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 β€˜Indu-Bala product’. Based on this product four sums were defined in the paper entitled β€œFour 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(𝑒) 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) = βˆ‘π‘’βˆˆπ‘‰(𝐺)𝑑𝐺2(𝑒)= βˆ‘π‘’βˆˆπ‘‰(𝐺)(𝑑𝐺(𝑒) + 𝑑𝐺(𝑣)).

The Second Zagreb index is defined as 𝑀2(𝐺) = βˆ‘π‘’π‘£βˆˆ 𝐸(𝐺)𝑑𝐺(𝑒)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 β€˜forgotten topological index’ which was reintroduced by Furtula and Gutman in7 is given as

𝑀3(𝐺) = βˆ‘π‘’βˆˆ 𝑉(𝐺)𝑑𝐺3(𝑒)= βˆ‘π‘’π‘£βˆˆ 𝐸(𝐺)(𝑑𝐺2(𝑒) + 𝑑𝐺2(𝑣)).

The reduced first Zagreb index is defined as 𝑅𝑀1(𝐺) = βˆ‘π‘’βˆˆπ‘‰(𝐺)(𝑑𝐺(𝑒) βˆ’ 1)2.

The Hyper Zagreb index12 of a simple graph G can be defined as 𝐻𝑀(𝐺) = βˆ‘π‘’π‘£βˆˆπΈ(𝑉)(𝑑𝐺(𝑒) + 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, 𝑃3βˆ‡π‘†π‘ƒ4, 𝑃3βˆ‡π‘…π‘ƒ4, 𝑃3βˆ‡π‘„π‘ƒ4 and 𝑃3βˆ‡π‘†π‘ƒ4 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

G1βˆ‡S G2 G1βˆ‡RG2

v v

w w

v v

w w

e f e f

u x u x

u x u x

e f e f

G1βˆ‡QG2 G1βˆ‡T 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) 𝑑𝐺1βˆ‡π‘†πΊ2(𝑒) = {

𝑑𝐺1(𝑒) + 𝑛2 𝑖𝑓 𝑒 ∈ 𝑉(𝐺1) 𝑛2+ 2 𝑖𝑓 𝑒 ∈ 𝑉(𝑆(𝐺1))\𝑉(𝐺1) 𝑑𝐺2(𝑒) + 𝑛1+ π‘š1+ 1 𝑖𝑓 𝑒 ∈ 𝑉(𝐺2).

(b) 𝑑𝐺1βˆ‡π‘…πΊ2(𝑒) = {

2𝑑𝐺1(𝑒) + 𝑛2 𝑖𝑓 𝑒 ∈ 𝑉(𝐺1) 𝑛2+ 2 𝑖𝑓 𝑒 ∈ 𝑉(𝑆(𝐺1))\𝑉(𝐺1) 𝑑𝐺2(𝑒) + 𝑛1+ π‘š1+ 1 𝑖𝑓 𝑒 ∈ 𝑉(𝐺2).

(c) 𝑑𝐺1βˆ‡π‘„πΊ2(𝑒) = {

𝑑𝐺1(𝑒) + 𝑛2 𝑖𝑓 𝑒 ∈ 𝑉(𝐺1) 𝑑𝐺1(𝑀) + 𝑑𝐺1(𝑀′) + 𝑛2 𝑖𝑓 𝑒 ∈ 𝑉(𝑆(𝐺1))\𝑉(𝐺1)

𝑑𝐺2(𝑒) + 𝑛1+ π‘š1+ 1 𝑖𝑓 𝑒 ∈ 𝑉(𝐺2), Where in the second case u is inserted into the edge 𝑀𝑀′ ∈ 𝐸(𝐺1).

(d) 𝑑𝐺1βˆ‡π‘‡πΊ2(𝑒) = {

2𝑑𝐺1(𝑒) + 𝑛2 𝑖𝑓 𝑒 ∈ 𝑉(𝐺1) 𝑑𝐺1(𝑀) + 𝑑𝐺1(𝑀′) + 𝑛2 𝑖𝑓 𝑒 ∈ 𝑉(𝑆(𝐺1))\𝑉(𝐺1)

𝑑𝐺2(𝑒) + 𝑛1+ π‘š1+ 1 𝑖𝑓 𝑒 ∈ 𝑉(𝐺2), Where in the second case u is inserted into the edge 𝑀𝑀′ ∈ 𝐸(𝐺1).

