DOI : 10.5281/zenodo.21833349
- Open Access
- Authors : Dr. M. G. Karunambigai, V. Gnaneswaran
- Paper ID : IJERTV15IS080026
- Volume & Issue : Volume 15, Issue 08 , August – 2026
- Published (First Online): 07-08-2026
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Software Development on Soft Graph Matrix
M.G.Karunambigai (1) and V. Gnaneswaran (2)
(1) Associate Professor & Head, PG & Research Department of Mathematics, Sri Vasavi College, Erode – 638 316, Tamilnadu, India.
(2) Assistant Professor, Department of Mathematics, Nandha Arts and Science College (Autonomous), Erode – 638 052, Tamilnadu, India.
Abstract – In this paper MATLAB-based software tool has been developed for the soft graph matrices. This system allows the key operations such as the Cartesian product, AND, OR, and complement on soft graph matrices. The effectiveness and applicability of the above operations in soft graph matrices are validated through illustrations and a real world application.
Keywords Soft Set, Soft Graph, Soft Graph Matrix, MATLAB Algorithm.
Definition 2.1
Let G = nonempty set. Let
-
PRELIMINARIES
be the simple graph, any be arbitrary relation between elements of
-
INTRODUCTION
Uncertainty is the unavoidable challenge in several fields, and the mathematical models are proposed to manage it. By addressing this uncertainty, [1] introduced the soft set theory, it is the more general and the adaptable framework which avoids the reliance on additional structures such as the membership functions or probabilities.
On this thought, the study [2] extended concept into the graph theory with the introduction of the soft graphs. Since the pioneering work of Euler [3], the graph theory plays the major role to solve the problems in computer science, operations
and. (i.e.)R . A set valued function F: A is defined as F(x)= .
The pair (F,A) is soft set over V.Then (F,A) is said to
be a Soft graph of G if the subgraph induced by F(x) in G(F%(x)) is a connected subgraph of G for all x A.
The set of all soft graph of G is denoted by SG(G).
Example 2.2
Consider simple graph G = as shown in Figure.1.
research, and network modeling[4-7]. The incorporation of the soft sets with graph theory provided the better way to represent and analyze uncertain relationships in the complex systems.
In the computational practice, graph structures are represented by utilizing the matrices like, the adjacency matrices, that translate the graphs into numerical form for better analysis. Although, the standard matrix operations are not equipped to handle the uncertainty. To fill this, soft graph matrices were proposed. The key advancement was made by Karunambigai and Gnaneswaran[8], who defined the operations such as the AND, OR, Cartesian product, and complement for the soft graphs, and shown their practical relevance in the decision-making issues.
Also theory of soft graph matrices has the considerable development, but the practical computational tools for implementation of these ideas were still lacking. To bridge this gap, this paper present the MATLAB-based software system which enables construction, manipulation and analysis of soft graph matrices. The system support the important operations like, the AND, OR, Cartesian product, and complement, and extending its capabilities with the additional
Let, V = { E = {
Fi
relation on .
} be a set of all vertices and
} be a set of edges which is
g – 1
analytical features. This paper provides the researchers and practitioners with the valuable tool for advancing studies and the real-world applications in uncertain data environments.
Let, we take a parameter set A ={ } Define set valued function F by,
F(x) = {y V| xRy }.
Then, F()={ }, F( )={}, F()
={ },
Here subgraph induced by F(x) in G, F%(x) is a connected subgraph of G, for all xA.
Fig 2
Hence, (F,A) SG(G) in Figure 2.
Definition 2.3
Let, G = (V, E) be the simple graph. A bethe parameter set and F be set valued function. (F, A) be a set of all soft graphs of G. The adjacency of matrix ,
where = 1 if there is an edge between vertex and vertex and = 0 otherwise. The matrix is called Soft Graph Matrix. Where, is set of all matrices of soft graphs in G.
Example 2.4
In Example 2.2, we saw about soft graphs. Here, we define the soft graph matrices for example 2.2.
