This is the presentation of graph theory we are different topics of theological indices are represented along with their applications.

2. TOPOLOGICAL INVARIANTS AND THEIR
BEHAVIOUR FOR CORONA PRODUCT GRAPH
KALEEM ARSHAD
Final thesis of BS Mathematics
May 25, 2023
UNIVERSITY OF EDUCATION LAHORE
VEHARI CAMPUS
2023

3. OBJECTIVES
Introduction.
Graph Theory And Its Types .
Topological Indices.
Addition based Topological Indices.
Corona Product Graph, Topological Indices for Corona Product Graph.
Conclusion.

4. Introduction
Graph theory is a powerful mathematical framework that enables the analysis and
understanding of various complex systems, ranging from social networks to
biological networks and communication networks. Topological invariants, which are
numerical characteristics associated with graphs, play a crucial role in capturing and
quantifying the structural properties of these networks. By studying the behavior of
topological invariants, researchers gain insights into the underlying connectivity
patterns, robustness, and other important attributes of the networks.
One particular graph operation that has attracted significant attention in recent
years is the corona product. The corona product combines two graphs, namely the
complete graph and the host graph, resulting in a composite graph that
incorporates the structural features of both individual graphs. This operation has
been widely employed in various graph theoretical studies and has shown promise
in applications such as network design, wireless communication systems, and social
network analysis.

5. Introduction
Furthermore, studying the behavior of topological invariants for corona product
graphs under different graph operations and transformations is another important
aspect. Investigating how these invariants change when the corona product is
combined with operations such as graph complementation or graph powers allows
researchers to explore the relationships and dependencies between different graph
operations and their collective effects on the topological invariants.

6. Graph
Suppose G is a graph with such an ordered pair (V, E) of vertices and
edges. We discuss 2D (two-dimensional graphs). Graphs are short
and simple ways to express complex mathematical data. The concept
of graph theory was first used by Euler in the 19th century.
Example:

7. Graph Theory
A graph consists of a collection of non-empty vertices or nodes V and a collection of
edges E. An edge's termination point is defined with vertex A. A collection of the
vertex is represented by an edge that connects two vertexes a, b.
Example:

8. Types of Graph:
• Simple Graph: A simple graph has one that has no loop, parallel edges, and
several edges. Example:
• Multiple Graphs: If there is more than one edge among the two adjacent
vertices, then that graph is called multiple graphs.
Example:

9. Types of Graph
• Regular graph: A graph that is regular if it’s all vertices carry equal degrees, is
known as a regular graph. Example:
• Corona product of graph: If G and H are graphs, the corona product G is
calculated by attaching each vertex of each copy to the 𝑖𝑡ℎ vertex of G.
Example

10. Types of Graph
• Complete Graph: If every two vertices are connected by an edge. A graph with
n vertices is referred to as "complete."
★ The complete graph on n vertices is represented by Kn.
★ There is no loop.
★ Every two vertices have exactly one edge.
★ Each vertex has an n-1 degree. K4
• Ladder Graph: The ladder graph is a 2n vertex, 3n-2 edge planar undirected
graph.
Example:

11. Topological Indices:
• A topological index is a mathematical formula that can be used to model any
molecular structure on any graph. In the field of mathematical chemistry
topological index is also called connective index. It is used to investigate the
physical and chemical properties of a molecule. It is a very beneficial way to avoid
expensive Laboratories that are time-consuming. The first topological index was
Wiener index proposed by Harold Wiener in 1947.

12. Addition Based Topological Indices
• Randic Index (ŔI): The ŔI is a molecular descriptor introduced by Milan Randic
in 1975 as. Example:
W4
Table 1.
Edges
point
Frequency (W4)
(3,4) 2
(2,4) 2
2 2
( )
(2 4) (3 4)
2 2
8 12
2 2
2 2 2 3
1 1
2 3
2 3 3 2
6
X G
X X
 
 
 
 



13. Addition Based Topological Indices
• Atom bond connectivity (ӐBҪ) index: Furtula et al created ӐBҪ index in
2009. It describes the stability of alkanes as well as the strain energy of
cycloalkanes. Example:
₭4
1
Table 2.
( )
2
( ) u v
uv E G u v
d d
ABC G
d d


