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

**J. Buragohain***^{1}**, A. Mahanta**^{2}** and A. Bharali**** ^{3 }**
Department of Mathematics,

Dibrugarh University, Assam-786004, INDIA.

email:j.bgohain75@yahoo.com^{1}, am02dib@gmail.com^{2}, a.bharali@dibru.ac.in^{3 }

(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 𝑑*_{𝐺}*(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

𝑀_{1}(𝐺) = ∑_{𝑢∈𝑉(𝐺)}𝑑_{𝐺}^{2}(𝑢)= ∑_{𝑢∈𝑉(𝐺)}(𝑑_{𝐺}(𝑢) + 𝑑_{𝐺}(𝑣)).* *

The Second Zagreb index is defined as
𝑀_{2}(𝐺) = ∑_{𝑢𝑣∈ 𝐸(𝐺)}𝑑_{𝐺}(𝑢)𝑑_{𝐺}(𝑣).

*Historically Zagreb index is considered to be the first degree based molecular structure *
descriptor which was invented in 1970^{9}. 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 in^{2,4,6,8,13}.

*Another version of Zagreb index called ‘forgotten topological index’ which was *
reintroduced by Furtula and Gutman in^{7} is given as

𝑀_{3}(𝐺) = ∑_{𝑢∈ 𝑉(𝐺)}𝑑_{𝐺}^{3}(𝑢)= ∑_{𝑢𝑣∈ 𝐸(𝐺)}(𝑑_{𝐺}^{2}(𝑢) + 𝑑_{𝐺}^{2}(𝑣)).

The reduced first Zagreb index is defined as
𝑅𝑀_{1}(𝐺) = ∑_{𝑢∈𝑉(𝐺)}(𝑑_{𝐺}(𝑢) − 1)^{2}.

The Hyper Zagreb index^{12} of a simple graph G can be defined as
𝐻𝑀(𝐺) = ∑_{𝑢𝑣∈𝐸(𝑉)}(𝑑_{𝐺}(𝑢) + 𝑑_{𝐺}(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 see^{5, 3,10,11}.

**2. THE F-sums OF GRAPHS **

*Four new operations of graphs based on the Indu-Bala product of two connected *
*graphs G**1** and G**2* are defined in^{1}. As an example, 𝑃_{3}∇_{𝑆}𝑃_{4}, 𝑃_{3}∇_{𝑅}𝑃_{4}, 𝑃_{3}∇_{𝑄}𝑃_{4} and 𝑃_{3}∇_{𝑆}𝑃_{4} are
shown in figure 1.

*S(G*1) *R(G*1) *Q(G*1) *T (G*1)

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

*a * *b * *c * *u * *v * *w * *x *

*G*1* = P*3 *G*2* = P*4

*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 *

*G*1∇*S **G*2 *G*1∇*R**G*2

*v *
*v *

*w *
*w *

*v *
*v *

*w *
*w *

*e * *f * *e * *f *

*u * *x * *u * *x *

*u * *x * *u * *x *

*e * *f * *e * *f *

*G*1∇*Q**G*2 *G*1∇*T **G*2

*v *
*v *

*w *
*w *

*v *
*v *

*w *
*w *

**Lemma 2.1: [1] Let G***1**, G**2** be two graphs with |V (G**i**)| = n**i** and |E(G**i**)| = m**i**, where i = 1, 2. *

*Then *

*(a) * 𝑑_{𝐺}_{1}_{∇}_{𝑆}_{𝐺}_{2}(𝑢) = {

𝑑_{𝐺}_{1}(𝑢) + 𝑛_{2} 𝑖𝑓 𝑢 ∈ 𝑉(𝐺_{1})
𝑛_{2}+ 2 𝑖𝑓 𝑢 ∈ 𝑉(𝑆(𝐺1))\𝑉(𝐺_{1})
𝑑_{𝐺}_{2}(𝑢) + 𝑛_{1}+ 𝑚_{1}+ 1 𝑖𝑓 𝑢 ∈ 𝑉(𝐺_{2}).

