DOI : 10.5281/zenodo.23054618
- Open Access
- Authors : Dr. N. Sarala, Mrs. K. Nirmaladevi
- Paper ID : IJERTV15IS090728
- Volume & Issue : Volume 15, Issue 09 , September – 2026
- Published (First Online): 30-09-2026
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Operations And Structural Properties of Fuzzy Signed Graphs
(1) Dr. N. Sarala & (2) K. Nirmaladevi
(1) Head/Associate Professor, A.D.M College for Women (Autonomous), Nagapattinam.
(2) Research Scholar, A.D.M College for Women (Autonomous), Nagapattinam.
Abstract – Fuzzy signed graphs have garnered significant attention in various fields such as social network analysis, decision- making processes, and system modelling due to their ability to represent complex relationships that involve both uncertainty and sign (positive or negative). This paper explores the operations on fuzzy signed graphs, focusing on the structural properties and mathematical foundations that underlie their manipulation. We propose novel operations, including union, intersection, Join within the context of fuzzy signed graphs, and provide detailed algorithms for each operation. Additionally, we analyze the impact of these operations on key graph-theoretic parameters such as connectivity, reachability, and the stability of the graph. Furthermore, we explore how these operations can be applied to real-world problems, such as modeling social influence, predicting interactions, and evaluating consensus in uncertain environments. The study highlights the flexibility of fuzzy signed graphs in capturing nuanced relationships and contributes to the development of more robust frameworks for handling uncertain and signed data in network-based analysis.
Keywords – Fuzzy signed graph, FSG union, FSG intersection, FSG join.
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BASIC DEFINITIONS
-
Fuzzy Graph
Let (, , µ) be a fuzzy graph corresponding to the Crisp graph is a non-empty set
together with the pair of Functions µ: [0,1] and : [0,1] such that for all , , (, ) [µ(), µ()].
-
Degree of vertex
The degree of a vertex of the FG G is defined as () =
(, ).
-
Signed graph:
Let (, , ) be a signed graph is a graph in which each edge has a positive (or) negative sign
-
Degree of signed graph
The difference between the positive degree and negative at a vertax is denoted by
() = +() ().
1.5 Fuzzy signed graph
Let µ be the SG it has ordered triple (, , µ) where is the set of all vertices and is the set of all positive and negative edges and µ is the membership values of vertices and edges and also satisfying the condition µ(+) = µ() µ() , µ() = µ() µ().
1.6. Degree of fuzzy signed graph:
Let µ be the fuzzy signed graph the degree of vertex is sum of the membership values of positive edges incident on subtract sum of the membership value of negative edges incident on .
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-
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Operations and Properties of Fuzzy Signed Graph
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Union:
Let 1(1, 1, 1) and 2(2, 2, 2) are two fuzzy signed graph. Then the union of fuzzy signed graph is denoted by
(1 2)(, , ) = (1 2)() = 1(1) 2(2)
1() 1
(1 2)() = { 2() 2
1() 2() 1 2
(1
2)() =
1() 1
2() 2
1() 2() 1 2( )
{1() 2() 1 2( )
and also must sign of the common edges are same.
1 2
-
Definition
Let 1(v1,1,µ1) and 2(v2,2,µ2) be a two fuzzy signed graph then the union of two FSG is = (1 2) = (1 2, 1 2, µ1 µ2). Then the degree of vertex in 1 2 is denoted by deg(12)() is determined by the Common Edge is positive
deg12 () = µ(1 2)(, )
= (+() ()) + +() ()
1
12
1 2 2
21
+ 1() 2()
12
The Common Edge is negative
deg(12)() = µ(1 2)()
= (+() ()) + +() ()
1
12
1 2 2
21
1() 2()
12
-
Theorem:
Let 1(1, 1, µ1) and 2(2, 2, µ2)be a two FSG then The common edge is positive
deg12 () = 1 () + 2 () £ 12 1() 2()
The Common edge is negative
deg12 () = 1 () + 2 () + £ 12 1() 2()
Proof
The common edge is positive from the definition
deg () = (+() ()) + +() ()
1 2 1 1
12
2 2
12
+ 1() 2()
12
= (+() ()) + (+() ())
1
12
1 2 2
12
+ 1() + 2()
12 12
1() 2()
12
= 1 () + 2 () 1() 2()
12
( Common edge is positive so negative edge membership value 0) The common edge is negative.
From the definition
deg () = (+() ()) + +() ()
1 2 1
12
1 2 2
21
1() 2()
12
= (+() ()) + +() ()
1
12
1 2 2
21
( 1() + 2()
12 12
1() 2())
12
= (+() ()) + +() ()
1
12
1 2 2
21
1() 2()
12 12
+ 1() 2()
12
= 1 () + 2 () 1() 2()
12
(common edge is negative so positive edge membership value 0)
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Example
(1) = 2.3
From definition:
1 + 2 £ 12 1() 2()
= 1.7 + 1.0 0.4
= 2.3
deg12 (1) = 1.1
From definition:
1 + 2 + (1() 2())
12
= 0.1 + 0.2 + 0.8
= 1.1
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Intersection of two fuzzy signed graph:
Let 1(1, 1, 1) and 2(2, 2, 2) is fuzzy two signed then the intersection of fuzzy signed graph is (1 2)(, , ) is
(1 2)() = 1() 2() 1 2
(1 2)() = 1() 2() 1 2
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Join of fuzzy signed graph
Let 1(1, 1, 1) and 2(2, 2, 2)be two fuzzy signed graphs.The join between them is defined as(1 + 2)(, , ) where one set of vertices, edges are given by
(1 + 2)() = 1 2
(1 + 2)() = {
1() 1
2() 2
1() 2() 1 2
(1 + 2)() = 1 2
Where is the set of all edges joining vertices of 1with vertices of 2such that
(1 + 2)() = {
1() 1
2() 2
1() 2()
The sign of the new edges is negative because we have to take the minimum value of the corresponding vertices
+
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Degree of a vertex in Join:
-
Here 1 2 = , hence 1 2 = (1 2) = 1() 1
= 2() 2
By definition
(1+2)() = (1 2)() 1() 2()
For any 1
12
(12)() = 1() 1() 2()
1
= (+() ()) 1() 2()
1
1
1
For any 2
= 1 () 1() 2()
2
1+2 () = 2() 1() 2()
2
2.8. Theorem
Let 1(1, 1, 1) and 2(2, 2, 2) be two fuzzy graphs
-
if 1() > 2() then 1+2 () = 1 () (2) 1
= 2 () 1() 2
-
if 2() 1() then (1+2)() = 1 () 21() 1
= 2 () (1) 2
Proof:
-
We have 1() 2() . For any 1
(12)() = 1 () 1() 2()
2
= 1 () + 2()
= 1 () (2)
For any 2
(12)() = 2 () 1() 2()
1
= 2 () 2()
1
= 2 () 2() 1
1
= 1 () 12()
(ii) proof is similar to the proof (i).
2.9 Theorem
Let 1(1, 1, 1) and 2(2, 2, 2) be two fuzzy graphs such that 1 2 is constant function then 1+2 () = 1 () 2 1
= 2 () 1 2
where 1 2 = is constant for every 1 2.
Proof:
Let 1() 2() = for every 1 and 2
where C is constant. For any 1
12 () = 1 () 1() 2()
2
= 2 ()
2
= 1 () 2
For any 2
12 () = 2 () 1() 2()
1
= 2 ()
1
= 2 () 1
CONCLUSION:
In this paper we have to found the operations of fuzzy signed graphs such as Union, Intersection, Join and the degree of vertices of these operations under some conditions and illustrated them through Examples.
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