 
Edges point Frequency ₭4
1
(1, 4) 1
(2, 4) 2
(3,4) 1
1
4
1 4 2 2 4 2 3 4 2
( ) 1 2 1
1 4 2 4 3 4
3 4 5
2
4 8 12
3 2 15
2
2 6
2 2
3 15
2
2 6
15 3 3
2
6
2.93
ABC K X X X
X X X
X
X
     
  
  
  
  

 


14. Addition Based Topological Indices
• Geometric arithmetic index: Geometric arithmetic index is a new topological
index, successor to Randic index used to determine a molecule's chemical
characteristics; it is also known as; Example:
₭8
Table 3.
Since
Edge Points Frequency
(₭8 )
(1, 7) 7
2 1 7
7
1 7
2 7
7
8
7 7
4
X
X
X





15. Corona Product Graph, Topological Indices for Corona
Product Graph.
• Corona product graph:
If G and H are graphs, the corona product G is calculated by
attaching each vertex of each copy to the ith vertex of G, where 11≤i≤v (G) is the number of
copies.
Example:
Formula of corona product graph:
G eKH =Ge( K-1) H e H

16. Corona Product Graph, Topological Indices for Corona
Product Graph.
• Randic Index of some Corona Product Graph :
Since
Table
Corona of P3 and P4
Total Number of Pair = 23
Edge Point Number of Pair
(2,5) 4
(3,5) 4
(5,6) 2
(6,2) 2
(6,3) 2
(2,3) 3
(3,3) 3
(3,2) 3

17. Corona Product Graph, Topological Indices for Corona
Product Graph
• Geometric Arithmetic Index of some Corona Product Graph:
Table:
Corona of W3 and P6
Total Number of Pair = 50
Edge Point Number of
Pair
(3,9) 16
(9,9) 6
(2,9) 8
(2,3) 4
(3,3) 12
(3,2) 4

18. Corona Product Graph, Topological Indices for Corona
Product Graph
• Geometric Arithmetic Index:
2 3 2 2 2 3 2 3 3 2 2 9 2 3 9 2 9 9
( ) 4 4 12 8 16 6
3 2 2 3 3 3 2 9 3 9 9 9
2 6 2 6 2 9 2 18 2 27 2 81
( ) 4 4 12 8 16 6
5 5 6 11 12 18
8 6 8 6 16 18 8 27 2 81
( ) 4 9
5 5 11 3 3
16 6 16 18
( ) 12 8 3 6
5 11
16 6 16 18
( ) 18 8 3
5 11
(
X X X X X X
GA G X X X X X X
GA G X X X X X X
GA G
GA G
GA G
GA
     
     
     
     
    
   
) 18 13.856 7.838 6.171
( ) 45.865
G
GA G
   


19. Corona Product Graph, Topological Indices for Corona
Product Graph
• Atom Bond Connectivity index of some Corona product Graph:
Table:
Corona of k4 and L4
Total Number of Pair = 78
Edge Point Number of Pair
(3,11) 16
(4,11) 16
(11,11) 6
(3,4) 8
(4,4) 16
(3,3) 8
(4,3) 8

20. Corona Product Graph, Topological Indices for Corona
Product Graph
• Atom-Bond Connectivity Index:
3 3 2 4 3 2 3 4 2 4 4 2 3 11 2 4 11 2 11 11 2
8 8 8 16 16 16 16
3 3 4 3 3 4 4 4 3 11 4 11 11 11
12 13 20 4 5 5 6
16 16 6 8 8 8 16
33 44 121 9 12 12 16
12 13 6 20 16 5 5
16 8 4 4 4 6
33 11 11 11 3 3
9.798 10.328 5.333 8.697 2.
X X X X X X x
X X X X X x x
X X X X X X X
             
      
      
      
     439 9.648
46.243



21. Conclusion
• The study on topological indices for corona product graphs has provided valuable
insights into the structural properties of these composite graphs. By investigating various
topological indices, researchers have been able to analyze and quantify important
characteristics of corona product graphs, such as connectivity, symmetry, and
complexity. These findings contribute to the broader field of graph theory and have
potential applications in diverse areas such as chemistry, biology, and computer science.
• In conclusion, the investigation into topological indices for corona product graphs has
yielded valuable insights and established a solid foundation for further research. The
findings enhance our understanding of the structural characteristics of these composite
graphs and open avenues for future exploration. By leveraging topological indices,
researchers can better analyze, predict, and design corona product graphs for a wide
range of applications.

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23. Thank
You