Clearly, the number of vertices in each graphs of these new operations is 2(𝑛1+ 𝑛2+ π‘š1) . 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)| = π‘šπ‘–, where i=1,2. Then,

|𝐸(𝐺1βˆ‡π‘†πΊ2)| = 2{𝑛2(𝑛1+ π‘š1) + 2π‘š1+ π‘š2} + 𝑛2.

|𝐸(𝐺1𝛻𝑅𝐺2)| = 2{𝑛2(𝑛1+ π‘š1) + 3π‘š1+ π‘š2} + 𝑛2.

|𝐸(𝐺1βˆ‡π‘†πΊ2)| = 2{𝑛2(𝑛1+ π‘š1) + 2π‘š1+ π‘š2+ βˆ‘ (𝑑𝐺1(𝑣) 2 )

π‘£βˆˆπ‘‰(𝐺1)

} + 𝑛2.

|𝐸(𝐺1βˆ‡π‘†πΊ2)| = 2{𝑛2(𝑛1+ π‘š1) + 3π‘š1+ π‘š2+ βˆ‘ (𝑑𝐺1(𝑣) 2 )

π‘£βˆˆπ‘‰(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)| = π‘šπ‘– where i=1,2. Then,

𝑀1(𝐺1βˆ‡S𝐺2) = 2[𝑀1(𝐺1) + 𝑀1(𝐺2) + π‘š1𝑛2(π‘š1+ 𝑛2+ 10) + 𝑛1𝑛2(𝑛1+ 𝑛2+ 2π‘š1+ 2 ) + 4π‘š2(𝑛1+ π‘š1) + 4(π‘š1+ π‘š2) + 𝑛2].

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

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𝑀2(𝐺1βˆ‡π‘†πΊ2) = 𝑀1(𝐺2) + 𝑛2(π‘š1+ 𝑛1+ 1 )2+ 4π‘š2(π‘š1+ 𝑛1+ 1) + 2[(𝑛2+ 2 )𝑀1(𝐺1) + (π‘š1+ 𝑛1+ 1)𝑀1(𝐺2) + 𝑀2(𝐺2)

+ (π‘š1+ 𝑛1+ 1){π‘š2(𝑛1+ π‘š1+ 1) + 4𝑛2π‘š1+ 𝑛22(π‘š1+ 𝑛1)} + 8π‘š1π‘š2 + 2π‘š2𝑛2(π‘š1+ 𝑛1) + 2π‘š1𝑛22+ 4π‘š1𝑛2].

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

𝑀1(𝐺1βˆ‡π‘…πΊ2) = 2[4𝑀1(𝐺1) + 𝑀1(𝐺2) + π‘š1𝑛2(π‘š1+ 𝑛2+ 14)

+ 𝑛1𝑛2(𝑛1+ 𝑛2+ 2π‘š1+ 2) + 4π‘š2(𝑛1+ π‘š1) + 4(π‘š1+ π‘š2) + 𝑛2].

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

𝑀2(𝐺1βˆ‡π‘…πΊ2) = 𝑀1(𝐺2) + 4π‘š2(π‘š1+ 𝑛1+ 1) + 𝑛2(π‘š1+ 𝑛1+ 1)2

+ 2[4(𝑛2+ 1)𝑀1(𝐺1) + (π‘š1+ 𝑛1+ 1)𝑀1(𝐺2) + 4𝑀2(𝐺1) + 𝑀2(𝐺2) + π‘š2(π‘š1+ 𝑛1+ 1)2+ {6π‘š1𝑛2+ 𝑛22(π‘š1+ 𝑛1)}(π‘š1+ 𝑛1+ 1) + 12π‘š1π‘š2+ 2π‘š2𝑛2(π‘š1+ 𝑛1) + 4π‘š1𝑛2(𝑛2+ 1)].

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

𝑀1(𝐺1βˆ‡π‘„πΊ2) = 2[(2𝑛2+ 1)𝑀1(𝐺1) + 𝑀1(𝐺2) + 2𝑀2(𝐺1) + 𝑀3(𝐺1) + 𝑛2(π‘š1𝑛2+ 4π‘š1+ 𝑛1𝑛2)].

