A graph is a non-linear data structure that consists of a set of non-empty Vertices with a set of Edges.Each edge joins two different Vertices.
- Undirected Graph
- Directed Graph
If an edge between any two nodes is not directly oriented, then it is undirected graph
(A,B) & (B,A) represents the same edge
If an edge between any two nodes is *directly oriented, then it is directed graph
(A,B) Shows the edge between A and B
Here A is called the Tail node and B is called the Head node
- A graph may not have an edge from a vertex V back to itself. If it has an eged to itself, then it is called Self edges.
- A graph may not have multiple occurrence of the same edges (multigraph)
A path is a sequence of distinct Vertices each adjacent ot the next.
Path from A to C is (A,B),(B,C)
A cycle is a simple path in which first and last vertices are same.
ABCD is cyclic as it starts and ends with the same vertex
In a digraph. a cycle is referred as Directed Cycle.
Note : The maximum number of edges in a graph with n vertices is n(n+1)/2
An n-vertex, undirected graph with exactly n(n+1)/2 edges is said to be a complete graph.
n=4, no of edges = 4(4+1)/2 = 6
A subgraph of G is a graph G' such that *V(G') is a subset of V(G)
Subgraphs of G
If there exists a path from any vertex to any other vertex, then that graph is called connected graph.
If for every pair of distinct vertices there is a directed path from every vertex to every other vertices, then that graph is called Strongly connected graph.
Strongly Connected Graph
If any vertex dosen't have a directed path to any other vertices, then the graph is called Weakly connected Graph.
Weakly Connected graph
Undirected graph only have one degree, ie. the The number of edges connected directly to a node
The degree of A is 2
Directed Graphs have two types of degrees, namely
- Indegree - The number of edges entering the node.
- Outdegree - The number of edges leaving the node.
| Node | Indegree | Outdegree |
|---|---|---|
| A | 1 | 1 |
| B | 1 | 1 |
A vertex whose indegree is zero is referred as source vertex.
A vertex whose outdegree is zero is referred as sink vertex.
If a graph has onlu one vertex in it, it is a isolated graph vertex.
If every edge in the graph is assigned some weight (or) value, then the graph is called Weighted Graph
Weighted Graph
A vertex whose indegree is 1 and outdegree is 0 is referred to as Pendant Vertex.
A graph containing m vertices and n edges can be represented by a matrix with m rows and n columns. The matrix is formed by storing 1 in the ith row and jth column corresponding to the matrix, if there exists a ith vertex connected to one end of the jth edge and a 0 if there is no ith vertex connected to any end of the jth edge of the graph, such a matrix is referred to as incidence matrix.
Consider the given weighted graph
incidence_matrix[i][j] = 1, if there is an edge
0, otherwise
The Incidence Matrix for this graph is given by:
| E1 | E2 | E3 | |
|---|---|---|---|
| V1 | 1 | 0 | 0 |
| V2 | 0 | 1 | 0 |
| V3 | 0 | 0 | 1 |
| V4 | 1 | 1 | 1 |
A graph containing n vertices can be represented by a matrix with n rows and n columns if there exists an edge between ith and jth vertex of the graph, then 1 is stored in the ith row and jth column of the matrix, otherwise 0 is stored.
The Adjacency matrix for the graph is given by:
| A | B | C | D | |
|---|---|---|---|---|
| A | 0 | 1 | 0 | 0 |
| B | 0 | 0 | 1 | 0 |
| C | 1 | 0 | 0 | 0 |
| D | 1 | 1 | 1 | 0 |
A graph containing m vertices and n edges can be represented using a linked list is referred to as adjacency list.
| V1 | V2 |
|---|
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