Edelweiss Applied Science and Technology 
ISSN: 2576-8484 
Vol. 8, No. 6, 5789-5799 
2024 
Publisher: Learning Gate 
DOI: 10.55214/25768484.v8i6.3258 
© 2024 by the authors; licensee Learning Gate 

© 2024 by the authors; licensee Learning Gate 
* Correspondence:  fuzzysansrmvcas@gmail.com 

 
 
 
 
s 

Total degree of maximal product of two constant intuitionistic fuzzy graphs 

 
P. Indumathi1*, S. Santhosh Kumar2, R., Buvaneswari3, S. K. Mala4 

1Department of Science and Humanities, Karpagam College of Engineering, Coimbatore, fuzzysansrmvcas@gmail.com (P.I.). 
2Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore. 
3Department of Mathematics, Sri Krishna Arts and Science College, Coimbatore. 
4Department of Mathematics, PSGR Krishnammal College for Women, Coimbatore.  

 

 
Abstract: This paper explains about the Total degree of Maximal product of Two constant IF graphs. 
Fuzzy graphs are derived from crisp graphs. Various properties of IF graphs are extended from Fuzzy 
graphs. Maximal product of Fuzzy graph structures with applications have been discussed in different 
papers and extended to IF graphs. Constant IF graphs are special type of IF graphs which have same 
degree for all its vertices. IF Graphs have many applications including the investigation of images by 
image segmentation, Brain mapping etc, Maximal product of fuzzy graphs is applied in various fields 
like effective logistic Management, Agricultural product mapping etc. Here in this paper, the Total 
degree of the vertices in maximal product of Constant IF graphs are studied in detail with definition and 
various examples and theorems. 
Keywords: Fuzzy graph, IFgraphs, Investigation of images, Maximal product, Theorems. 

 
1. Overview  

Zadeh introduced the principle of fuzzy sets in the year 1965. After his, introduction, many 
generalisations of this fundamental concepts have been developed.by many Mathematicians in different 
field related to the Fuzzy. In 1999 Atanassov introduced the notion of an IF set.  In 2002 Atanassov 
along with Shannon further explained about generalisation of an IF Fuzzy graphs. Subsequently, in 
2006 & 2009, various properties of IF graph have been discussed by Parvathy & Karunambigai on 
identical fields. In 2012 Karunambigai, Parvathy & Bhuvaneswari explained in details, the structure of 
an IF graph on its arcs and the properties of complete IF graph and Constant IF graph.  In 2019, Sitara, 
Muhannad Akram and Muhammad Yusaf introduced maximal products of fuzzy graph structure and 
analysed the properties with examples.  In 2021, Mala, Shanmugapriya & Santhosh Kumar explained the 

degrees of vertices and edges for the maximal product of an IF Ideals of M𝛤groups in Near rings. 
 
2. Preliminaries 

In this part of the article, few descriptions of an IF graphs, Constant IF graphs and maximal 
product of an IF graphs are presented. 
 
Definition: 2.1 [1]  

Let The Set E Be Fixed. An If Set A In E Takes The Form A =  {𝛼,ΜA(Α), ΓA(Α)/Αϵe} Where The 

Degrees Of Membership And Non – Membership Of The Element Α∈ E Are Indicated By The 

Functions ΜA: E → [0,1] And               ΓA: E → [0,1] Where 0 ≤ ΜA(Α) + ΓA(Α) ≤ 1 
 
Definition: 2.2 [2]  

The set G = {< α, β >, μG(α, β), γG(α, β)/< α, β > ϵ VxV} is said to be an IF graph if these 

functions μG: VxV → [0,1] and γG: VxV → [0,1] define the corresponding degrees of membership and 

https://orcid.org/0009-0007-5195-8744
https://orcid.org/0000-0003-2276-3706
https://orcid.org/0009-0009-0822-6761


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non – membership of the elements (α, β) ∈ VxV over IFSs for all (α, β) ∈ VxV such that 0 ≤ μG(α, β) +
γG(α, β) ≤ 1. 
Using one of this Cartesian Product the following definition is obtained. 
 
