Graphs can be represented in several ways, each with its own advantages and disadvantages depending on the specific application and the properties of the graph. Here are some common methods for representing graphs:
1. Adjacency Matrix
An adjacency matrix is a 2D array of size $$V \times V$$ where $$V$$ is the number of vertices in the graph. Each cell $$(i, j)$$ in the matrix indicates whether there is an edge between vertex $$i$$ and vertex $$j$$.
Advantages:
Simple and easy to implement.
Efficient for dense graphs where the number of edges is close to $$V^2$$.
Fast edge lookup: Checking if there is an edge between two vertices is $$O(1)$$.
Disadvantages:
Requires $$O(V^2)$$ space, which can be inefficient for sparse graphs.
Iterating over all edges takes $$O(V^2)$$ time.
2. Adjacency List
An adjacency list represents a graph as an array of lists. The array has one entry for each vertex, and each entry is a list of all adjacent vertices (i.e., vertices connected by an edge).
Advantages:
Space-efficient for sparse graphs, requiring $$O(V + E)$$ space where $$E$$ is the number of edges.
Efficient for iterating over all edges of a vertex.
Disadvantages:
Edge lookup is slower compared to adjacency matrix, taking $$O(V)$$ in the worst case.
3. Edge List
An edge list is a list of all edges in the graph. Each edge is represented as a pair (or triplet for weighted graphs) of vertices.
Advantages:
Simple and straightforward representation.
Space-efficient for sparse graphs.
Disadvantages:
Edge lookup is slow, requiring $$O(E)$$ time.
Not efficient for algorithms that need quick access to the neighbors of a vertex.
4. Incidence Matrix
An incidence matrix is a 2D array of size $$V \times E$$ where $$V$$ is the number of vertices and $$E$$ is the number of edges. Each cell $$(i, j)$$ indicates whether vertex $$i$$ i...
