝑀(𝐺) = ∑𝑢𝑣∈𝐸(𝑉)(𝑑𝐺(𝑢) + 𝑑𝐺(𝑣))2.

The F-sums of a graph G, namely subdivision graph S(G), vertex-semitotal graph Q(G), edge- semitotal graph R(G) and total graph T(G) can deal with the property that along with the atom- atom interactions the intermolecular forces also exists between the atoms and bonds of the molecule. Let G be a finite and simple connected graph. Then, these four related graphs of G can be given as

S(G) is the graph obtained by inserting an additional vertex in each edge of G.

Equivalently, each edge of G is replaced by a path of length 2.

R(G) is obtained from G by adding a new vertex corresponding to each edge of G, then joining each new vertex to the end vertices of the corresponding edge.

Q(G) is obtained from G by inserting a new vertex into each edge of G, then joining with edges those pairs of new vertices on adjacent edges of G.

T(G) has as its vertices the edges and vertices of G. Adjacency in T(G) is defined as adjacency or incidence for the corresponding elements of G.

Various research works on these four related graphs has been reported in connection with Wiener indices and Zagreb indices. For details see5, 3,10,11.

2. THE F-sums OF GRAPHS

Four new operations of graphs based on the Indu-Bala product of two connected graphs G1 and G2 are defined in1. As an example, 𝑃3𝑆𝑃4, 𝑃3𝑅𝑃4, 𝑃3𝑄𝑃4 and 𝑃3𝑆𝑃4 are shown in figure 1.

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S(G1) R(G1) Q(G1) T (G1)

Figure 1: The four new operations of graphs based on Indu-Bala product

a b c u v w x

G1 = P3 G2 = P4

e f e f e f e f

a b c a b c a b c a b c

e f e f

u x u x

u x u x

e f e f

G1S G2 G1RG2

v v

w w

v v

w w

e f e f

u x u x

u x u x

e f e f

G1QG2 G1T G2

v v

w w

v v

w w

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Lemma 2.1: [1] Let G1, G2 be two graphs with |V (Gi)| = ni and |E(Gi)| = mi, where i = 1, 2.

Then

(a) 𝑑𝐺1𝑆𝐺2(𝑢) = {

𝑑𝐺1(𝑢) + 𝑛2 𝑖𝑓 𝑢 ∈ 𝑉(𝐺1) 𝑛2+ 2 𝑖𝑓 𝑢 ∈ 𝑉(𝑆(𝐺1))\𝑉(𝐺1) 𝑑𝐺2(𝑢) + 𝑛1+ 𝑚1+ 1 𝑖𝑓 𝑢 ∈ 𝑉(𝐺2).

(b) 𝑑𝐺1𝑅𝐺2(𝑢) = {

2𝑑𝐺1(𝑢) + 𝑛2 𝑖𝑓 𝑢 ∈ 𝑉(𝐺1) 𝑛2+ 2 𝑖𝑓 𝑢 ∈ 𝑉(𝑆(𝐺1))\𝑉(𝐺1) 𝑑𝐺2(𝑢) + 𝑛1+ 𝑚1+ 1 𝑖𝑓 𝑢 ∈ 𝑉(𝐺2).

(c) 𝑑𝐺1𝑄𝐺2(𝑢) = {

𝑑𝐺1(𝑢) + 𝑛2 𝑖𝑓 𝑢 ∈ 𝑉(𝐺1) 𝑑𝐺1(𝑤) + 𝑑𝐺1(𝑤′) + 𝑛2 𝑖𝑓 𝑢 ∈ 𝑉(𝑆(𝐺1))\𝑉(𝐺1)

𝑑𝐺2(𝑢) + 𝑛1+ 𝑚1+ 1 𝑖𝑓 𝑢 ∈ 𝑉(𝐺2), Where in the second case u is inserted into the edge 𝑤𝑤 ∈ 𝐸(𝐺1).

(d) 𝑑𝐺1𝑇𝐺2(𝑢) = {

2𝑑𝐺1(𝑢) + 𝑛2 𝑖𝑓 𝑢 ∈ 𝑉(𝐺1) 𝑑𝐺1(𝑤) + 𝑑𝐺1(𝑤′) + 𝑛2 𝑖𝑓 𝑢 ∈ 𝑉(𝑆(𝐺1))\𝑉(𝐺1)

𝑑𝐺2(𝑢) + 𝑛1+ 𝑚1+ 1 𝑖𝑓 𝑢 ∈ 𝑉(𝐺2), Where in the second case u is inserted into the edge 𝑤𝑤′ ∈ 𝐸(𝐺1).