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

𝑀2(𝐺1βˆ‡π‘„πΊ2) = 𝑀1(𝐺2) + 4π‘š2(π‘š1+ 𝑛1+ 1) + 𝑛2(π‘š1+ 𝑛1+ 1)2 + 2[(2π‘š2+ 𝑛2(π‘š1+ 𝑛1+ 1) +𝑛22

2 βˆ’ 2 + 3𝑛2) 𝑀1(𝐺1) + (π‘š1+ 𝑛1+ 1)𝑀1(𝐺2) + 3𝑀2(𝐺1) + 𝑀2(𝐺2) +1

2(3𝑀3(𝐺1) + 𝑀4(𝐺1)) + π‘š2(π‘š1+ 𝑛1+ 1)2+ π‘š1𝑛2(2π‘š2+ 𝑛2(π‘š1+ 𝑛1+ 1))

+ π‘š1𝑛2(𝑛2+ 2)(π‘š1+ 𝑛1+ 1) + 𝑛2(4π‘š1+ 𝑛1𝑛2)(π‘š1+ 𝑛1+ 1) + 2π‘š1π‘š2(𝑛2+ 6) + 2𝑛2(π‘š1𝑛2+ π‘š2𝑛1) βˆ’ π‘š1𝑛22

+ βˆ‘ 𝛾𝑒𝑀𝑑𝐺1(𝑒)𝑑𝐺1(𝑀) + βˆ‘ 𝑑𝐺21(𝑣) βˆ‘ 𝑑(𝑒)]

π‘’βˆˆπ‘‰(𝐺1),π‘’π‘£βˆˆπΈ(𝐺1) π‘£βˆˆπ‘‰(𝐺1)

𝑒,π‘€βˆˆπ‘‰(𝐺1)

, Where 𝛾𝑒𝑀 is the number of common neighbors of 𝑒, 𝑀 ∈ 𝑉(𝐺1).

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

𝑀1(𝐺1βˆ‡π‘‡πΊ2) = 2[(2𝑛2+ 4)𝑀1(𝐺1) + 𝑀1(𝐺2) + 2𝑀2(𝐺1) + 𝑀3(𝐺1) + 𝑛22(𝑛1+ π‘š1) + 8π‘š1𝑛2+ 𝑛2(π‘š1+ 𝑛1+ 1)2+ 4π‘š2(π‘š1+ 𝑛1+ 1)].

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

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𝑀2(𝐺1βˆ‡π‘‡πΊ2) = 𝑀1(𝐺2) + 4π‘š2(π‘š1+ 𝑛1+ 1) + 𝑛2(π‘š1+ 𝑛1+ 1)2 + 2[{2(π‘š2βˆ’ 1) + 𝑛2(π‘š1+ 𝑛1+ 7 +𝑛2

2)} 𝑀1(𝐺1) + (π‘š1+ 𝑛1+ 1)𝑀1(𝐺2) + 7𝑀2(𝐺1) + 𝑀2(𝐺2) +5

2𝑀3(𝐺1) +1

4𝑀4(𝐺1) + π‘š1𝑛22(π‘š1+ 𝑛1+ 1) + 2π‘š2(4π‘š1+ 𝑛1𝑛2+ π‘š1𝑛2)

+ βˆ‘ 𝛾𝑒𝑀𝑑𝐺1(𝑒)𝑑𝐺1(𝑀) + βˆ‘ 𝑑𝐺21(𝑣) βˆ‘ 𝑑(𝑒)]

π‘’βˆˆπ‘‰(𝐺1),π‘’π‘£βˆˆπΈ(𝐺1) π‘£βˆˆπ‘‰(𝐺1)

𝑒,π‘€βˆˆπ‘‰(𝐺1)

, Where 𝛾𝑒𝑀 is the number of common neighbors of 𝑒, 𝑀 ∈ 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)| = π‘šπ‘– be the number of vertices and number of edges respectively in 𝐺𝑖; 𝑖 = 1,2. Then,

𝑀3(𝐺1βˆ‡π‘†πΊ2) = 2[3{𝑛2𝑀1(𝐺1) + (π‘š1+ 𝑛1+ 1 )𝑀1(𝐺2)} + 𝑀3(𝐺1) + 𝑀3(𝐺2) + (π‘š1+ 𝑛1+ 1)2{𝑛2(π‘š1+ 𝑛1+ 1) + 6π‘š2} + 𝑛22(𝑛1𝑛2+ 6π‘š1) + π‘š1(𝑛2+ 2)3].

