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Chapter 6: Problem 264

Give examples of the following concepts a) Graph b) Digraph c) Matrix of a digraph d) Incidence matrix.

Short Answer

Expert verified
a) Graph example: \(G = (A - B - C)\), an undirected graph with 3 vertices and 2 edges. b) Digraph example: \(D = (A \rightarrow B \rightarrow C)\), a directed graph with 3 vertices and 2 directed edges. c) Matrix of a digraph example: \(M = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}\), representing connections between vertices in digraph D. d) Incidence matrix example: \(N = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ 0 & 1 \end{bmatrix}\), representing connections between vertices and edges in graph G.

Step by step solution

01

Example of a Graph

Consider a graph G with three vertices, A, B, and C. Let there be edges connecting vertex A to vertex B, and vertex B to vertex C. The graph can be represented as follows: ``` A -- B -- C ``` In this example, the graph is undirected, as the edges do not have an associated direction.
02

Example of a Digraph

Now, let's create a digraph D with the same vertices A, B, and C. This time, there is a directed edge from vertex A to vertex B, and a directed edge from vertex B to vertex C. The digraph can be represented as follows: ``` A -> B -> C ``` In this example, the edges have a direction, making it a directed graph.
03

Matrix of a Digraph

Next, let's find the adjacency matrix (matrix of a digraph) for the digraph D from the previous example. The adjacency matrix is a square matrix that represents the connections between the vertices. For digraph D with vertices A, B, and C, the adjacency matrix can be defined as: \[ M = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} \] The matrix entry M[i][j] is 1 if there is a directed edge from vertex i to vertex j, and 0 otherwise. In this example, there is a directed edge from vertex A to vertex B, and from vertex B to vertex C.
04

Incidence Matrix

Finally, let's find the incidence matrix for the original undirected graph G. The incidence matrix of a graph is a rectangular matrix that defines the connection between vertices and edges. In graph G with vertices A, B, and C, and edges e1(A-B) and e2(B-C), the incidence matrix can be defined as: \[ N = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ 0 & 1 \end{bmatrix} \] The matrix entry N[i][j] is 1 if vertex i is incident with edge j, and 0 otherwise. In this example, vertex A is incident with edge e1, vertex B is incident with edges e1 and e2, and vertex C is incident with edge e2.

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Most popular questions from this chapter

Draw the multi graphs whose adjacency matrices are given below: (1) \(\begin{array}{rrrrrr} & 1 & 1 & 1 & 1 & 2 \\ & 11 & 3 & 1 & 3 & 1 \\\ \mathrm{G}_{1}= & \mid 1 & 1 & 0 & 1 & 1 \mid \\ & 1 & 3 & 1 & 0 & 1 \\ & 2 & 1 & 1 & 1 & 0\end{array}\) (2) \(\quad \begin{array}{rrrrr} & 0 & 2 & 2 & 3 \\ \mathrm{G}_{2}= & 2 & 0 & 3 & 2 \\ & 2 & 3 & 0 & 0 \\ & 2 & 2 & 0 & 0\end{array}\)

Prove the following theorem: "A graph \(\mathrm{G}=(\mathrm{V}, \mathrm{E})\) has a partial graph that is a tree if, and only if, \(G\) is connected."

Convert (a) - (c) from Infix to Prefix and Postfix expressions and \((\mathrm{d})-(\mathrm{f})\) from Prefix to Infix and Postfix expressions. (a) \(\mathrm{A}+\mathrm{B}^{*} \mathrm{C} / \mathrm{D}\) (b) \(A-C \& D+B \uparrow E\) (c) \(\mathrm{X} \uparrow \mathrm{Y} \uparrow \mathrm{Z}\) (d) \(+*-\mathrm{ABCD}\) (e) \(+\mathrm{mA}^{*} \mathrm{BmC}\) (f) \(\uparrow \mathrm{X}+{ }^{*} \mathrm{YZW}\)

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Draw the following: (1) \(\mathrm{K}_{1}\) (2) a connected regular graph of degree 0 (3) a graph that is 1 -regular (4) \(\mathrm{K}_{2,8}\) (5) a bipartite graph of 10 vertices (6) a connected regular graph of degree 3 with 8 vertices.
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