Clearly, the number of vertices in each graphs of these new operations is 2(𝑛1+ 𝑛2+ 𝑚1) . The cardinality of the edge sets of these new graph operations can be given as below.

Lemma 2.2: [1] Let G1 and G2 be two graphs with |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖, where i=1,2. Then,

|𝐸(𝐺1𝑆𝐺2)| = 2{𝑛2(𝑛1+ 𝑚1) + 2𝑚1+ 𝑚2} + 𝑛2.

|𝐸(𝐺1𝛻𝑅𝐺2)| = 2{𝑛2(𝑛1+ 𝑚1) + 3𝑚1+ 𝑚2} + 𝑛2.

|𝐸(𝐺1𝑆𝐺2)| = 2{𝑛2(𝑛1+ 𝑚1) + 2𝑚1+ 𝑚2+ ∑ (𝑑𝐺1(𝑣) 2 )

𝑣∈𝑉(𝐺1)

} + 𝑛2.

|𝐸(𝐺1𝑆𝐺2)| = 2{𝑛2(𝑛1+ 𝑚1) + 3𝑚1+ 𝑚2+ ∑ (𝑑𝐺1(𝑣) 2 )

𝑣∈𝑉(𝐺1)

} + 𝑛2.

The first Zagreb index and the second Zagreb index of the new F-sums of graphs are given below.

Theorem 2.3: [1] Let G1 and G2 be two graphs with |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 where i=1,2. Then,

𝑀1(𝐺1S𝐺2) = 2[𝑀1(𝐺1) + 𝑀1(𝐺2) + 𝑚1𝑛2(𝑚1+ 𝑛2+ 10) + 𝑛1𝑛2(𝑛1+ 𝑛2+ 2𝑚1+ 2 ) + 4𝑚2(𝑛1+ 𝑚1) + 4(𝑚1+ 𝑚2) + 𝑛2].

Theorem 2.4: [1] Let G1 and G2 be two graphs with |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 where i=1,2. Then,

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𝑀2(𝐺1𝑆𝐺2) = 𝑀1(𝐺2) + 𝑛2(𝑚1+ 𝑛1+ 1 )2+ 4𝑚2(𝑚1+ 𝑛1+ 1) + 2[(𝑛2+ 2 )𝑀1(𝐺1) + (𝑚1+ 𝑛1+ 1)𝑀1(𝐺2) + 𝑀2(𝐺2)

+ (𝑚1+ 𝑛1+ 1){𝑚2(𝑛1+ 𝑚1+ 1) + 4𝑛2𝑚1+ 𝑛22(𝑚1+ 𝑛1)} + 8𝑚1𝑚2 + 2𝑚2𝑛2(𝑚1+ 𝑛1) + 2𝑚1𝑛22+ 4𝑚1𝑛2].

Theorem 2.5: [1] Let G1 and G2 be two graphs with |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 where i=1,2. Then,

𝑀1(𝐺1𝑅𝐺2) = 2[4𝑀1(𝐺1) + 𝑀1(𝐺2) + 𝑚1𝑛2(𝑚1+ 𝑛2+ 14)

+ 𝑛1𝑛2(𝑛1+ 𝑛2+ 2𝑚1+ 2) + 4𝑚2(𝑛1+ 𝑚1) + 4(𝑚1+ 𝑚2) + 𝑛2].

Theorem 2.6: [1] Let G1 and G2 be two graphs with |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 where i=1,2. Then,

𝑀2(𝐺1𝑅𝐺2) = 𝑀1(𝐺2) + 4𝑚2(𝑚1+ 𝑛1+ 1) + 𝑛2(𝑚1+ 𝑛1+ 1)2

+ 2[4(𝑛2+ 1)𝑀1(𝐺1) + (𝑚1+ 𝑛1+ 1)𝑀1(𝐺2) + 4𝑀2(𝐺1) + 𝑀2(𝐺2) + 𝑚2(𝑚1+ 𝑛1+ 1)2+ {6𝑚1𝑛2+ 𝑛22(𝑚1+ 𝑛1)}(𝑚1+ 𝑛1+ 1) + 12𝑚1𝑚2+ 2𝑚2𝑛2(𝑚1+ 𝑛1) + 4𝑚1𝑛2(𝑛2+ 1)].