*(b) * 𝑑_{𝐺}_{1}_{∇}_{𝑅}_{𝐺}_{2}(𝑢) = {

2𝑑_{𝐺}_{1}(𝑢) + 𝑛_{2} 𝑖𝑓 𝑢 ∈ 𝑉(𝐺_{1})
𝑛_{2}+ 2 𝑖𝑓 𝑢 ∈ 𝑉(𝑆(𝐺1))\𝑉(𝐺_{1})
𝑑_{𝐺2}(𝑢) + 𝑛_{1}+ 𝑚_{1}+ 1 𝑖𝑓 𝑢 ∈ 𝑉(𝐺_{2}).

*(c) * 𝑑_{𝐺}_{1}_{∇}_{𝑄}_{𝐺}_{2}(𝑢) = {

𝑑_{𝐺1}(𝑢) + 𝑛_{2} 𝑖𝑓 𝑢 ∈ 𝑉(𝐺_{1})
𝑑_{𝐺}_{1}(𝑤) + 𝑑_{𝐺1}(𝑤′) + 𝑛_{2} 𝑖𝑓 𝑢 ∈ 𝑉(𝑆(𝐺_{1}))\𝑉(𝐺_{1})

𝑑_{𝐺2}(𝑢) + 𝑛_{1}+ 𝑚_{1}+ 1 𝑖𝑓 𝑢 ∈ 𝑉(𝐺_{2}),
*Where in the second case u is inserted into the edge 𝑤𝑤*^{′} ∈ 𝐸(𝐺_{1}*). *

*(d) * 𝑑_{𝐺}_{1}_{∇}_{𝑇}_{𝐺}_{2}(𝑢) = {

2𝑑_{𝐺1}(𝑢) + 𝑛_{2} 𝑖𝑓 𝑢 ∈ 𝑉(𝐺_{1})
𝑑_{𝐺}_{1}(𝑤) + 𝑑_{𝐺1}(𝑤′) + 𝑛_{2} 𝑖𝑓 𝑢 ∈ 𝑉(𝑆(𝐺_{1}))\𝑉(𝐺_{1})

𝑑_{𝐺2}(𝑢) + 𝑛_{1}+ 𝑚_{1}+ 1 𝑖𝑓 𝑢 ∈ 𝑉(𝐺_{2}),
*Where in the second case u is inserted into the edge 𝑤𝑤′ ∈ 𝐸(𝐺*_{1}*). *

Clearly, the number of vertices in each graphs of these new operations is 2(𝑛_{1}+ 𝑛_{2}+ 𝑚_{1}) .
The cardinality of the edge sets of these new graph operations can be given as below.

**Lemma 2.2: [1] Let G***1** and G**2** be two graphs with |𝑉(𝐺*_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}*, where *
*i=1,2. Then, *

|𝐸(𝐺_{1}∇_{𝑆}𝐺_{2})| = 2{𝑛_{2}(𝑛_{1}+ 𝑚_{1}) + 2𝑚_{1}+ 𝑚_{2}} + 𝑛_{2}*. *

|𝐸(𝐺_{1}𝛻_{𝑅}𝐺_{2})| = 2{𝑛_{2}(𝑛_{1}+ 𝑚_{1}) + 3𝑚_{1}+ 𝑚_{2}} + 𝑛_{2}*. *

|𝐸(𝐺_{1}∇_{𝑆}𝐺_{2})| = 2{𝑛_{2}(𝑛_{1}+ 𝑚_{1}) + 2𝑚_{1}+ 𝑚_{2}+ ∑ (𝑑_{𝐺}_{1}(𝑣)
2 )

𝑣∈𝑉(𝐺1)

} + 𝑛_{2}*. *

|𝐸(𝐺_{1}∇_{𝑆}𝐺_{2})| = 2{𝑛_{2}(𝑛_{1}+ 𝑚_{1}) + 3𝑚_{1}+ 𝑚_{2}+ ∑ (𝑑_{𝐺}_{1}(𝑣)
2 )

𝑣∈𝑉(𝐺1)

} + 𝑛_{2}*. *

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

**Theorem 2.3: [1] Let G***1** and G**2** be two graphs with |𝑉(𝐺*_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* where *
*i=1,2. Then, *