Definition: 2.3 [4]  

Maximum IF graph takes the form G = (V, E) where V = {v1, v2, … . vn} such that μ: V → [0,1] and            

γ: V → [0,1] represent the degrees of membership and non – membership of the element v1 ∈ V 

respectively with                      0 ≤ μ(vi) + γ(vi) ≤ 1 for i =  1,2, … … . n. 
If E <  VxV where μ: VxV → [0,1] and γ: VxV → [0,1]  
such that μ(vi, vj) ≤ max[μ(vi), μ(vj)] 

        and γ(vi, vj) ≤ min[γ(vi), γ(vj)] 

with 0 ≤ μ(vi, vj) + γ(vi, vj) ≤ 1 for every vi, vj ∈ E for i, j = 1,2, . . . . . n. 

 
Definition: 2.4 [7]  

Let G(μ, γ) be an IF graph, the μ − degree of a vertex viis 

𝑑𝜇(vi)  = ∑ μ(vi,

(vi,vj)∈E

vj)  

and the γ − degree of the vertex vi is 

𝑑𝛾(vi)  = ∑ γ(vi,

(vi,vj)∈E

vj) 

the degree of the vertex is 

d(vi)  =  { ∑ μ(vi,

(vi,vj)∈E

vj), ∑ γ(vi,

(vi,vj)∈E

vj)} 

and μ(vi, vj) = γ(vi, vj) = 0 if (vi, vj) ∉ E. 
 
Definition: 2.5. [5]  

Let G(μ, γ) be an IF graph with dμ(vi) = ki and dγ(vj) = Kj for all vi, vj ∈ V of the IF graph 

G(V, E), the graph is denoted as (ki, kj) - IFG (or) Constant IFG of degree (ki, kj) 
Definition: 2.6. [7]  

Let G(V, E) be an IF graph with G(μ, γ), the total degree of a vertex v ∈ V is defined as  

td(u) =  ∑ dμ(vi,

(vi,vj)∈E

vj) + μ(vi), ∑ dγ(vi,

(vi,vj)∈E

vj) + γ(vi) 

 

If the total degree of each vertex in G is the same and it is denoted as (r1, r2) , then G is called an IF 

graph of total degree (r1, r2) or a (r1, r2) totally Constant IF graph. 
 
Definition: 2.6. [6]  

Let GI1(V𝐼2
E𝐼2

μ𝐼2
γ𝐼2

) and G𝐼2
(V𝐼2

E𝐼2
μ𝐼2

γ𝐼2
) be two graphs of IFIMFGNR I1 and I2 is near ring N∗ 

then  GI1 ∗ GI2 = (VIEIμIγI) is called maximal product structure of IFMFGNR . The set of vertices 

VI =  V𝐼1
 x V𝐼2

 exist with  

 μI(ri, si) = μ𝐼1
(ri) ⋁ μ𝐼2

(si) and γI(ri, si)  =  γ𝐼1
(ri) ⋀ γ𝐼2

(si) for all (ri, si) ∈  VI 
The set of edges 

EI  =  {(r1, s1)(r2, s2)} / r1  =  r2 and s1s2 ∈  E𝐼2
 (or)  

s1  =  s2 and r1 r2  ∈ E𝐼1
 exist with μI(r1, s1) (r2, s2) 

 = {μ𝐼1
(r1) ⋁ μ𝐼2

(s1s2) where r1  =  r2 & s1s2  ∈ E𝐼2
  



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   {μ𝐼2
(s2) ⋁ μ𝐼1

(r1r2) where s1  =  s2 & r1r2  ∈ E𝐼2
 

and  γI(r1, s1) (r2, s2) 

 = {γ𝐼1
(r1) ⋀ γ𝐼2

(s1s2) where r1  =  r2 & s1s2  ∈ E𝐼2
  

   {γ𝐼2
(s2) ⋁ γ𝐼1

(r1r2) where s1  =  s2 & r1r2  ∈ E𝐼1
 

 
Definition: 2.7. [6]  

The vertex degree of maximal product of IFMFGNR  

 GI1(V𝐼1
E𝐼1

μ𝐼1
γ𝐼1

) and GI2(V𝐼2
E𝐼2

μ𝐼2
γ𝐼2

)  is given by: 

D(G1 ∗ G2) μI(rj, sj)  =  ∑ μ𝐼1
(rjrk)  ⋁ μ𝐼2

(sj)  + ∑ μ𝐼2
(sjsi)  ⋁ μ𝐼2

(rj) and 

D(G1 ∗ G2) γI(rj, sj)  =  ∑ γ𝐼1
(rjrk)  ⋀ γ𝐼2

(sj)  +  ∑ γ𝐼2
(sjsi)  ⋀ γ𝐼1

(rj)    
 