Theorem 2.7: [1] Let G1 and G2 be two graphs with |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 where i=1,2. Then,

𝑀1(𝐺1𝑄𝐺2) = 2[(2𝑛2+ 1)𝑀1(𝐺1) + 𝑀1(𝐺2) + 2𝑀2(𝐺1) + 𝑀3(𝐺1) + 𝑛2(𝑚1𝑛2+ 4𝑚1+ 𝑛1𝑛2)].

Theorem 2.8: [1] Let G1 and G2 be two graphs with |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 where i=1,2. Then,

𝑀2(𝐺1𝑄𝐺2) = 𝑀1(𝐺2) + 4𝑚2(𝑚1+ 𝑛1+ 1) + 𝑛2(𝑚1+ 𝑛1+ 1)2 + 2[(2𝑚2+ 𝑛2(𝑚1+ 𝑛1+ 1) +𝑛22

2 − 2 + 3𝑛2) 𝑀1(𝐺1) + (𝑚1+ 𝑛1+ 1)𝑀1(𝐺2) + 3𝑀2(𝐺1) + 𝑀2(𝐺2) +1

2(3𝑀3(𝐺1) + 𝑀4(𝐺1)) + 𝑚2(𝑚1+ 𝑛1+ 1)2+ 𝑚1𝑛2(2𝑚2+ 𝑛2(𝑚1+ 𝑛1+ 1))

+ 𝑚1𝑛2(𝑛2+ 2)(𝑚1+ 𝑛1+ 1) + 𝑛2(4𝑚1+ 𝑛1𝑛2)(𝑚1+ 𝑛1+ 1) + 2𝑚1𝑚2(𝑛2+ 6) + 2𝑛2(𝑚1𝑛2+ 𝑚2𝑛1) − 𝑚1𝑛22

+ ∑ 𝛾𝑢𝑤𝑑𝐺1(𝑢)𝑑𝐺1(𝑤) + ∑ 𝑑𝐺21(𝑣) ∑ 𝑑(𝑢)]

𝑢∈𝑉(𝐺1),𝑢𝑣∈𝐸(𝐺1) 𝑣∈𝑉(𝐺1)

𝑢,𝑤∈𝑉(𝐺1)

, Where 𝛾𝑢𝑤 is the number of common neighbors of 𝑢, 𝑤 ∈ 𝑉(𝐺1).

Theorem 2.9: [1] Let G1 and G2 be two graphs with |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 where i=1,2. Then,

𝑀1(𝐺1𝑇𝐺2) = 2[(2𝑛2+ 4)𝑀1(𝐺1) + 𝑀1(𝐺2) + 2𝑀2(𝐺1) + 𝑀3(𝐺1) + 𝑛22(𝑛1+ 𝑚1) + 8𝑚1𝑛2+ 𝑛2(𝑚1+ 𝑛1+ 1)2+ 4𝑚2(𝑚1+ 𝑛1+ 1)].

Theorem 2.10: [1] Let G1 and G2 be two graphs with |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 where i=1,2. Then,

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𝑀2(𝐺1𝑇𝐺2) = 𝑀1(𝐺2) + 4𝑚2(𝑚1+ 𝑛1+ 1) + 𝑛2(𝑚1+ 𝑛1+ 1)2 + 2[{2(𝑚2− 1) + 𝑛2(𝑚1+ 𝑛1+ 7 +𝑛2

2)} 𝑀1(𝐺1) + (𝑚1+ 𝑛1+ 1)𝑀1(𝐺2) + 7𝑀2(𝐺1) + 𝑀2(𝐺2) +5

2𝑀3(𝐺1) +1

4𝑀4(𝐺1) + 𝑚1𝑛22(𝑚1+ 𝑛1+ 1) + 2𝑚2(4𝑚1+ 𝑛1𝑛2+ 𝑚1𝑛2)

+ ∑ 𝛾𝑢𝑤𝑑𝐺1(𝑢)𝑑𝐺1(𝑤) + ∑ 𝑑𝐺21(𝑣) ∑ 𝑑(𝑢)]

𝑢∈𝑉(𝐺1),𝑢𝑣∈𝐸(𝐺1) 𝑣∈𝑉(𝐺1)

𝑢,𝑤∈𝑉(𝐺1)

, Where 𝛾𝑢𝑤 is the number of common neighbors of 𝑢, 𝑤 ∈ 𝑉(𝐺1).

The forgotten topological indices of the four new operations of graphs are given below.