Adjacency matrix for a graph G is,
MG =
Soft graph matrices are defined as in the example 1
M =
%SG[F ( )]
M =
%SG[F ( )]
M =
%SG[F ()]
Definition 2.5
Let, G = be the simple graph, A be a parameter set with a set valued function F. (F, A) be a set of all soft graphs of the G. be a set of all matrices of soft graphs of G. Let and
are two soft graph matrices of G, where . The Cartesian Product for two soft graph matrices is defined in the following way:
Where, is also a soft graph matrix of same order
that of and each element is
defined by .
Definition 2.6
Let, be the set of all matrices of soft graphs of
G. Let and are
two soft graph matrices of G, where. Then, AND of two soft graph matrices is defined in the following way:
Where, is also a soft graph matrix of same order
that of and each element is
defined by
Definition 2.7
Let, be the set of all matrices of soft graphs of
G. Let and are
two soft graph matrices of G, where . Then, OR of two soft graph matrices is defined in the following way:
Where, is also a soft graph matrix of same order
that of and each element is
defined by .
Definition 2.8
Let, be a set of all matrices of soft graphs of G. Let be the soft graph matrix of G, where . Complement of a soft graph matrix is defined in the following way:
Where, is also a soft graph matrix of same order that of .
-
-
MATLAB IMPLEMENTATION FOR SOFT
GRAPH MATRIX
In this, a MATLAB program was developed to create soft graph matrices using a specified base graph and a set of parameters. Each parameter is linked to a subset of vertices, and the program generates matrices that depict the interconnections within these subsets. The dimensions of the original graph are preserved in all the matrices produced, with irrelevant entries being assigned a zero value. This technique offers a practical means of applying soft set theory to graphs in a computational setting.
Code:
%% Step 1: Define the base graph G (Adjacency Matrix)
MG
= [0 1 0
1 0;
1 0 1 0
0;
0
1 0 1 1;
1 0 1 0
1;
0 0 1 1 0];
%% Step 2: Define soft set mappings F = struct();
F.a1 = [1 2 4];
F.a2 = [2 3 5];
F.a3 = [3 4 5];
parameters = fieldnames(F);
%% Step 3: Create subplot layout (2×2) figure;
% — Base Graph —
subplot(2,2,1); Gbase = graph(MG); plot(Gbase, …
‘Layout’,’force’, …
‘NodeLabel’,1:5, …
‘LineWidth’,1.5, …
‘MarkerSize’,7, …
‘NodeColor’,’b’); % Blue for base graph
title(‘Base Graph (MG)’);
% — Soft Graphs —
for i = 1:length(parameters) paramName = parameters[4]; vertexSet = F.(paramName);
% Construct softMatrix for this parameter softMatrix = zeros(5);
for r = 1:length(vertexSet) for c = 1:length(vertexSet) vr = vertexSet(r);
vc = vertexSet(c); softMatrix(vr, vc) = MG(vr, vc); nd
MSG[F%(a1)] =
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
MSG[F%(a2)] =
0 0 0 0 0
0
0
1
0
0
0
1
0
0
1
0
0
0
0
0
0
0
1
0
0
end
% Graph object
Gsoft = graph(softMatrix);
MSG[F%(a3)] =
subplot(2,2,i+1);
0
0
0
0
0
0
0
0
0
0
plot(Gsoft, …
0
0
0
1
1
‘Layout’,’force’, …
‘NodeLabel’,1:5, …
‘LineWidth’,1.5, …
‘MarkerSize’,7, …
‘NodeColor’,’r’); title([‘Soft Graph ‘, paramName]);
0 0 1 0 1
0 0 1 1 0
Soft Graph a1
end
Output:
Base Graph Adjacency Matrix (MG):
0
1
0
1
0
1
0
1
0
0
0
1
0
1
1
1
0
1
0
1
0
0
1
1
0
Base Graph (MG)
Soft Graph Matrices MSG[F%(x)]:
Soft Graph a2
Soft Graph a3
-
MATLAB Programming for Matrix Operations
-
Cartesian Product
Computes the Cartesian product of two soft graph matrices by performing element-wise multiplication.