𝑀_{1}(𝐺_{1}∇_{S}𝐺_{2}) = 2[𝑀_{1}(𝐺_{1}) + 𝑀_{1}(𝐺_{2}) + 𝑚_{1}𝑛_{2}(𝑚_{1}+ 𝑛_{2}+ 10) + 𝑛_{1}𝑛_{2}(𝑛_{1}+ 𝑛_{2}+ 2𝑚_{1}+ 2 )
+ 4𝑚_{2}(𝑛_{1}+ 𝑚_{1}) + 4(𝑚_{1}+ 𝑚_{2}) + 𝑛_{2}*]. *

**Theorem 2.4: [1] Let G***1** and G**2** be two graphs with |𝑉(𝐺*_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* where *
*i=1,2. Then, *

𝑀_{2}(𝐺_{1}∇_{𝑆}𝐺_{2}) = 𝑀_{1}(𝐺_{2}) + 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1 )^{2}+ 4𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1)
+ 2[(𝑛_{2}+ 2 )𝑀_{1}(𝐺_{1}) + (𝑚_{1}+ 𝑛_{1}+ 1)𝑀_{1}(𝐺_{2}) + 𝑀_{2}(𝐺_{2})

+ (𝑚_{1}+ 𝑛_{1}+ 1){𝑚_{2}(𝑛_{1}+ 𝑚_{1}+ 1) + 4𝑛_{2}𝑚_{1}+ 𝑛_{2}^{2}(𝑚_{1}+ 𝑛_{1})} + 8𝑚_{1}𝑚_{2}
+ 2𝑚_{2}𝑛_{2}(𝑚_{1}+ 𝑛_{1}) + 2𝑚_{1}𝑛_{2}^{2}+ 4𝑚_{1}𝑛_{2}**]. **

**Theorem 2.5: [1] Let G***1** and G**2** be two graphs with |𝑉(𝐺*_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* where *
*i=1,2. Then, *

𝑀_{1}(𝐺_{1}∇_{𝑅}𝐺_{2}) = 2[4𝑀_{1}(𝐺_{1}) + 𝑀_{1}(𝐺_{2}) + 𝑚_{1}𝑛_{2}(𝑚_{1}+ 𝑛_{2}+ 14)

+ 𝑛_{1}𝑛_{2}(𝑛_{1}+ 𝑛_{2}+ 2𝑚_{1}+ 2) + 4𝑚_{2}(𝑛_{1}+ 𝑚_{1}) + 4(𝑚_{1}+ 𝑚_{2}) + 𝑛_{2}*]. *

**Theorem 2.6: [1] Let G***1** and G**2** be two graphs with |𝑉(𝐺*_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* where *
*i=1,2. Then, *

𝑀_{2}(𝐺_{1}∇_{𝑅}𝐺_{2}) = 𝑀_{1}(𝐺_{2}) + 4𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1) + 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{2}

+ 2[4(𝑛_{2}+ 1)𝑀_{1}(𝐺_{1}) + (𝑚_{1}+ 𝑛_{1}+ 1)𝑀_{1}(𝐺_{2}) + 4𝑀_{2}(𝐺_{1}) + 𝑀_{2}(𝐺_{2})
+ 𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{2}+ {6𝑚_{1}𝑛_{2}+ 𝑛_{2}^{2}(𝑚_{1}+ 𝑛_{1})}(𝑚_{1}+ 𝑛_{1}+ 1)
+ 12𝑚_{1}𝑚_{2}+ 2𝑚_{2}𝑛_{2}(𝑚_{1}+ 𝑛_{1}) + 4𝑚_{1}𝑛_{2}(𝑛_{2}+ 1)].

**Theorem 2.7: [1] Let G***1** and G**2** be two graphs with |𝑉(𝐺*_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* where *
*i=1,2. Then, *

𝑀_{1}(𝐺_{1}∇_{𝑄}𝐺_{2}) = 2[(2𝑛_{2}+ 1)𝑀_{1}(𝐺_{1}) + 𝑀_{1}(𝐺_{2}) + 2𝑀_{2}(𝐺_{1}) + 𝑀_{3}(𝐺_{1})
+ 𝑛_{2}(𝑚_{1}𝑛_{2}+ 4𝑚_{1}+ 𝑛_{1}𝑛_{2})].