Theorem 2.11: [1] Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges respectively in 𝐺𝑖; 𝑖 = 1,2. Then,

𝑀3(𝐺1𝑆𝐺2) = 2[3{𝑛2𝑀1(𝐺1) + (𝑚1+ 𝑛1+ 1 )𝑀1(𝐺2)} + 𝑀3(𝐺1) + 𝑀3(𝐺2) + (𝑚1+ 𝑛1+ 1)2{𝑛2(𝑚1+ 𝑛1+ 1) + 6𝑚2} + 𝑛22(𝑛1𝑛2+ 6𝑚1) + 𝑚1(𝑛2+ 2)3].

Theorem 2.12: [1] Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges respectively in 𝐺𝑖; 𝑖 = 1,2. Then,

𝑀3(𝐺1𝑅𝐺2) = 2[12𝑛2𝑀1(𝐺1) + 3(𝑚1+ 𝑛1+ 1)𝑀1(𝐺2) + 8𝑀3(𝐺1) + 𝑀3(𝐺2) + 𝑛2(𝑚1+ 𝑛1+ 1)3+ 6𝑚2(𝑚1+ 𝑛1+ 1)2+ 𝑚1(𝑛2+ 2)3 + 𝑛22(12𝑚1+ 𝑛1𝑛2)].

Theorem 2.13: [1] Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges respectively in 𝐺𝑖; 𝑖 = 1,2. Then,

𝑀3(𝐺1𝑄𝐺2) = 2[3𝑛2(𝑛2+ 1)𝑀1(𝐺1) + 3(𝑚1+ 𝑛1+ 1)𝑀1(𝐺2) + 6𝑛2𝑀2(𝐺1) + (3𝑛2+ 1)𝑀3(𝐺1) + 𝑀3(𝐺2) + 𝑀4(𝐺1) + 𝑛22(𝑛1𝑛2+ 6𝑚1) + 𝑛2(𝑚1+ 𝑛1+ 1)3+ 6𝑚2(𝑚1+ 𝑛1+ 1 )2+ 3 ∑𝑤𝑤∈𝐸(𝐺1)𝑑𝐺1(𝑤)𝑑𝐺1(𝑤){𝑑𝐺1(𝑤) + 𝑑𝐺1(𝑤)}].

Theorem 2.14: [1] Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges respectively in 𝐺𝑖; 𝑖 = 1,2. Then,

𝑀3(𝐺1𝑇𝐺2) = 2 [𝑛2(3𝑛2+ 8𝑚1)𝑀1(𝐺1) + 3(𝑚1+ 𝑛1+ 1)𝑀1(𝐺2) + 6𝑛2𝑀2(𝐺1) + (3𝑛2+ 8)𝑀3(𝐺1) + 𝑀3(𝐺2) + 𝑀4(𝐺1) + 𝑚1𝑛22(𝑛2+ 8)

+ 𝑛2(𝑚1+ 𝑛1+ 1)3+ 6𝑚2(𝑚1+ 𝑛1+ 1)2+ 𝑛1𝑛23 + 3 ∑ 𝑑𝐺1(𝑤)𝑑𝐺1(𝑤){𝑑𝐺1(𝑤) + 𝑑𝐺1(𝑤)}

𝑤𝑤∈𝐸(𝐺1)

].

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Now we propose the following theorems.

3. REDUCED FIRST ZAGREB INDEX OF THE NEW F-sums

Theorem 3.1: Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges in 𝐺𝑖 ; i=1,2. Then,

𝑅𝑀1(𝐺1𝑆𝐺2) = 2𝑀1(𝐺1) + 2𝑀1(𝐺2) + 4𝑚1𝑛2(𝑚1+ 𝑛2) + 2𝑛1𝑛2(𝑛1+ 𝑛2+ 2𝑚1) + 8𝑚2(𝑛1+ 𝑚1) + 32𝑚1𝑛2− 4𝑛1𝑛2− 6𝑚1+ 2𝑛1− 4𝑛2.

Theorem 3.2: Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges in 𝐺𝑖 ; i=1,2. Then,

𝑅𝑀1(𝐺1𝑅𝐺2) = 8𝑀1(𝐺1) + 2𝑀1(𝐺2) + 2𝑚1𝑛2(𝑚1+ 𝑛2) + 2𝑛1𝑛2(𝑛1+ 𝑛2+ 2𝑚1) + 8𝑚2(𝑛1+ 𝑚1) + 20𝑚1𝑛2− 4𝑛1𝑛2− 2𝑚1+ 4𝑚2+ 2𝑛1

− 8𝑛2(𝑚1+ 𝑛1).