Code:
functionresult = softMatrixCartesian(M1, M2)
% Cartesian product (element-wise multiplication)
result = M1 .* M2; end
Example:
A1 = [0 1 0 1 0;
1 0 0 0 0;
0 0 0 0 0;
1 0 0 0 0;
0 0 0 0 0];
A2 = [0 0 0 0 0;
0
0
1
0
0;
0
1
0
0
1;
0
0
0
0
0;
0 0 1 0 0];
A3 = [ 0 0 0 0 0;
0
0
0
0
0;
0
0
0
1
1;
0
0
1
0
1;
0 0 1 1 0];
% Call Cartesian product function
cartesianResult = softMatrixCartesian(A2, A3);
disp(‘Cartesian Product (A2, A3) =’); disp(cartesianResult);
% Plot Cartesian Product Graph G = graph(cartesianResult);
figure; plot(G, …
‘Layout’,’force’, …
‘NodeLabel’,1:numnodes(G), …
‘LineWidth’,1.5, …
‘MarkerSize’,7, … ‘NodeColor’,’m’); % Magenta nodes
title(‘Cartesian Product Graph (A2 × A3)’);
Output:
Cartesian Product (A2, A3) =
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
Cartesian Product Graph (A2 × A3)
-
AND Operation
Performs element-wise AND operation between two soft graph matrices. For binary matrices, this is equivalent to taking the minimum of corresponding elements.
Code:
functionresult = softMatrixAND(M1, M2)
% Performs AND operation (element-wise minimum)
result = min(M1, M2); end
Example:
% Using A1 and A2 from above
A1 = [0 1 0 1 0;
1 0 0 0 0;
0 0 0 0 0;
1 0 0 0 0;
0 0 0 0 0];
A2 = [0 0 0 0 0;
0
0
1
0
0;
0
1
0
0
1;
0
0
0
0
0;
0 0 1 0 0];
% Call AND function
andResult = softMatrixAND(A1, A2); disp(‘AND(A1, A2) =’);
disp(andResult);
Output:
AND(A1, A2) =
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
-
OR Operation
Performs element-wise OR operation between two soft graph matrices. This takes the maximum of each corresponding element.
Code:
functionresult = softMatrixOR(M1, M2)
% Performs OR operation (element-wise maximum)
result = max(M1, M2); end
Example:
% Using A1 and A3 from above
A1 = [0 1 0 1 0;
1 0 0 0 0;
0 0 0 0 0;
1 0 0 0 0;
0 0 0 0 0];
A3 = [0 0 0 0 0;
0 0 0 0 0;
0 0 0 1 1;
0 0 1 0 1;
0 0 1 1 0];
% Call OR function
orResult = softMatrixOR(A1, A3); disp(‘OR(A1, A3) =’);
disp(orResult);
% Plot the graph
G = graph(orResult); % create graph from adjacency matrix
figure;
plot(G, ‘Layout’, ‘force’); % force- directed layout
title(‘Graph of OR(A1, A3)’);
Output:
-
Complement
Returns the complement of a binary soft graph matrix, replacing 1 with 0 and 0 with 1.
Code:
functionresult = softMatrixComplement(M)
% Complement of a binary matrix (1 – M) result = 1 – M;
end
Example:
% Using A1 from above
A1 = [0 1 0 1 0;
1 0 0 0 0;
0 0 0 0 0;
1 0 0 0 0;
0 0 0 0 0];
% Call complement function
complementResult = softMatrixComplement(A1); disp(‘Complement of A1 =’); disp(complementResult);
% Plot the graph
G = graph(complementResult); % create graph from adjacency matrix figure;
plot(G, ‘Layout’, ‘force’); % force- directed layout
title(‘Graph of Complement of A1’);
Output:
Complement of A1 =
OR(A1, A3) =
1
0
1
0
1
0
1
0
1
0
0
1
1
1
1
1
0
0
0
0
1
1
1
1
1
0
0
0
1
1
0
1
1
1
1
1
0
1
0
1
1
1
1
1
1
0
0
1
1
0
Graph of OR(A1, A3)
Graph of Complement of A1
-
-
-
APPLICATION
A textile manufacturer, who was located in the Coimbatore, distributes the products to international markets. To verify the quality, company runs 2 centers in Salem and
Karur and utilizes2 primary shipping ports – Chennai and Tuticorinto deliver to its overseas destinations: such as the Dubai, London, and New York. Every shipping route includes the specific investment costs and the anticipated profit margins. To increase the profitability when managing the uncertainties in cost and quality, it uses the soft graph theory. This model took 3 main parameters: a (Low Cost), a (High Quality), and a (High Profit).