**Theorem 2.8: [1] Let G***1** and G**2** be two graphs with |𝑉(𝐺*_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* where *
*i=1,2. Then, *

𝑀_{2}(𝐺_{1}∇_{𝑄}𝐺_{2}) = 𝑀_{1}(𝐺_{2}) + 4𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1) + 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{2}
+ 2[(2𝑚_{2}+ 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1) +𝑛_{2}^{2}

2 − 2 + 3𝑛_{2}) 𝑀_{1}(𝐺_{1})
+ (𝑚_{1}+ 𝑛_{1}+ 1)𝑀_{1}(𝐺_{2}) + 3𝑀_{2}(𝐺_{1}) + 𝑀_{2}(𝐺_{2}) +1

2(3𝑀_{3}(𝐺_{1}) + 𝑀_{4}(𝐺_{1}))
+ 𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{2}+ 𝑚_{1}𝑛_{2}(2𝑚_{2}+ 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1))

+ 𝑚_{1}𝑛_{2}(𝑛_{2}+ 2)(𝑚_{1}+ 𝑛_{1}+ 1) + 𝑛_{2}(4𝑚_{1}+ 𝑛_{1}𝑛_{2})(𝑚_{1}+ 𝑛_{1}+ 1)
+ 2𝑚_{1}𝑚_{2}(𝑛_{2}+ 6) + 2𝑛_{2}(𝑚_{1}𝑛_{2}+ 𝑚_{2}𝑛_{1}) − 𝑚_{1}𝑛_{2}^{2}

+ ∑ 𝛾_{𝑢𝑤}𝑑_{𝐺1}(𝑢)𝑑_{𝐺1}(𝑤) + ∑ 𝑑_{𝐺}^{2}_{1}(𝑣) ∑ 𝑑(𝑢)]

𝑢∈𝑉(𝐺1),𝑢𝑣∈𝐸(𝐺1) 𝑣∈𝑉(𝐺1)

𝑢,𝑤∈𝑉(𝐺1)

,
Where 𝛾_{𝑢𝑤} is the number of common neighbors of 𝑢, 𝑤 ∈ 𝑉(𝐺_{1}).

**Theorem 2.9: [1] Let G***1** and G**2** be two graphs with |𝑉(𝐺*_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* where *
*i=1,2. Then, *

𝑀_{1}(𝐺_{1}∇_{𝑇}𝐺_{2}) = 2[(2𝑛_{2}+ 4)𝑀_{1}(𝐺_{1}) + 𝑀_{1}(𝐺_{2}) + 2𝑀_{2}(𝐺_{1}) + 𝑀_{3}(𝐺_{1}) + 𝑛_{2}^{2}(𝑛_{1}+ 𝑚_{1})
+ 8𝑚_{1}𝑛_{2}+ 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{2}+ 4𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1)].

**Theorem 2.10: [1] Let G***1** and G**2** be two graphs with |𝑉(𝐺*_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* where *
*i=1,2. Then, *

𝑀_{2}(𝐺_{1}∇_{𝑇}𝐺_{2}) = 𝑀_{1}(𝐺_{2}) + 4𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1) + 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{2}
+ 2[{2(𝑚_{2}− 1) + 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 7 +𝑛_{2}

2)} 𝑀_{1}(𝐺_{1})
+ (𝑚_{1}+ 𝑛_{1}+ 1)𝑀_{1}(𝐺_{2}) + 7𝑀_{2}(𝐺_{1}) + 𝑀_{2}(𝐺_{2}) +5

2𝑀_{3}(𝐺_{1}) +1

4𝑀_{4}(𝐺_{1})
+ 𝑚_{1}𝑛_{2}^{2}(𝑚_{1}+ 𝑛_{1}+ 1) + 2𝑚_{2}(4𝑚_{1}+ 𝑛_{1}𝑛_{2}+ 𝑚_{1}𝑛_{2})

+ ∑ 𝛾_{𝑢𝑤}𝑑_{𝐺1}(𝑢)𝑑_{𝐺1}(𝑤) + ∑ 𝑑_{𝐺}^{2}_{1}(𝑣) ∑ 𝑑(𝑢)]

𝑢∈𝑉(𝐺_{1}),𝑢𝑣∈𝐸(𝐺1)
𝑣∈𝑉(𝐺_{1})

𝑢,𝑤∈𝑉(𝐺_{1})

,
Where 𝛾_{𝑢𝑤} is the number of common neighbors of 𝑢, 𝑤 ∈ 𝑉(𝐺_{1}).