Theorem 3.3: Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges in 𝐺𝑖 ; i=1,2. Then,

𝑅𝑀1(𝐺1𝑄𝐺2) = 2[(2𝑛2+ 1)𝑀1(𝐺1) + 𝑀1(𝐺2) + 2𝑀2(𝐺1) + 𝑀3(𝐺1)

+ 𝑛2(𝑚1𝑛2+ 4𝑚1+ 𝑛1𝑛2)] − 8𝑛2(𝑛1+ 𝑚1) − 14𝑚1− 8𝑚2+ 2𝑛1

− 2𝑛2− 8 ∑ (𝑑𝐺1(𝑣) 2 )

𝑣∈𝑉(𝐺1)

.

Theorem 3.4: Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges in 𝐺𝑖 ; i=1,2. Then,

𝑅𝑀1(𝐺1𝑇𝐺2) = 2 [(2𝑛2+ 4)𝑀1(𝐺1) + 𝑀1(𝐺2) + 2𝑀2(𝐺1) + 𝑀3(𝐺1) + 8𝑚1𝑛2− 𝑛2 + (𝑛1+ 𝑚1)(𝑛22− 4𝑛2+ 1) + 𝑛2(𝑚1+ 𝑛1+ 1)2+ 4𝑚2(𝑚1+ 𝑛1+ 1)

− 4 (3𝑚1+ 𝑚2+ ∑ (𝑑𝐺1(𝑣) 2 )

𝑣∈𝑉(𝐺1)

)]

Theorem 3.1 to 3.4 can be proved using relation 𝑅𝑀1(𝐺) = 𝑀1(𝐺) + |𝑉(𝐺)| − 4|𝐸(𝐺)| and the lemma 2.2.

4. HYPER ZAGREB INDEX OF THE NEW F-sums

Theorem 4.1: Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and edges in 𝐺𝑖

; i=1,2. Then,

𝐻𝑀(𝐺1𝑆𝐺2) = 2(3𝑛2+ 2)𝑀1(𝐺1) + 2[3(𝑚1+ 𝑛1+ 1) + 2]𝑀1(𝐺2) + 2𝑀3(𝐺1) + 2𝑀3(𝐺2) + 2(𝑚1+ 𝑛1+ 1)2[𝑛2(𝑚1+ 𝑛1+ 1) + 6𝑚2]

+ (𝑚1+ 𝑛1)(2𝑛23+ 16𝑚2) + 28𝑚1𝑛2(𝑛2+ 2) + 4𝑛2(𝑚1+ 𝑛2)2 + 4𝑛1𝑛2(𝑛2+ 2𝑚1+ 2) + 16(2𝑚1+ 𝑚2) + 𝑛2

(8)

Proof: 𝐻𝑀(𝐺1𝑆𝐺2) = ∑𝑢𝑣∈𝐸(𝐺1𝑆𝐺2)(𝑑(𝑢) + 𝑑(𝑣))2 = ∑ (𝑑2(𝑢) + 𝑑2(𝑣))

𝑢𝑣∈𝐸(𝐺1SG2)

+ 2 ∑ 𝑑(𝑢)𝑑(𝑣)

𝑢𝑣∈𝐸(𝐺1SG2)

= 𝑀3(𝐺1𝑆𝐺2) + 2𝑀2(𝐺1𝑆𝐺2)

Now putting the values of 𝑀2 𝑎𝑛𝑑 𝑀3 and by suitable manipulation we get the above result.

Hence proved. □

Theorem 4.2: Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges in 𝐺𝑖 ; i=1,2. Then,

𝐻𝑀(𝐺1𝑅𝐺2) = (40𝑛2+ 16)𝑀1(𝐺1) + 2[5(𝑚1+𝑛1+ 1) + 1]𝑀1(𝐺2) + 8𝑀2(𝐺1) + 2𝑀2(𝐺2) + 24𝑀3(𝐺1) + 2𝑀3(𝐺2) + 2𝑛2(𝑚1+𝑛1+ 1)3

+ 14𝑚2(𝑚1+𝑛1+ 1)2+ [12𝑚1𝑛2+ 2𝑛22 (𝑚1+ 𝑛1)](𝑚1+𝑛1+ 1) + 2𝑛23(𝑚1+ 𝑛1) + 24𝑚1(𝑚2+ 𝑛2) + 4𝑚2𝑛2(𝑚1+ 𝑛1)

+ 2𝑚1(22𝑛22+ 4𝑛2+ 27).