3. Maximal Product of Two Constant If Graph 
Definition 3.1. 

Let 𝐺1(𝑉′, 𝐸′, 𝜇′, 𝛾′) and 𝐺2(𝑉′′, 𝐸′′, 𝜇′′, 𝛾′′) be two constant IF graphs then 𝐺1 ∗ 𝐺2 =
(𝑉∗, 𝐸∗, 𝜇∗, 𝛾∗) is the maximal product structure of 𝐺1 and 𝐺2 with 𝑉′𝑋 𝑉′′ = 𝑉∗, the set of vertices 
exist with  

 
Example 3.2. 

Consider 𝐺1(𝑉′, 𝐸′, 𝜇′, 𝛾′) and 𝐺2(𝑉′′, 𝐸′′, 𝜇′′, 𝛾′′) be to constant IF graphs and 𝐺(𝑉, 𝐸, 𝜇, 𝛾) is their 
maximal product of Constant IF graphs 



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Example 3.3. 

Consider the following two Constant IF Graphs G1 and G2.  



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Figure 1. 

 

 
Figure 2. 

 
Let us find the maximal product of these two graphs as G. 



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Consider the above graph, the degree of the maximal product of the CONSTANT IF graphs are 

calculated using the above definition as given below: 

D(G1 ∗ G2)μ(v1xv1′) = μ1(v1v2)⋁μ2(v1′) + μ1(v1v4)⋁μ2(v1′)  + μ2(v1′v2′) ⋁μ1(v1) +

                                               μ2(v1′v3′)⋁μ1(v1′) = 1.7  

D(G1 ∗  G2)γ(v1xv1′) = γ1(v1v2)⋀γ2(v1′) + γ1(v1v4)⋀γ2(v1′) + γ2(v1′v2′) ⋀γ1(v1) +

                                              γ2(v1′v3′)⋀γ1(v1′) = 0.7  
(i. e)   D(G1 ∗  G2)(v1xv1′) = (1.7,0.7) 

D(G1 ∗  G2)μ(v1xv2′) =  μ1(v1v2)⋁μ2(v2′) + μ1(v1v4)⋁μ2(v2′)  + μ2(v2′v1′) ⋁μ1(v1) +

                                               μ2(v2′v3′)⋁μ1(v1′) = 1.8                                   

D(G1 ∗  G2)γ(v1xv2′) =  γ1(v1v2)⋀γ2(v2′) + γ1(v1v4)⋀γ2(v2′)  + γ2(v2′v1′) ⋀γ1(v1) +

                                              γ2(v2′v3′)⋀γ1(v1′) = 0.7   
D(G1 ∗  G2)(v1xv2′) = (1.8,0.7) 



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D(G1 ∗  G2)(v1xv3′) =  μ1(v1v2)⋁μ2(v3′) + μ1(v1v4)⋁μ2(v3′)  + μ2(v3′v1′) ⋁μ1(v1) +
                                             μ2(v3′v2′)⋁μ1(v1′) = 1.6     
D(G1 ∗  G2)γ(v1xv3′) =  γ1(v1v2)⋀γ2(v3′) + γ1(v1v4)⋀γ2(v3′)  + γ2(v3′v1′) ⋀γ1(v1) +

                                               γ2(v3′v2′)⋀γ1(v1′) = 0.6     
D(G1 ∗  G2)(v1xv3′) = (1.6,0.6) 
 

Likewise apply the same technique to determine the degree of each vertex in the maximal product.  

D(G1 ∗  G2)(v2xv1′) = (1.9,0.7), 

D(G1 ∗  G2)(v2xv2′) = (2.0,0.7), 

D(G1 ∗  G2)(v2xv3′) = (1.8,0.6), 

D(G1 ∗  G2)(v3xv1′) = (1.7,0.5), 

D(G1 ∗  G2)(v3xv2′) = (1.8,0.5), 

D(G1 ∗  G2)(v3xv3′) = (1.6,0.4), 

D(G1 ∗  G2)(v4xv1′) = (1.9,0.7), 

D(G1 ∗  G2)(v4xv2′) = (2.0,0.7), 

D(G1 ∗  G2)(v4xv3′) = (1.8,0.6). 
The above example implies the maximal product of two constant IF graphs need not be a constant 

IF graphs. 
 
