Graph Representation Using Adjacency Matrix
Representing a graph in computer memory is crucial for performing various operations like searching, traversing, and analysis. A graph consists of vertices (nodes) and edges connecting them. There are several ways to store this information.
Understanding Adjacency Matrix Representation
One common method is the adjacency matrix. This approach uses a 2D array (or matrix), say A, where the size of the matrix is proportional to the number of vertices (V) in the graph. If the graph has V vertices, the adjacency matrix will be a V x V matrix.
- For an unweighted graph, A[i][j] = 1 if there is an edge between vertex i and vertex j, and A[i][j] = 0 otherwise.
- For a weighted graph, A[i][j] stores the weight of the edge between vertex i and vertex j. If no edge exists, it might store a special value like infinity or 0, depending on the implementation.
Adjacency Matrix as Sequential Representation
When we implement an adjacency matrix, we typically use a 2D array. In most programming languages, arrays are stored in contiguous blocks of memory. This means the entire matrix, representing the connections between all pairs of vertices, is laid out sequentially in memory. Therefore, maintaining a graph in memory using its adjacency matrix is known as Sequential Representation. This structure allows for quick lookup of whether an edge exists between two specific vertices (O(1) time complexity) but can consume significant memory (O(V^2)) especially for sparse graphs.
Evaluating Other Representation Options
- Complete Representation: This term is too general and doesn't specifically describe the adjacency matrix method. While an adjacency matrix fully describes the graph's structure, "Complete Representation" isn't the standard terminology for this storage technique.
- Linked Representation: This typically refers to the adjacency list method, where each vertex maintains a list (often a linked list) of its adjacent vertices. This is fundamentally different from the matrix-based approach and is not stored sequentially in a single 2D block.
- Circular Representation: This term doesn't align with standard graph storage techniques like adjacency matrices or adjacency lists. It might relate to specific data structures but not the general concept of matrix representation.
In summary, the adjacency matrix, being a 2D array stored contiguously, exemplifies a Sequential Representation of a graph in memory.
Important Questions from Graph Theory
Let $G(V, E)$ be an undirected and unweighted graph with 100 vertices. Let $d(u, v)$ denote the number of edges in a shortest path between vertices $u$ and $v$ in $V$. Let the maximum value of $d(u, v)$, $u, v \in V$ such that $u \neq v$, be 30. Let T be any breadth-first-search tree of G. Which ONE of the given options is CORRECT for every such graph G?
Let G be a simple undirected planar graph on 10 vertices with 15 edges. If G is a connected graph, then the number of bounded faces in any embedding of G on plane is equal to_________.
- The number of possible rooted trees in a phylogeny of three species is ________.
Consider the following statements regarding graph theory :
1. Chord is that branch of the graph that does not belong to the particular tree.
2. Cut-set is a unique set with respect to a given tree of a connected graph containing one chord and all of the free branches contained in the free path formed between two vertices of the chord.
3. If M represents the number of branches and N the number of nodes, the minimum number of variables involved in analyzing a network is equal to $(M + N − 1)$.
Which of the above statements are not correct?
The incidence matrix of the graph shown
