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Sub graph
It consists of less or equal number of verticals (nodes) & edges, as in its complete graph.
True & Co-tree
A connected sub-graph of a network which has its nodes same as original graph but does not
contain any closed path is called tree of network.
A tree always has (n - 1) branches.
Eg. The following trees can be made from graph shown before.
The set of branches of a network which are remove to form a tree is called co-tree of graph.
Twigs & Links
The branches of a tree are called as its twigs & branches of a co-tree are called as chords or links.
Page 3
Sub graph
It consists of less or equal number of verticals (nodes) & edges, as in its complete graph.
True & Co-tree
A connected sub-graph of a network which has its nodes same as original graph but does not
contain any closed path is called tree of network.
A tree always has (n - 1) branches.
Eg. The following trees can be made from graph shown before.
The set of branches of a network which are remove to form a tree is called co-tree of graph.
Twigs & Links
The branches of a tree are called as its twigs & branches of a co-tree are called as chords or links.
Incidence Matrix
The dimension of incidence matrix is (nxb)
N = no. of nodes
B = no. of branches
It is represented by A
aij = + 1 , If
th
j branch is oriented away from
th
i node
aij = -1 , If
th
j branch is oriented into
th
i node.
aij = 0 , If
th
j branch is not connected to
th
i node
??
??
??
??
? ? ? ?
??
?
??
??
? ? ?
??
a b c d e f
1 0 1 0 0 1 1
A 1 1 0 1 0 0 2
0 1 0 0 1 1 3
4 0 0 1 1 1 0
If one of nodes is considered as ground & that particulars row is neglected while writing the
incidence matrix, then it is reduced incidence matrix. ? ? Order n 1 b ? ? ?
Number of trees of any graph ? ? ? ?
T
rr
det A A
??
?
??
r
A = reduced incidence matrix
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FAQs on Short Notes: Graph Theory - Short Notes for Electrical Engineering - Electrical Engineering (EE)
| 1. What is graph theory? |  |
Ans. Graph theory is a branch of mathematics that deals with the study of graphs, which are mathematical structures used to model pairwise relations between objects. It focuses on understanding the properties and characteristics of graphs and their applications in various fields such as computer science, social networks, and operations research.
| 2. How is a graph defined in graph theory? | |
Ans. In graph theory, a graph is defined as a set of vertices (also known as nodes) connected by edges. The vertices represent the objects or entities, while the edges represent the relationships or connections between them. A graph can be either directed (where the edges have a specific direction) or undirected (where the edges have no direction).
| 3. What are the common types of graphs studied in graph theory? | |
Ans. There are several common types of graphs studied in graph theory, including:
- Complete graph: A graph in which every pair of distinct vertices is connected by an edge.
- Bipartite graph: A graph whose vertices can be divided into two disjoint sets such that no two vertices within the same set are connected by an edge.
- Tree: A connected graph with no cycles.
- Directed acyclic graph (DAG): A directed graph with no directed cycles.
- Weighted graph: A graph in which each edge is assigned a weight or value.
| 4. What are some real-world applications of graph theory? | |
Ans. Graph theory has numerous real-world applications, including:
- Social networks: Graph theory is used to model and analyze social networks, such as Facebook and Twitter, to understand patterns of relationships between individuals.
- Transportation networks: Graphs are used to model and optimize transportation networks, such as road networks, airline routes, and public transportation systems.
- Internet and web analysis: Graph theory is used to study the structure and connectivity of the internet and analyze web pages' relationships (web graphs) for search engine algorithms.
- Operations research: Graph theory is used in operations research to solve optimization problems, such as finding the shortest path, scheduling, and resource allocation.
- Biological networks: Graph theory is used to model biological systems such as protein-protein interaction networks, genetic networks, and ecological networks.
| 5. What is the significance of graph theory in computer science? | |
Ans. Graph theory plays a crucial role in computer science. Some significant applications include:
- Data structures: Graphs are used as fundamental data structures to represent relationships between objects or entities in various algorithms and data structures like trees, hash tables, and adjacency lists.
- Network analysis: Graph theory is essential in network analysis, such as routing algorithms, network flow optimization, and network reliability analysis.
- Graph algorithms: Various graph algorithms, such as Dijkstra's algorithm for shortest path, Kruskal's algorithm for minimum spanning tree, and Ford-Fulkerson algorithm for maximum flow, are widely used in computer science.
- Compiler optimization: Graph theory is used in compiler optimization to analyze and optimize control flow graphs and data dependency graphs.
- Database management: Graph databases utilize graph theory concepts to store and query highly interconnected data efficiently.
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