Definition 3.4. 

If G(𝑉∗, 𝐸∗, 𝜇∗, 𝛾∗)  is the maximal product of two constant IF graphs 𝐺1(𝑉′, 𝐸′, 𝜇′, 𝛾′) and 

𝐺2(𝑉′′, 𝐸′′, 𝜇′′, 𝛾′′) then the total degree of the vertices of G (𝑢𝑖
′, 𝑢𝑗

′′) ∈ 𝑉∗ is defined as, 

 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢𝑖
′, 𝑢𝑗

′′)

= ∑ 𝑚𝑎𝑥 {𝜇′(𝑢𝑖
′𝑢𝑘

′), 𝜇′′(𝑢𝑗
′′)}  + ∑ 𝑚𝑎𝑥 {𝜇′′(𝑢𝑗

′′𝑢𝑘
′′) , 𝜇′(𝑢𝑖

′)} + 𝜇∗(𝑢𝑖
′, 𝑢𝑗

′′) 

Where (𝑢𝑖
′𝑢𝑘

′) ∈ 𝐸′ and (𝑢𝑗
′′𝑢𝑘

′′) ∈ 𝐸′′ and  

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢𝑖
′, 𝑢𝑗

′′) = ∑ min {𝛾′(𝑢𝑖
′𝑢𝑘

′), 𝛾′′(𝑢𝑗
′′)} + ∑ 𝑚𝑖𝑛{𝛾′′(𝑢𝑗

′′𝑢𝑘
′′) , 𝛾′(𝑢𝑖

′)} + 𝛾∗(𝑢𝑖
′, 𝑢𝑗

′′), 

wher 𝑖, 𝑗, 𝑘 = 1,2,3 … . 𝑛   

If each vertex of G has the unique total degree (𝑘′, 𝑘′′) then G is said to a maximal product of IF 

graphs of total degree (𝑘′, 𝑘′′) or (𝑘′, 𝑘′′)- totally constant maximal product IF graphs. 
 
Example 3.5 

Consider the maximal product of two constant IF graphs which is obtained in Example 3.2. The 

total degree of all 9 vertices of G has calculated as follows: 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢1
′, 𝑢1

′′) = (0.4 + 0.4 + 0.3 + 0.3) + 0.3 = 1.7 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢1
′, 𝑢1

′′) = (0.2 + 0.2 + 0.1 + 0.1) + 0.1 = 0.7 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢1
′, 𝑢2

′′) = (0.4 + 0.4 + 0.4 + 0.4) + 0.4 = 2.0 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢1
′, 𝑢2

′′) = (0.1 + 0.1 + 0.2 + 0.2) + 0.1 = 0.7 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢1
′, 𝑢3

′′) = (0.4 + 0.4 + 0.5 + 0.5) + 0.5 = 2.3 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢1
′, 𝑢3

′′) = (0.2 + 0.2 + 0.1 + 0.1) + 0.2 = 0.8 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢2
′, 𝑢1

′′) = 2.1 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢2
′, 𝑢1

′′) = 0.9 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢2
′, 𝑢2

′′) = 2.3 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢2
′, 𝑢2

′′) = 0.9 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢2
′, 𝑢3

′′) = 2.5 



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𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢2
′, 𝑢3

′′) = 1.0 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢3
′, 𝑢1

′′) = 1.7 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢3
′, 𝑢1

′′) = 0.5 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢3
′, 𝑢2

′′) = 2.0 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢3
′, 𝑢2

′′) = 0.5 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢3
′, 𝑢3

′′) = 2.3 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢3
′, 𝑢3

′′) = 0.5 
 
Example 3.6 

The following example explains the maximal product of two totally constant IF graph need not be 
totally constant maximal product IF graphs. 

