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  Concept of Adjacency Matrix, Adjacency List and Implicit Representation

Adjacency Matrix:

A second manner to symbolize a graph is to utilize an adjacency matrix. This is an N by N array (N is the no. of vertices). The i,j entry comprises a 1 if the edge (i,j) is in the graph; or else it contains a 0. For an undirected graph, the matrix is symmetric.

This representation is simple to code. It is less space efficient, particularly for big, sparse graphs. Debugging is harder, as the matrix is big. Finding all edges incident to a given vertex is fairly costly (that is, linear in the number of vertices), however checking if two vertices are adjacent is extremely quick. Adding up and eliminating edges are as well very inexpensive operations.

For weighted graphs, the value of (i,j) entry is employed to store the weight of edge. For an unweighted multigraph, the (i,j) entry can sustain the number of edges among the vertices. For a weighted multigraph, it is harder to extend this.

Example:

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The sample undirected graph would be symbolized by the adjacency matrix shown below:
 
It is sometimes obliging to use the fact that the (i,j) entry of adjacency matrix increased to the k-th power provides the number of paths from vertex i to vertex j comprising of exactly k edges. 

Adjacency List:

The third representation of a matrix is to maintain track of all edges incident to a given vertex. This can be completed by using an array of length N, where N is the number of vertices. The i th entry in this array is the list of edges incident to i th vertex (that is, edges are symbolized by the index of other vertex incident to that edge).

This representation is much harder to code, particularly if the number of edges incident to each and every vertex is not bounded, therefore the lists should be linked lists (or dynamically assigned). Debugging is difficult, as following linked lists is much more difficult.

Though, this representation employs about as much memory as the edge list. Determining the vertices adjacent to each node is very inexpensive in this structure, however checking if two vertices are adjacent needs checking all edges adjacent to one of the vertices. Adding an edge is simple, however deleting an edge is hard, if the positions of edge in the suitable lists are not known.

Extend this symbolization to handle weighted graphs by maintaining the weight and other incident vertex for each edge rather than just other incident vertex. Multigraphs are already representable. The directed graphs are as well simply handled by this representation, in one of some ways: store only edges in one direction, keep a separate list of incoming and outgoing arcs, or represent the direction of each arc in the list.

Example:

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The adjacency list symbolization of an illustration undirected graph is as shown below: 
 
Implicit Representation:

For certain graphs, the graph itself doesn’t have to be stored at all. For illustration, for the Knight moves and over fencing problems, it is simple to compute the neighbors of a vertex, check adjacency, and find out all the edges devoid of really storing that information, therefore, there is no reason to really store that information; the graph is implicit in data itself.

When it is possible to store the graph in this format, it is usually the accurate thing to do, as it saves a lot on storage and decreases the complexity of your code, making it simple to both write and debug.

When N is the number of vertices, M the number of edges and d max the maximum degree of a node, the table shown below summarizes the differences among the representations:

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