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

𝑀3(𝐺1βˆ‡π‘…πΊ2) = 2[12𝑛2𝑀1(𝐺1) + 3(π‘š1+ 𝑛1+ 1)𝑀1(𝐺2) + 8𝑀3(𝐺1) + 𝑀3(𝐺2) + 𝑛2(π‘š1+ 𝑛1+ 1)3+ 6π‘š2(π‘š1+ 𝑛1+ 1)2+ π‘š1(𝑛2+ 2)3 + 𝑛22(12π‘š1+ 𝑛1𝑛2)].

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

𝑀3(𝐺1βˆ‡π‘„πΊ2) = 2[3𝑛2(𝑛2+ 1)𝑀1(𝐺1) + 3(π‘š1+ 𝑛1+ 1)𝑀1(𝐺2) + 6𝑛2𝑀2(𝐺1) + (3𝑛2+ 1)𝑀3(𝐺1) + 𝑀3(𝐺2) + 𝑀4(𝐺1) + 𝑛22(𝑛1𝑛2+ 6π‘š1) + 𝑛2(π‘š1+ 𝑛1+ 1)3+ 6π‘š2(π‘š1+ 𝑛1+ 1 )2+ 3 βˆ‘π‘€π‘€β€²βˆˆπΈ(𝐺1)𝑑𝐺1(𝑀)𝑑𝐺1(𝑀′){𝑑𝐺1(𝑀) + 𝑑𝐺1(𝑀′)}].

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

𝑀3(𝐺1βˆ‡π‘‡πΊ2) = 2 [𝑛2(3𝑛2+ 8π‘š1)𝑀1(𝐺1) + 3(π‘š1+ 𝑛1+ 1)𝑀1(𝐺2) + 6𝑛2𝑀2(𝐺1) + (3𝑛2+ 8)𝑀3(𝐺1) + 𝑀3(𝐺2) + 𝑀4(𝐺1) + π‘š1𝑛22(𝑛2+ 8)

+ 𝑛2(π‘š1+ 𝑛1+ 1)3+ 6π‘š2(π‘š1+ 𝑛1+ 1)2+ 𝑛1𝑛23 + 3 βˆ‘ 𝑑𝐺1(𝑀)𝑑𝐺1(𝑀′){𝑑𝐺1(𝑀) + 𝑑𝐺1(𝑀′)}

π‘€π‘€β€²βˆˆπΈ(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)| = π‘šπ‘– be the number of vertices and number of edges in 𝐺𝑖 ; i=1,2. Then,

𝑅𝑀1(𝐺1βˆ‡π‘†πΊ2) = 2𝑀1(𝐺1) + 2𝑀1(𝐺2) + 4π‘š1𝑛2(π‘š1+ 𝑛2) + 2𝑛1𝑛2(𝑛1+ 𝑛2+ 2π‘š1) + 8π‘š2(𝑛1+ π‘š1) + 32π‘š1𝑛2βˆ’ 4𝑛1𝑛2βˆ’ 6π‘š1+ 2𝑛1βˆ’ 4𝑛2.

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

𝑅𝑀1(𝐺1βˆ‡π‘…πΊ2) = 8𝑀1(𝐺1) + 2𝑀1(𝐺2) + 2π‘š1𝑛2(π‘š1+ 𝑛2) + 2𝑛1𝑛2(𝑛1+ 𝑛2+ 2π‘š1) + 8π‘š2(𝑛1+ π‘š1) + 20π‘š1𝑛2βˆ’ 4𝑛1𝑛2βˆ’ 2π‘š1+ 4π‘š2+ 2𝑛1

βˆ’ 8𝑛2(π‘š1+ 𝑛1).

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

𝑅𝑀1(𝐺1βˆ‡π‘„πΊ2) = 2[(2𝑛2+ 1)𝑀1(𝐺1) + 𝑀1(𝐺2) + 2𝑀2(𝐺1) + 𝑀3(𝐺1)

+ 𝑛2(π‘š1𝑛2+ 4π‘š1+ 𝑛1𝑛2)] βˆ’ 8𝑛2(𝑛1+ π‘š1) βˆ’ 14π‘š1βˆ’ 8π‘š2+ 2𝑛1

βˆ’ 2𝑛2βˆ’ 8 βˆ‘ (𝑑𝐺1(𝑣) 2 )

π‘£βˆˆπ‘‰(𝐺1)

.