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

**Theorem 2.11: [1] Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number *
*of edges respectively in 𝐺*_{𝑖}*; 𝑖 = 1,2. Then, *

𝑀_{3}(𝐺_{1}∇_{𝑆}𝐺_{2}) = 2[3{𝑛_{2}𝑀_{1}(𝐺_{1}) + (𝑚_{1}+ 𝑛_{1}+ 1 )𝑀_{1}(𝐺_{2})} + 𝑀_{3}(𝐺_{1}) + 𝑀_{3}(𝐺_{2})
+ (𝑚_{1}+ 𝑛_{1}+ 1)^{2}{𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1) + 6𝑚_{2}} + 𝑛_{2}^{2}(𝑛_{1}𝑛_{2}+ 6𝑚_{1})
+ 𝑚_{1}(𝑛_{2}+ 2)^{3}*]. *

**Theorem 2.12: [1] Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number *
*of edges respectively in 𝐺*_{𝑖}*; 𝑖 = 1,2. Then, *

𝑀_{3}(𝐺_{1}∇_{𝑅}𝐺_{2}) = 2[12𝑛_{2}𝑀_{1}(𝐺_{1}) + 3(𝑚_{1}+ 𝑛_{1}+ 1)𝑀_{1}(𝐺_{2}) + 8𝑀_{3}(𝐺_{1}) + 𝑀_{3}(𝐺_{2})
+ 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{3}+ 6𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{2}+ 𝑚_{1}(𝑛_{2}+ 2)^{3}
+ 𝑛_{2}^{2}(12𝑚_{1}+ 𝑛_{1}𝑛_{2}**)]. **

**Theorem 2.13: [1] Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number *
*of edges respectively in 𝐺*_{𝑖}*; 𝑖 = 1,2. Then, *

𝑀_{3}(𝐺_{1}∇_{𝑄}𝐺_{2}) = 2[3𝑛_{2}(𝑛_{2}+ 1)𝑀_{1}(𝐺_{1}) + 3(𝑚_{1}+ 𝑛_{1}+ 1)𝑀_{1}(𝐺_{2}) + 6𝑛_{2}𝑀_{2}(𝐺_{1}) + (3𝑛_{2}+
1)𝑀_{3}(𝐺_{1}) + 𝑀_{3}(𝐺_{2}) + 𝑀_{4}(𝐺_{1}) + 𝑛_{2}^{2}(𝑛_{1}𝑛_{2}+ 6𝑚_{1}) + 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{3}+ 6𝑚_{2}(𝑚_{1}+
𝑛_{1}+ 1 )^{2}+ 3 ∑_{𝑤𝑤}^{′}_{∈𝐸(𝐺}_{1}_{)}𝑑_{𝐺1}(𝑤)𝑑_{𝐺1}(𝑤^{′}){𝑑_{𝐺1}(𝑤) + 𝑑_{𝐺1}(𝑤^{′})}].

**Theorem 2.14: [1] Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number *
*of edges respectively in 𝐺*_{𝑖}*; 𝑖 = 1,2. Then, *

𝑀_{3}(𝐺_{1}∇_{𝑇}𝐺_{2}) = 2 [𝑛_{2}(3𝑛_{2}+ 8𝑚_{1})𝑀_{1}(𝐺_{1}) + 3(𝑚_{1}+ 𝑛_{1}+ 1)𝑀_{1}(𝐺_{2}) + 6𝑛_{2}𝑀_{2}(𝐺_{1})
+ (3𝑛_{2}+ 8)𝑀_{3}(𝐺_{1}) + 𝑀_{3}(𝐺_{2}) + 𝑀_{4}(𝐺_{1}) + 𝑚_{1}𝑛_{2}^{2}(𝑛_{2}+ 8)

+ 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{3}+ 6𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{2}+ 𝑛_{1}𝑛_{2}^{3}
+ 3 ∑ 𝑑_{𝐺1}(𝑤)𝑑_{𝐺1}(𝑤^{′}){𝑑_{𝐺1}(𝑤) + 𝑑_{𝐺1}(𝑤^{′})}

𝑤𝑤^{′}∈𝐸(𝐺_{1})

].

