Finding the eigenvalue for a given graph

In summary, the characteristic polynomial of a simple graph with n vertices is the determinant of the matrix (A-\lambdaI), where A is the adjacency matrix and I is the n X n identity matrix. The characteristic equation of a matrix can also be written as the determinant of (xI-A). Both equations are correct. The eigenvalue is represented by the variable \lambda or x and is the root of the characteristic equation.
  • #1
Falnom
5
0
I'm having trouble finding the eigenvalue for a given graph; but more specifically I can't seem to find the characteristic polynomial. My book tells me that the characteristic polynomial of a simple graph with n vertices is the determinant of the matrix (A-[tex]\lambda[/tex]I), where A is the adjaceny matrix and I is the n X n identity matrix. What is [tex]\lambda[/tex]? And I've read someplace else that the characteristic equation of a matrix is the determinant of (xI-A) - which is right, or are they both?
And isn't the root of the equation the eigenvalue?
 
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  • #2
lambda is a variable, use x if you prefer. It's just a letter.
 
  • #3
Ah yes of course, I had thought as much. Thank you very much. (It works now!)
 

Related to Finding the eigenvalue for a given graph

What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used to describe the behavior of a linear transformation or matrix. Eigenvalues represent the scaling factor of the eigenvector when it is multiplied by the transformation or matrix.

How do you find the eigenvalues for a given graph?

To find the eigenvalues for a given graph, you first need to convert the graph into an adjacency matrix. Then, using a matrix calculator or by hand, you can solve for the eigenvalues by finding the roots of the characteristic polynomial of the adjacency matrix.

What information can eigenvalues provide about a graph?

Eigenvalues can provide information about the connectivity and structure of a graph. They can also be used to determine important nodes or clusters within a graph.

Can a graph have multiple eigenvalues?

Yes, a graph can have multiple eigenvalues. The number of distinct eigenvalues a graph has is equal to its number of connected components.

How are eigenvalues used in real-world applications?

Eigenvalues have various applications in fields such as physics, engineering, and computer science. They are used in image and signal processing, network analysis, and data compression, among others.

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