**Original Graph Nodes (V):**
-
Coimbatore
-
Salem
-
Karur
-
Chennai Port
-
Tuticorin Port
-
Dubai
-
London
-
New York
**Original Graph Edges (E) with Cost and Profit:**
-
Coimbatore Salem: Cost = 100
-
Coimbatore Karur: Cost = 90
-
Salem Chennai Port: Cost = 60
-
Karur Tuticorin Port: Cost = 70
-
Chennai Port Dubai: Profit = 500
-
Tuticorin Port London: Profit = 600
-
Tuticorin Port New York: Profit = 750
0 0 0 0 0 0 0 0
]
**MSG2 (a2 – High Quality):**
[ 0-100
0
0
0
0
0
0
0
-60
0
0
0
0
0
0
0
0
0
0
0
0
0
+500
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0;
0
0;
0
0;
0
0;
0
0;
0
0;
0
0;
0
0 ]
-90 0 0 0 0 0;
0
0
0
0
0
0
0
0
0
-70
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
**MSG3 (a3 – High Profit):** [ 0 0
0
0;
0
0;
0
0;
**Soft Parameters and the Corresponding Routes:**
-
a1: Low Cost Route = Coimbatore Karur
Tuticorin Port London
-
a2: High Quality Route = Coimbatore Salem Chennai Port Dubai
-
a3: High Profit Route = Coimbatore Karur Tuticorin Port New York
-
**Soft Graph Matrices:**
**MSG1 (a1 – Low Cost):**
[ 0 0;0
-90
0
0
0
0
0
0
0
0
0
0
0
0
0
-70
0
0
0
0
0
0
0
0
0
0
0
0
0
+600
0
0
0
0
0
0
0
0
0
0
0
0
0
0
+750;
0
0;
0
0;
0
]
MATLAB Code:
% Nodes nodes =
{‘Coimbatore’,’Salem’,’Karur’,’Chennai Port’,’TuticorinPort’,’Dubai’,’London’,’N ew York’};
n = length(nodes);
0;
0 %% ———- Base Graph (Original Graph)
0; ———-
0 BaseG = zeros(n);
0;
0 % Costs
0; BaseG(1,2) = -100; % Coimbatore Salem
0 BaseG(1,3) = -90; % Coimbatore Karur
0; BaseG(2,4) = -60; % Salem Chennai
0 Port
0; BaseG(3,5) = -70; % Karur Tuticorin
Port
% Profits
BaseG(4,6) = 500; % Chennai Port
Dubai
BaseG(5,7) = 600; % Tuticorin Port
London
BaseG(5,8) = 750; % Tuticorin Port
New York
% —– Plot Base Graph —– figure;
G_base = digraph(BaseG, nodes); p1 =
plot(G_base,’Layout’,’layered’,’EdgeLabel ‘,G_base.Edges.Weight);
title(‘Base Graph (Original Routes)’); p1.NodeColor = ‘magenta’; p1.MarkerSize = 7;
p1.LineWidth = 1.5;
%% ———- Soft Graph MSG1 (Low Cost Route a1)
MSG1 = zeros(n);
MSG1(1,3) = -90; % Coimbatore Karur MSG1(3,5) = -70; % Karur Tuticorin Port
MSG1(5,7) = 600; % Tuticorin Port
London
% —– Plot MSG1 —–
figure;
G1 = digraph(MSG1, nodes); p1 =
plot(G1,’Layout’,’layered’,’EdgeLabel’,G1
.Edges.Weight);
title(‘Soft Graph MSG1 (Low Cost Route)’);
p1.NodeColor = ‘red’; p1.MarkerSize = 7;
p1.LineWidth = 1.5;
%% ———- Soft Graph MSG2 (High
Quality Route a2) ———- MSG2 = zeros(n);
MSG2(1,2) = -100; % Coimbatore Salem MSG2(2,4) = -60; % Salem Chennai Port
MSG2(4,6) = 500; % Chennai Port
Dubai
% —– Plot MSG2 —–
figure;
G2 = digraph(MSG2, nodes); p2 =
plot(G2,’Layout’,’layered’,’EdgeLabel’,G2
.Edges.Weight);
