2. In graph theory, an adjacency matrix is nothing but a square matrix utilized to describe
a finite graph. The components of the matrix express whether the pairs of a finite set
of vertices (also called nodes) are adjacent in the graph or not. In graph
representation, the networks are expressed with the help of nodes and edges, where
nodes are the vertices and edges are the finite set of ordered pairs .
ADJENCY MATRIX
3. In graph theory and computer science an adjacency list is a collection of unordered lists used to
represent a finite graph. Each unordered list within an adjacency list describes the set of
neighbors of a particular vertex in the graph. This is one of several commonly used
representations of graphs for use in computer programs.
ADJACENCY LIST
4. PROPERTIES OF ADJACENCY MATRIX
The vertex matrix is an array of numbers
which is used to represent the information
about the graph. Some of the properties of
the graph correspond to the properties of
the adjacency matrix, and vice versa.
6. DIFFERENCES BETWEEN ADJACENCY MATRIX AND ADJACENCY LIST
Operations Adjacency Matrix Adjacency List
Storage Space
This representation makes use of VxV
matrix, so space required in worst case
is O(|V|2).
In this representation, for every vertex we
store its neighbours. In the worst case, if a
graph is connected O(V) is required for a
vertex and O(E) is required for storing
neighbours corresponding to every vertex
.Thus, overall space complexity is
O(|V|+|E|).
Adding a vertex
In order to add a new vertex to VxV matrix
the storage must be increases to (|V|+1)2.
To achieve this we need to copy the whole
matrix. Therefore the complexity is O(|V|2).
There are two pointers in adjacency list first
points to the front node and the other one
points to the rear node.Thus insertion of a
vertex can be done directly in O(1) time.
Adding an edge
To add an edge say from i to j, matrix[i][j] =
1 which requires O(1) time.
Similar to insertion of vertex here also two
pointers are used pointing to the rear and
front of the list. Thus, an edge can be
inserted in O(1)time.
Removing a vertex
In order to remove a vertex from V*V matrix
the storage must be decreased to |V|2 from
(|V|+1)2. To achieve this we need to copy
the whole matrix. Therefore the complexity
is O(|V|2).
In order to remove a vertex, we need to
search for the vertex which will require
O(|V|) time in worst case, after this we
need to traverse the edges and in worst case
it will require O(|E|) time.Hence, total time
complexity is O(|V|+|E|).
7. APPLICATION OF ADJACENCY MATRIX
The adjacency matrix is a matrix used to represent
finite graphs. The values in the matrix show
whether pairs of nodes are adjacent to each other
in the graph structure. If the graph is undirected,
then the adjacency matrix will be a symmetric one.
8. APPLICATION OF ADJACENCY LIST
In graph theory and computer science, an adjacency list
is a collection of unordered lists used to represent a
finite graph. Each unordered list within an adjacency list
describes the set of neighbors of a particular vertex in
the graph.
9. THE RELATIONSHIP BETWEEN ADJACENCY LIST AND ADJACENCY MATRIX
If a graph has n number of vertices, then the adjacency matrix of
that graph is n x n, and each entry of the matrix represents the
number of edges from one vertex to another. An adjacency matrix
is also called as connection matrix. Sometimes it is also called a
Vertex matrix.
10. CONCLUSION
• Adjacency matrix is good for dense
graphs.
• Adjecency lists is good for sparse
graphs and also for changing the no
of nodes.