                                    



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Theorem 3.7 

If 𝐺1(𝑉′, 𝐸′, 𝜇′, 𝛾′) and 𝐺2(𝑉′′, 𝐸′′, 𝜇′′, 𝛾′′) are two constant IF graphs such that 𝜇′(𝑢𝑖
′) ≤

𝜇′′(𝑢𝑗
′′𝑢𝑘

′′) , 𝜇′(𝑢𝑖
′𝑢𝑗

′) ≤ 𝜇′′(𝑢𝑘
′′) and   𝛾′(𝑢𝑖

′) ≥ 𝛾′′(𝑢𝑗
′′𝑢𝑘

′′), 𝛾′(𝑢𝑖
′𝑢𝑗

′) ≥ 𝛾′′(𝑢𝑘
′′) then the total 

degree of their maximal product 𝐺∗(𝑉∗, 𝐸∗, 𝜇∗, 𝛾∗) = 𝐺1 ∗ 𝐺2 is given by  

 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢𝑖
′, 𝑢𝑗

′′) = 𝑛0(𝑢𝑖
′) 𝜇′′(𝑢𝑗

′′) + 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇′′(𝑢𝑗
′′) and  

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢𝑖
′, 𝑢𝑗

′′) = 𝑛0(𝑢𝑖
′) 𝛾′′(𝑢𝑗

′′) + 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾′′(𝑢𝑗
′′) for  𝑖, 𝑗, 𝑘 = 1,2,3 … . 𝑛 

Proof: 

If 𝐺1(𝑉′, 𝐸′, 𝜇′, 𝛾′) and 𝐺2(𝑉′′, 𝐸′′, 𝜇′′, 𝛾′′) are two constant IF graphs such that 𝜇′(𝑢𝑖
′) ≤

𝜇′′(𝑢𝑗
′′𝑢𝑘

′′) , 𝜇′(𝑢𝑖
′𝑢𝑗

′) ≤ 𝜇′′(𝑢𝑘
′′) then the total degree of the vertices in their maximal product is 

defined for 𝑖, 𝑗, 𝑘 = 1,2,3 … . 𝑛 as, 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢𝑖
′, 𝑢𝑗

′′)

= ∑ 𝑚𝑎𝑥 {𝜇′(𝑢𝑖
′𝑢𝑘

′), 𝜇′′(𝑢𝑗
′′)}  + ∑ 𝑚𝑎𝑥 {𝜇′′(𝑢𝑗

′′𝑢𝑘
′′) , 𝜇′(𝑢𝑖

′)} + 𝜇∗(𝑢𝑖
′, 𝑢𝑗

′′) 

Where (𝑢𝑖
′𝑢𝑘

′) ∈ 𝐸′, (𝑢𝑗
′′𝑢𝑘

′′) ∈ 𝐸′′ and 𝑢𝑖
′ =

𝑢𝑗
′, 𝑢𝑖

′′ = 𝑢𝑗′′ 

= ∑ 𝜇′′(𝑢𝑗
′′) + ∑ 𝜇′′(𝑢𝑗

′′𝑢𝑘
′′) + 𝜇∗(𝑢𝑖

′, 𝑢𝑗
′′) 

= 𝑛0(𝑢𝑖
′) 𝜇′′(𝑢𝑗

′′) + 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇′′(𝑢𝑗
′′) 

 



5798 

 

 
Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 6: 5789-5799, 2024 
DOI: 10.55214/25768484.v8i6.3258 
© 2024 by the authors; licensee Learning Gate 

 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢𝑖
′, 𝑢𝑗

′′) = ∑ 𝑚𝑖𝑛 {𝛾′(𝑢𝑖
′𝑢𝑘

′), 𝛾′′(𝑢𝑗
′′)}  + ∑ 𝑚𝑖𝑛 {𝛾′′(𝑢𝑗

′′𝑢𝑘
′′) , 𝛾′(𝑢𝑖

′)} + 𝛾∗(𝑢𝑖
′, 𝑢𝑗

′′) 

Where (𝑢𝑖
′𝑢𝑘

′) ∈ 𝐸′, (𝑢𝑗
′′𝑢𝑘

′′) ∈ 𝐸′′ and 𝑢𝑖
′ =

𝑢𝑗
′, 𝑢𝑖

′′ = 𝑢𝑗′′ 

= ∑ 𝛾′′(𝑢𝑗
′′) + ∑ 𝛾′′(𝑢𝑗

′′𝑢𝑘
′′) + 𝛾∗(𝑢𝑖

′, 𝑢𝑗
′′) 

= 𝑛0(𝑢𝑖
′) 𝛾′′(𝑢𝑗

′′) + 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾′′(𝑢𝑗
′′) 

Here 𝑛0(𝑢𝑖
′) is the number of edges incident at 𝑢𝑖

′ in 𝐺1(𝑉′, 𝐸′, 𝜇′, 𝛾′) 
 