Now we propose the following theorems.

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

**Theorem 3.1: Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number of *
*edges in 𝐺*_{𝑖}* ; i=1,2. Then, *

𝑅𝑀_{1}(𝐺_{1}∇_{𝑆}𝐺_{2}) = 2𝑀_{1}(𝐺_{1}) + 2𝑀_{1}(𝐺_{2}) + 4𝑚_{1}𝑛_{2}(𝑚_{1}+ 𝑛_{2}) + 2𝑛_{1}𝑛_{2}(𝑛_{1}+ 𝑛_{2}+ 2𝑚_{1})
+ 8𝑚_{2}(𝑛_{1}+ 𝑚_{1}) + 32𝑚_{1}𝑛_{2}− 4𝑛_{1}𝑛_{2}− 6𝑚_{1}+ 2𝑛_{1}− 4𝑛_{2}.

**Theorem 3.2: Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number of *
*edges in 𝐺*_{𝑖}* ; i=1,2. Then, *

𝑅𝑀_{1}(𝐺_{1}∇_{𝑅}𝐺_{2}) = 8𝑀_{1}(𝐺_{1}) + 2𝑀_{1}(𝐺_{2}) + 2𝑚_{1}𝑛_{2}(𝑚_{1}+ 𝑛_{2}) + 2𝑛_{1}𝑛_{2}(𝑛_{1}+ 𝑛_{2}+ 2𝑚_{1})
+ 8𝑚_{2}(𝑛_{1}+ 𝑚_{1}) + 20𝑚_{1}𝑛_{2}− 4𝑛_{1}𝑛_{2}− 2𝑚_{1}+ 4𝑚_{2}+ 2𝑛_{1}

− 8𝑛_{2}(𝑚_{1}+ 𝑛_{1}).

**Theorem 3.3: Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number of *
*edges in 𝐺*_{𝑖}* ; i=1,2. Then, *

𝑅𝑀_{1}(𝐺_{1}∇_{𝑄}𝐺_{2}) = 2[(2𝑛_{2}+ 1)𝑀_{1}(𝐺_{1}) + 𝑀_{1}(𝐺_{2}) + 2𝑀_{2}(𝐺_{1}) + 𝑀_{3}(𝐺_{1})

+ 𝑛_{2}(𝑚_{1}𝑛_{2}+ 4𝑚_{1}+ 𝑛_{1}𝑛_{2})] − 8𝑛_{2}(𝑛_{1}+ 𝑚_{1}) − 14𝑚_{1}− 8𝑚_{2}+ 2𝑛_{1}

− 2𝑛_{2}− 8 ∑ (𝑑_{𝐺}_{1}(𝑣)
2 )

𝑣∈𝑉(𝐺1)

.

**Theorem 3.4: Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number of *
*edges in 𝐺*_{𝑖}* ; i=1,2. Then, *

𝑅𝑀_{1}(𝐺_{1}∇_{𝑇}𝐺_{2}) = 2 [(2𝑛_{2}+ 4)𝑀_{1}(𝐺_{1}) + 𝑀_{1}(𝐺_{2}) + 2𝑀_{2}(𝐺_{1}) + 𝑀_{3}(𝐺_{1}) + 8𝑚_{1}𝑛_{2}− 𝑛_{2}
+ (𝑛_{1}+ 𝑚_{1})(𝑛_{2}^{2}− 4𝑛_{2}+ 1) + 𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{2}+ 4𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1)

− 4 (3𝑚_{1}+ 𝑚_{2}+ ∑ (𝑑_{𝐺}_{1}(𝑣)
2 )

𝑣∈𝑉(𝐺_{1})

)]

*Theorem 3.1 to 3.4 can be proved using relation 𝑅𝑀*_{1}(𝐺) = 𝑀_{1}(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(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and edges in 𝐺*_{𝑖}