Theorem 4.3: Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges in 𝐺𝑖 ; i=1,2. Then,

𝐻𝑀(𝐺1𝑄𝐺2) = [4{4𝑛2+ 2𝑚2+ 𝑛2(𝑚1+ 𝑛1) − 1} + 7]𝑀1(𝐺1) + 8(𝑚1+𝑛1+ 1)𝑀1(𝐺2) + 6(𝑛2+ 1)𝑀2(𝐺1) + 2𝑀2(𝐺2) + (6𝑛2+ 5)𝑀3(𝐺1) + 2𝑀3(𝐺2)

+ 3𝑀4(𝐺1) + 2𝑛2(𝑚1+𝑛1+ 1)3+ 14𝑚2(𝑚1+𝑛1+ 1)2

+ 6𝑚1𝑚2(𝑛2+ 2) + 𝑛2[(3𝑚1𝑛2+ 𝑛1𝑛2) + 6𝑚1](𝑚1+𝑛1+ 1) + 𝑛2(𝑚1𝑛2+ 2𝑚2𝑛1)

+ 6 ∑ 𝑑𝐺1(𝑤)𝑑𝐺1(𝑤′){𝑑𝐺1(𝑤) + 𝑑𝐺1(𝑤′)}

𝑤𝑤′∈𝐸(𝐺1)

+ 2 ∑ 𝛾𝑢𝑤𝑑𝐺1(𝑢)

𝑢,𝑤∈𝑉(𝐺1)

𝑑𝐺1(𝑤) + 2 ∑ 𝑑2𝐺1(𝑣) ∑ 𝑑(𝑢).

𝑢∈𝑉(𝐺1),𝑢𝑣∈𝐸(𝐺1) 𝑣∈𝑉(𝐺1)

Theorem 4.4: Let |𝑉(𝐺𝑖)| = 𝑛𝑖 and |𝐸(𝐺𝑖)| = 𝑚𝑖 be the number of vertices and number of edges in 𝐺𝑖 ; i=1,2. Then,

𝐻𝑀(𝐺1𝑇𝐺2) = [20𝑛22+ 4𝑛2(13𝑚1+ 𝑛1) + 4(2𝑚2+ 7𝑛2− 2)]𝑀1(𝐺1) + 8(𝑚1+𝑛1+ 1)𝑀1(𝐺2) + 2(6𝑛2+ 7)𝑀2(𝐺1) + 2𝑀2(𝐺2) + (6𝑛2+ 21)𝑀3(𝐺1) + 2𝑀3(𝐺2) +5

2𝑀4(𝐺1) + 2𝑛2(𝑚1+ 𝑛1+ 1)3 + 12𝑚2(𝑚1+ 𝑛1+ 1)2+ 2𝑚1𝑛22(𝑚1+ 𝑛1+ 2)

+ 2𝑛22(𝑚1𝑛2+ 8𝑚1+ 𝑛1𝑛2) + 4𝑚2(4𝑚1+ 𝑛1𝑛2+ 𝑚1𝑛2) + 6 ∑ 𝑑𝐺1(𝑤)𝑑𝐺1(𝑤′){𝑑𝐺1(𝑤) + 𝑑𝐺1(𝑤′)}

𝑤𝑤∈𝐸(𝐺1)

+ 2 ∑ 𝛾𝑢𝑤𝑑𝐺1(𝑢)

𝑢,𝑤∈𝑉(𝐺1)

𝑑𝐺1(𝑤) + 2 ∑ 𝑑2𝐺1(𝑣) ∑ 𝑑𝐺1(𝑢)

𝑢∈𝑉(𝐺1),𝑢𝑣∈𝐸(𝐺1) 𝑣∈𝑉(𝐺1)

(9)

Where 𝑢𝑤 is the number of common neighbours of 𝑢, 𝑤 ∈ 𝑉(𝐺1).

The proofs of the theorems 4.2 to 4.4 are similar to theorem 4.1 with mutatis mutandis.

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10. Sarala, D., et al., ‘The Zagreb indices of graphs based on four new operations related to the lexicographic product’, Applied Mathematics and Computation 309, 156-169(2017).

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Chem. 4, 213-220 (2013).

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