Theorem 3.8 

If 𝐺1(𝑉′, 𝐸′, 𝜇′, 𝛾′) and 𝐺2(𝑉′′, 𝐸′′, 𝜇′′, 𝛾′′) are two constant IF graphs such that 𝜇′(𝑢𝑖
′) ≤

𝜇′′(𝑢𝑗
′′𝑢𝑘

′′) , 𝜇′(𝑢𝑖
′𝑢𝑗

′) ≤ 𝜇′′(𝑢𝑘
′′) and   𝛾′(𝑢𝑖

′) ≥ 𝛾′′(𝑢𝑗
′′𝑢𝑘

′′), 𝛾′ (𝑢𝑖
′𝑢𝑗

′) ≥  𝛾′′(𝑢𝑘
′′) with second 

constant IF graph is (𝐶1
∗∗, 𝐶2

∗∗) then the total degree of the vertices of their maximal product is  

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢𝑖
′, 𝑢𝑗

′′) = 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇′′(𝑢𝑗
′′) + 𝑛0(𝑢𝑖

′)𝐶1
∗∗  and 

         𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢𝑖
′, 𝑢𝑗

′′) = 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾′′(𝑢𝑗
′′) + 𝑛0(𝑢𝑖

′)𝐶2
∗∗  

 
Theorem 3.9 

If 𝐺1(𝑉′, 𝐸′, 𝜇′, 𝛾′) and 𝐺2(𝑉′′, 𝐸′′, 𝜇′′, 𝛾′′) are two constant IF graphs such that 𝜇′′(𝑢𝑖
′′) ≤

𝜇′(𝑢𝑗
′𝑢𝑘

′) , 𝜇′′(𝑢𝑖
′′𝑢𝑗

′′) ≤ 𝜇′(𝑢𝑘
′) and   𝛾′′(𝑢𝑖

′′) ≥ 𝛾′(𝑢𝑗
′𝑢𝑘

′), 𝛾′′(𝑢𝑖
′′𝑢𝑗

′′) ≥ 𝛾′(𝑢𝑘
′) then the total 

degree of the vertices in their maximal product G(𝑉∗, 𝐸∗, 𝜇∗, 𝛾∗) = 𝐺1 ∗ 𝐺2 is given by  

 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢𝑖
′, 𝑢𝑗

′′) = 𝑛0(𝑢𝑗
′′) 𝜇′(𝑢𝑖

′) + 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇′(𝑢𝑖
′) and  

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢𝑖
′, 𝑢𝑗

′′) = 𝑛0(𝑢𝑗
′′) 𝛾′(𝑢𝑖

′) + 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾′(𝑢𝑖
′) for  𝑖, 𝑗, 𝑘 = 1,2,3 … . 𝑛 

 
 
 
 
Proof: 

Let 𝐺1(𝑉′, 𝐸′, 𝜇′, 𝛾′) and 𝐺2(𝑉′′, 𝐸′′, 𝜇′′, 𝛾′′) are two constant IF graphs such that 𝜇′′(𝑢𝑖
′′) ≤

𝜇′(𝑢𝑗
′𝑢𝑘

′) , 𝜇′′(𝑢𝑖
′′𝑢𝑗

′′) ≤ 𝜇′(𝑢𝑘
′) and   𝛾′′(𝑢𝑖

′′) ≥ 𝛾′(𝑢𝑗
′𝑢𝑘

′), 𝛾′′(𝑢𝑖
′′𝑢𝑗

′′) ≥ 𝛾′(𝑢𝑘
′) then by 

definition of total degree, their maximal product 𝐺1 ∗ 𝐺2 = G(𝑉∗, 𝐸∗, 𝜇∗, 𝛾∗) is obtained as 

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢𝑖
′, 𝑢𝑗

′′)

= ∑ 𝑚𝑎𝑥 {𝜇′(𝑢𝑖
′𝑢𝑘

′), 𝜇′′(𝑢𝑗
′′)}  + ∑ 𝑚𝑎𝑥 {𝜇′′(𝑢𝑗

′′𝑢𝑘
′′) , 𝜇′(𝑢𝑖

′)} + 𝜇∗(𝑢𝑖
′, 𝑢𝑗

′′) 