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

𝑅𝑀1(𝐺1βˆ‡π‘‡πΊ2) = 2 [(2𝑛2+ 4)𝑀1(𝐺1) + 𝑀1(𝐺2) + 2𝑀2(𝐺1) + 𝑀3(𝐺1) + 8π‘š1𝑛2βˆ’ 𝑛2 + (𝑛1+ π‘š1)(𝑛22βˆ’ 4𝑛2+ 1) + 𝑛2(π‘š1+ 𝑛1+ 1)2+ 4π‘š2(π‘š1+ 𝑛1+ 1)

βˆ’ 4 (3π‘š1+ π‘š2+ βˆ‘ (𝑑𝐺1(𝑣) 2 )

π‘£βˆˆπ‘‰(𝐺1)

)]

Theorem 3.1 to 3.4 can be proved using relation 𝑅𝑀1(𝐺) = 𝑀1(𝐺) + |𝑉(𝐺)| βˆ’ 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)| = π‘šπ‘– be the number of vertices and edges in 𝐺𝑖

; i=1,2. Then,

𝐻𝑀(𝐺1βˆ‡π‘†πΊ2) = 2(3𝑛2+ 2)𝑀1(𝐺1) + 2[3(π‘š1+ 𝑛1+ 1) + 2]𝑀1(𝐺2) + 2𝑀3(𝐺1) + 2𝑀3(𝐺2) + 2(π‘š1+ 𝑛1+ 1)2[𝑛2(π‘š1+ 𝑛1+ 1) + 6π‘š2]

+ (π‘š1+ 𝑛1)(2𝑛23+ 16π‘š2) + 28π‘š1𝑛2(𝑛2+ 2) + 4𝑛2(π‘š1+ 𝑛2)2 + 4𝑛1𝑛2(𝑛2+ 2π‘š1+ 2) + 16(2π‘š1+ π‘š2) + 𝑛2

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Proof: 𝐻𝑀(𝐺1βˆ‡π‘†πΊ2) = βˆ‘π‘’π‘£βˆˆπΈ(𝐺1βˆ‡π‘†πΊ2)(𝑑(𝑒) + 𝑑(𝑣))2 = βˆ‘ (𝑑2(𝑒) + 𝑑2(𝑣))

π‘’π‘£βˆˆπΈ(𝐺1βˆ‡SG2)

+ 2 βˆ‘ 𝑑(𝑒)𝑑(𝑣)

π‘’π‘£βˆˆπΈ(𝐺1βˆ‡SG2)

= 𝑀3(𝐺1βˆ‡π‘†πΊ2) + 2𝑀2(𝐺1βˆ‡π‘†πΊ2)

Now putting the values of 𝑀2 π‘Žπ‘›π‘‘ 𝑀3 and by suitable manipulation we get the above result.

Hence proved. β–‘

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

𝐻𝑀(𝐺1βˆ‡π‘…πΊ2) = (40𝑛2+ 16)𝑀1(𝐺1) + 2[5(π‘š1+𝑛1+ 1) + 1]𝑀1(𝐺2) + 8𝑀2(𝐺1) + 2𝑀2(𝐺2) + 24𝑀3(𝐺1) + 2𝑀3(𝐺2) + 2𝑛2(π‘š1+𝑛1+ 1)3

+ 14π‘š2(π‘š1+𝑛1+ 1)2+ [12π‘š1𝑛2+ 2𝑛22 (π‘š1+ 𝑛1)](π‘š1+𝑛1+ 1) + 2𝑛23(π‘š1+ 𝑛1) + 24π‘š1(π‘š2+ 𝑛2) + 4π‘š2𝑛2(π‘š1+ 𝑛1)

+ 2π‘š1(22𝑛22+ 4𝑛2+ 27).

