**Graph Theory Introduction**Consider the following graph:**Tasks:**a. Calculate the Laplacian of the graph.b. Calculate the eigenvalues of the graph.c. Use Kirchhoff’s matrix tree theorem to count the number of spanning trees in the graph.**Graph Description:**Below is a visual representation of the graph, which consists of six vertices and eight edges. Each vertex is connected by edges in a layout resembling two parallel vertical rhombuses sharing a common edge, thus forming a shape similar to a sideways "H".Diagram (Graph vii): O---O---O \ / \ / O O / \ / \ O---O---OThis graph will serve as the basis for the following calculations:1. **Laplacian Calculation:** - The Laplacian matrix of a graph is defined as \( L = D - A \), where \( D \) is the degree matrix and \( A \) is the adjacency matrix of the graph.2. **Eigenvalue Calculation:** - The eigenvalues are derived from the Laplacian matrix and provide significant insight into the graph's properties, such as its connectivity.3. **Spanning Trees Calculation using Kirchhoff’s Matrix Tree Theorem:** - Kirchhoff’s theorem states that the number of spanning trees in a graph can be found by calculating the determinant of any cofactor of the Laplacian matrix.These concepts will be elaborated further as we delve into the mathematical intricacies involved in solving the tasks mentioned above.

Since the graph has 8 vertices it will have a laplacian matrix of order 8.

Consider the following graph. a. Calculate the Laplacian of the graph. b. Calculate the eigenvalues of the graph. c. Use Kirchhoff's matrix tree theorem to count the number of spanning trees in the graph. (vii)

Consider the following graph. a. Calculate the Laplacian of the graph. b. Calculate the eigenvalues of the graph. c. Use Kirchhoff's matrix tree theorem to count the number of spanning trees in the graph. (vii)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**Graph Theory Introduction**

Consider the following graph:

**Tasks:**
a. Calculate the Laplacian of the graph.
b. Calculate the eigenvalues of the graph.
c. Use Kirchhoff’s matrix tree theorem to count the number of spanning trees in the graph.

**Graph Description:**

Below is a visual representation of the graph, which consists of six vertices and eight edges. Each vertex is connected by edges in a layout resembling two parallel vertical rhombuses sharing a common edge, thus forming a shape similar to a sideways "H".

Diagram (Graph vii):

O---O---O
\ / \ /
O O
/ \ / \
O---O---O

This graph will serve as the basis for the following calculations:

1. **Laplacian Calculation:**
- The Laplacian matrix of a graph is defined as \( L = D - A \), where \( D \) is the degree matrix and \( A \) is the adjacency matrix of the graph.

2. **Eigenvalue Calculation:**
- The eigenvalues are derived from the Laplacian matrix and provide significant insight into the graph's properties, such as its connectivity.

3. **Spanning Trees Calculation using Kirchhoff’s Matrix Tree Theorem:**
- Kirchhoff’s theorem states that the number of spanning trees in a graph can be found by calculating the determinant of any cofactor of the Laplacian matrix.

These concepts will be elaborated further as we delve into the mathematical intricacies involved in solving the tasks mentioned above.

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