p2.NodeColor = ‘cyan’; p2.MarkerSize = 7;
p2.LineWidth = 1.5;
%% ———- Soft Graph MSG3 (High
Profit Route a3) ———- MSG3 = zeros(n);
MSG3(1,3) = -90; % Coimbatore Karur MSG3(3,5) = -70; % Karur Tuticorin Port
MSG3(5,8) = 750; % Tuticorin Port
New York
% —– Plot MSG3 —–
figure;
G3 = digraph(MSG3, nodes); p3 =
plot(G3,’Layout’,’layered’,’EdgeLabel’,G3
.Edges.Weight);
title(‘Soft Graph MSG3 (High Profit Route)’);
p3.NodeColor = ‘green’; p3.MarkerSize = 7;
p3.LineWidth = 1.5;
%% ———- OR Operation (MSG2 MSG3)
———-
MSG_OR = zeros(n); for i = 1:n
for j = 1:n
if MSG2(i,j) == 0
MSG_OR(i,j) = MSG3(i,j);
elseif MSG3(i,j) == 0
MSG_OR(i,j) = MSG2(i,j);
else
if MSG2(i,j) > 0 && MSG3(i,j) > 0
MSG_OR(i,j) =
max(MSG2(i,j), MSG3(i,j)); % max profit else
MSG_OR(i,j) =
min(MSG2(i,j), MSG3(i,j)); % min cost end
end end end
% —– Plot OR Graph —– figure;
G_or = digraph(MSG_OR, nodes); p4 =
plot(G_or,’Layout’,’layered’,’EdgeLabel’, G_or.Edges.Weight);
title(‘Soft Graph OR Operation (High Quality High Profit)’); p4.NodeColor = ‘yellow’; p4.MarkerSize = 7;
p4.LineWidth = 1.5;
%% ———- Net Profit Calculation —-
——
net_profit = sum(MSG_OR(MSG_OR ~= 0)); fprintf(‘Net Profit (High Quality High Profit): %d\n’, net_profit);
Output:
Base Graph
Soft Graph MSG1 (Low Cost Route)
Soft Graph MSG2 (High Quality Route)
Soft Graph MSG3 (High Profit Route)
Soft Graph OR Operation (High Quality
High Profit)
Net Profit (High Quality High Profit): 930
The case study highlightedsoft graph matrices role in enabling the decision-making when faced with the uncertainty. By analyzing the parameters such as the cost, quality, and profit, MATLAB implementation identifies optimal trade route, which yields the net profit of Rs. 930. This demonstrated practical application of the soft graph theory by solving the real-world supply chain and the logistics issues.
-
-
CONCLUSION
This paper presented the effective and simple MATLAB framework for working with soft graph matrices, and it is built on soft set theory & graph theory. Also, it creates the matrices from the base graph with given parameters and then supports the key operations such as the AND, OR, Cartesian product, and complement. The full- matrix method keeps the analysis consistent when leaving room for upgrades. The work links the theory with practice, useful in uncertain environments, and it can be extended with the user-friendly tools, real-time scalability, and applications in decision-making, social networks, and recommendations. The Future scope include fuzzy or hybrid soft graphs for the more complex uses.
-
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