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

𝐻𝑀(𝐺1βˆ‡π‘„πΊ2) = [4{4𝑛2+ 2π‘š2+ 𝑛2(π‘š1+ 𝑛1) βˆ’ 1} + 7]𝑀1(𝐺1) + 8(π‘š1+𝑛1+ 1)𝑀1(𝐺2) + 6(𝑛2+ 1)𝑀2(𝐺1) + 2𝑀2(𝐺2) + (6𝑛2+ 5)𝑀3(𝐺1) + 2𝑀3(𝐺2)

+ 3𝑀4(𝐺1) + 2𝑛2(π‘š1+𝑛1+ 1)3+ 14π‘š2(π‘š1+𝑛1+ 1)2

+ 6π‘š1π‘š2(𝑛2+ 2) + 𝑛2[(3π‘š1𝑛2+ 𝑛1𝑛2) + 6π‘š1](π‘š1+𝑛1+ 1) + 𝑛2(π‘š1𝑛2+ 2π‘š2𝑛1)

+ 6 βˆ‘ 𝑑𝐺1(𝑀)𝑑𝐺1(𝑀′){𝑑𝐺1(𝑀) + 𝑑𝐺1(𝑀′)}

π‘€π‘€β€²βˆˆπΈ(𝐺1)

+ 2 βˆ‘ 𝛾𝑒𝑀𝑑𝐺1(𝑒)

𝑒,π‘€βˆˆπ‘‰(𝐺1)

𝑑𝐺1(𝑀) + 2 βˆ‘ 𝑑2𝐺1(𝑣) βˆ‘ 𝑑(𝑒).

π‘’βˆˆπ‘‰(𝐺1),π‘’π‘£βˆˆπΈ(𝐺1) π‘£βˆˆπ‘‰(𝐺1)

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

𝐻𝑀(𝐺1βˆ‡π‘‡πΊ2) = [20𝑛22+ 4𝑛2(13π‘š1+ 𝑛1) + 4(2π‘š2+ 7𝑛2βˆ’ 2)]𝑀1(𝐺1) + 8(π‘š1+𝑛1+ 1)𝑀1(𝐺2) + 2(6𝑛2+ 7)𝑀2(𝐺1) + 2𝑀2(𝐺2) + (6𝑛2+ 21)𝑀3(𝐺1) + 2𝑀3(𝐺2) +5

2𝑀4(𝐺1) + 2𝑛2(π‘š1+ 𝑛1+ 1)3 + 12π‘š2(π‘š1+ 𝑛1+ 1)2+ 2π‘š1𝑛22(π‘š1+ 𝑛1+ 2)

+ 2𝑛22(π‘š1𝑛2+ 8π‘š1+ 𝑛1𝑛2) + 4π‘š2(4π‘š1+ 𝑛1𝑛2+ π‘š1𝑛2) + 6 βˆ‘ 𝑑𝐺1(𝑀)𝑑𝐺1(𝑀′){𝑑𝐺1(𝑀) + 𝑑𝐺1(𝑀′)}

π‘€π‘€β€²βˆˆπΈ(𝐺1)

+ 2 βˆ‘ 𝛾𝑒𝑀𝑑𝐺1(𝑒)

𝑒,π‘€βˆˆπ‘‰(𝐺1)

𝑑𝐺1(𝑀) + 2 βˆ‘ 𝑑2𝐺1(𝑣) βˆ‘ 𝑑𝐺1(𝑒)

π‘’βˆˆπ‘‰(𝐺1),π‘’π‘£βˆˆπΈ(𝐺1) π‘£βˆˆπ‘‰(𝐺1)

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Where 𝑒𝑀 is the number of common neighbours of 𝑒, 𝑀 ∈ 𝑉(𝐺1).

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

REFERENCES

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7. Furtula, B., Gutman, I.,β€˜A forgotten topological index’, J. Math. Chem. 53, 1187-1190 (2015).

8. Furtula, B., Gutman, I., SΒ¨uleyman, E., ’On difference of Zagreb indices’, Discrete Applied Mathematices 178, 83-88 (2014).

9. Gutman, I., TrinajstiΒ΄c, N., β€˜Graph theory and molecular orbitals. Total πœ‹-electron energy of alternant hydrocarbons’, Chem. Phys. Lett. 17(4), 535-538 (1972).

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

11. Sarkar, P., De, N., Pal A., β€˜The Zagreb indices of graphs based on new operations related to the join of graphs’, Journal of The International Mathematical Virtual Institute, 7, 181- 209 (2017).

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Chem. 4, 213-220 (2013).

13. Zhou, B., Gutman, I., β€˜Further properties of Zagreb indices’, MATCH Commun. Math.

Comput. Chem. 54, 233-239 (2005).

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