Where (𝑢𝑖
′𝑢𝑘

′) ∈ 𝐸′, (𝑢𝑗
′′𝑢𝑘

′′) ∈ 𝐸′′ and 𝑢𝑖
′ =

𝑢𝑗
′, 𝑢𝑖

′′ = 𝑢𝑗′′ 

= ∑ 𝜇′(𝑢𝑗
′𝑢𝑘

′) + ∑ 𝜇′(𝑢𝑖
′) + 𝜇∗(𝑢𝑖

′, 𝑢𝑗
′′) 

= ∑ 𝜇′(𝑢𝑗
′𝑢𝑘

′) + ∑ 𝜇′(𝑢𝑖
′) +  max {𝜇′(𝑢𝑖

′) , 𝜇′′(𝑢𝑗
′′)}   

= 𝑛0(𝑢𝑗
′′)𝜇′(𝑢𝑖

′) + 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇′(𝑢𝑖
′) 

Similarly,  

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢𝑖
′, 𝑢𝑗

′′) = ∑ min {𝛾′(𝑢𝑖
′𝑢𝑘

′), 𝛾′′(𝑢𝑗
′′)} + ∑ 𝑚𝑖𝑛{𝛾′′(𝑢𝑗

′′𝑢𝑘
′′) , 𝛾′(𝑢𝑖

′)} + 𝛾∗(𝑢𝑖
′, 𝑢𝑗

′′), 

Where (𝑢𝑖
′𝑢𝑘

′) ∈ 𝐸′, (𝑢𝑗
′′𝑢𝑘

′′) ∈ 𝐸′′ and 𝑢𝑖
′ =

𝑢𝑗
′, 𝑢𝑖

′′ = 𝑢𝑗′′ 

= ∑ 𝛾′(𝑢𝑖
′𝑢𝑘

′) + ∑ 𝛾′(𝑢𝑖
′) + 𝛾∗(𝑢𝑖

′, 𝑢𝑗
′′) 



5799 

 

 
Edelweiss Applied Science and Technology 
ISSN: 2576-8484 

Vol. 8, No. 6: 5789-5799, 2024 
DOI: 10.55214/25768484.v8i6.3258 
© 2024 by the authors; licensee Learning Gate 

 

= ∑ 𝛾′(𝑢𝑖
′𝑢𝑘

′) + ∑ 𝛾′(𝑢𝑖
′) + min{𝜇′(𝑢𝑖

′) , 𝜇′′(𝑢𝑗
′′)}  

= 𝑛0(𝑢𝑗
′′) 𝛾′(𝑢𝑖

′) + 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾′(𝑢𝑖
′) 

Hence 𝑛0(𝑢𝑗
′′) is the number of edges incident at 𝑢𝑗

′′ in 𝐺2(𝑉′′, 𝐸′′, 𝜇′′, 𝛾′′) constant IF graphs. 
 

Theorem 3.10 

 If 𝐺1(𝑉′, 𝐸′, 𝜇′, 𝛾′) and 𝐺2(𝑉′′, 𝐸′′, 𝜇′′, 𝛾′′) are two constant IF graphs such that 𝜇′′(𝑢𝑖
′′) ≤

𝜇′(𝑢𝑗
′𝑢𝑘

′) , 𝜇′′(𝑢𝑖
′′𝑢𝑗

′′) ≤ 𝜇′(𝑢𝑘
′) and   𝛾′′(𝑢𝑖

′′) ≥ 𝛾′(𝑢𝑗
′𝑢𝑘

′), 𝛾′′ (𝑢𝑖
′′𝑢𝑗

′′) ≥  𝛾′(𝑢𝑘
′) with second 

constant IF graph is (𝐶1
∗, 𝐶2

∗) then their maximal product has the total degree as  

𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇∗(𝑢𝑖
′, 𝑢𝑗

′′) = 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝜇′(𝑢𝑖
′) + 𝑛0(𝑢𝑗

′′)𝐶1
∗  and 

         𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾∗(𝑢𝑖
′, 𝑢𝑗

′′) = 𝑡𝑜𝑡𝑎𝑙𝑑𝑒𝑔𝛾′(𝑢𝑖
′) + 𝑛0(𝑢𝑗

′′)𝐶2
∗  

 
4. Conclusion 

The total degree of the vertices of the maximal product of two constant IF graphs has been 
explained with definitions, examples and theorems. Also, theorems on different conditions related to two 
different constant IF graphs  on membership and non - membership values are proved explicitly. 
 
Copyright:  
© 2024 by the authors. This article is an open access article distributed under the terms and conditions 
of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). 
 
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