*; i=1,2. Then, *

𝐻𝑀(𝐺_{1}∇_{𝑆}𝐺_{2}) = 2(3𝑛_{2}+ 2)𝑀_{1}(𝐺_{1}) + 2[3(𝑚_{1}+ 𝑛_{1}+ 1) + 2]𝑀_{1}(𝐺_{2}) + 2𝑀_{3}(𝐺_{1})
+ 2𝑀_{3}(𝐺_{2}) + 2(𝑚_{1}+ 𝑛_{1}+ 1)^{2}[𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1) + 6𝑚_{2}]

+ (𝑚_{1}+ 𝑛_{1})(2𝑛_{2}^{3}+ 16𝑚_{2}) + 28𝑚_{1}𝑛_{2}(𝑛_{2}+ 2) + 4𝑛_{2}(𝑚_{1}+ 𝑛_{2})^{2}
+ 4𝑛_{1}𝑛_{2}(𝑛_{2}+ 2𝑚_{1}+ 2) + 16(2𝑚_{1}+ 𝑚_{2}) + 𝑛_{2}

**Proof: 𝐻𝑀(𝐺**_{1}∇_{𝑆}𝐺_{2}) = ∑_{𝑢𝑣∈𝐸(𝐺}_{1}_{∇}_{𝑆}_{𝐺}_{2}_{)}(𝑑(𝑢) + 𝑑(𝑣))^{2}
= ∑ (𝑑^{2}(𝑢) + 𝑑^{2}(𝑣))

𝑢𝑣∈𝐸(𝐺_{1}∇_{S}G_{2})

+ 2 ∑ 𝑑(𝑢)𝑑(𝑣)

𝑢𝑣∈𝐸(𝐺_{1}∇_{S}G_{2})

= 𝑀_{3}(𝐺_{1}∇_{𝑆}𝐺_{2}) + 2𝑀_{2}(𝐺_{1}∇_{𝑆}𝐺_{2})

Now putting the values of 𝑀_{2} 𝑎𝑛𝑑 𝑀_{3} and by suitable manipulation we get the above result.

Hence proved. □

**Theorem 4.2: Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number of *
*edges in 𝐺*_{𝑖}* ; i=1,2. Then, *

𝐻𝑀(𝐺_{1}∇_{𝑅}𝐺_{2}) = (40𝑛_{2}+ 16)𝑀_{1}(𝐺_{1}) + 2[5(𝑚_{1}+𝑛_{1}+ 1) + 1]𝑀_{1}(𝐺_{2}) + 8𝑀_{2}(𝐺_{1})
+ 2𝑀_{2}(𝐺_{2}) + 24𝑀_{3}(𝐺_{1}) + 2𝑀_{3}(𝐺_{2}) + 2𝑛_{2}(𝑚_{1}+𝑛_{1}+ 1)^{3}

+ 14𝑚_{2}(𝑚_{1}+𝑛_{1}+ 1)^{2}+ [12𝑚_{1}𝑛_{2}+ 2𝑛_{2}^{2} (𝑚_{1}+ 𝑛_{1})](𝑚_{1}+𝑛_{1}+ 1)
+ 2𝑛_{2}^{3}(𝑚_{1}+ 𝑛_{1}) + 24𝑚_{1}(𝑚_{2}+ 𝑛_{2}) + 4𝑚_{2}𝑛_{2}(𝑚_{1}+ 𝑛_{1})

+ 2𝑚_{1}(22𝑛_{2}^{2}+ 4𝑛_{2}+ 27).

**Theorem 4.3: Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number of *
*edges in 𝐺*_{𝑖}* ; i=1,2. Then, *

𝐻𝑀(𝐺1∇_{𝑄}𝐺_{2}) = [4{4𝑛_{2}+ 2𝑚_{2}+ 𝑛_{2}(𝑚_{1}+ 𝑛_{1}) − 1} + 7]𝑀_{1}(𝐺_{1}) + 8(𝑚_{1}+𝑛_{1}+ 1)𝑀_{1}(𝐺_{2})
+ 6(𝑛_{2}+ 1)𝑀_{2}(𝐺_{1}) + 2𝑀_{2}(𝐺_{2}) + (6𝑛_{2}+ 5)𝑀_{3}(𝐺_{1}) + 2𝑀_{3}(𝐺_{2})

