The incidence matrix of the graph shown
\(\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ -1 & 1 & 0 & 1 & 0 \\ 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & -1 & -1 \end{bmatrix}\)
The provided question requires determining the incidence matrix of the given directed graph. The incidence matrix is a way of representing a directed graph using a matrix. Each row of the matrix corresponds to a vertex, and each column corresponds to an edge. The elements of the matrix are typically 0, 1, or -1:
- 1 if the edge originates from the vertex.
- -1 if the edge terminates at the vertex.
- 0 if the edge is not incident with the vertex.
From the diagram, we can number the edges as follows:
- Edge 1: from vertex 1 to vertex 2
- Edge 2: from vertex 2 to vertex 3
- Edge 3: from vertex 3 to vertex 4
- Edge 4: from vertex 2 to vertex 4
- Edge 5: from vertex 4 to vertex 1
Based on this, we can construct the incidence matrix as:
| Edge 1 | Edge 2 | Edge 3 | Edge 4 | Edge 5 | |
|---|---|---|---|---|---|
| Vertex 1 | 1 | 0 | 0 | 0 | 1 |
| Vertex 2 | -1 | 1 | 0 | 1 | 0 |
| Vertex 3 | 0 | -1 | 1 | 0 | 0 |
| Vertex 4 | 0 | 0 | -1 | -1 | -1 |
Therefore, the correct incidence matrix is:
\(\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ -1 & 1 & 0 & 1 & 0 \\ 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & -1 & -1 \end{bmatrix}\)
This matches the given correct option.
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?
- Maintaining a graph in memory by means of its adjacency matrix is known as
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?