+ 3𝑀_{4}(𝐺_{1}) + 2𝑛_{2}(𝑚_{1}+𝑛_{1}+ 1)^{3}+ 14𝑚_{2}(𝑚_{1}+𝑛_{1}+ 1)^{2}

+ 6𝑚_{1}𝑚_{2}(𝑛_{2}+ 2) + 𝑛_{2}[(3𝑚_{1}𝑛_{2}+ 𝑛_{1}𝑛_{2}) + 6𝑚_{1}](𝑚_{1}+𝑛_{1}+ 1)
+ 𝑛_{2}(𝑚_{1}𝑛_{2}+ 2𝑚_{2}𝑛_{1})

+ 6 ∑ 𝑑_{𝐺}_{1}(𝑤)𝑑_{𝐺}_{1}(𝑤′){𝑑_{𝐺}_{1}(𝑤) + 𝑑_{𝐺}_{1}(𝑤′)}

𝑤𝑤′∈𝐸(𝐺_{1})

+ 2 ∑ 𝛾_{𝑢𝑤}𝑑_{𝐺}_{1}(𝑢)

𝑢,𝑤∈𝑉(𝐺_{1})

𝑑_{𝐺}_{1}(𝑤) + 2 ∑ 𝑑^{2}_{𝐺}_{1}(𝑣) ∑ 𝑑(𝑢).

𝑢∈𝑉(𝐺_{1}),𝑢𝑣∈𝐸(𝐺_{1})
𝑣∈𝑉(𝐺_{1})

**Theorem 4.4: Let |𝑉(𝐺**_{𝑖})| = 𝑛_{𝑖}* and |𝐸(𝐺*_{𝑖})| = 𝑚_{𝑖}* be the number of vertices and number of *
*edges in 𝐺*_{𝑖}* ; i=1,2. Then, *

𝐻𝑀(𝐺_{1}∇_{𝑇}𝐺_{2}) = [20𝑛_{2}^{2}+ 4𝑛_{2}(13𝑚_{1}+ 𝑛_{1}) + 4(2𝑚_{2}+ 7𝑛_{2}− 2)]𝑀_{1}(𝐺_{1})
+ 8(𝑚_{1}+𝑛_{1}+ 1)𝑀_{1}(𝐺_{2}) + 2(6𝑛_{2}+ 7)𝑀_{2}(𝐺_{1}) + 2𝑀_{2}(𝐺_{2})
+ (6𝑛_{2}+ 21)𝑀_{3}(𝐺_{1}) + 2𝑀_{3}(𝐺_{2}) +5

2𝑀_{4}(𝐺_{1}) + 2𝑛_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{3}
+ 12𝑚_{2}(𝑚_{1}+ 𝑛_{1}+ 1)^{2}+ 2𝑚_{1}𝑛_{2}^{2}(𝑚_{1}+ 𝑛_{1}+ 2)

+ 2𝑛_{2}^{2}(𝑚_{1}𝑛_{2}+ 8𝑚_{1}+ 𝑛_{1}𝑛_{2}) + 4𝑚_{2}(4𝑚_{1}+ 𝑛_{1}𝑛_{2}+ 𝑚_{1}𝑛_{2})
+ 6 ∑ 𝑑_{𝐺}_{1}(𝑤)𝑑_{𝐺}_{1}(𝑤′){𝑑_{𝐺}_{1}(𝑤) + 𝑑_{𝐺}_{1}(𝑤′)}

𝑤𝑤^{′}∈𝐸(𝐺1)

+ 2 ∑ 𝛾_{𝑢𝑤}𝑑_{𝐺}_{1}(𝑢)

𝑢,𝑤∈𝑉(𝐺_{1})

𝑑_{𝐺}_{1}(𝑤) + 2 ∑ 𝑑^{2}_{𝐺}_{1}(𝑣) ∑ 𝑑_{𝐺}_{1}(𝑢)

𝑢∈𝑉(𝐺_{1}),𝑢𝑣∈𝐸(𝐺_{1})
𝑣∈𝑉(𝐺_{1})

Where 𝑢𝑤 is the number of common neighbours of 𝑢, 𝑤 ∈ 𝑉(𝐺_{1}).

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

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