Understanding Adjacency Matrices for Digraphs?

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In summary, the conversation discusses the use of adjacency matrices for digraphs in linear algebra. The conversation also explores how to determine the number of pathways between nodes and how to find the total number of possible paths.
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Marie11
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In linear algebra, we are now learning about digraphs and I am somewhat confused as to how we can obtain and use these.

How do adjacency matrices for digraphs work?
If given that a31 is a certain number (ex. a31 = 2), how could you tell which pathways account for it? Also, if the adjacency matrix has rows ABCD and columns ABCD, how would one tell in how many transitions information travels from B to all of the others? Thanks. I really need help in understanding this. :-p
 
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You can't tell what pathways 'account for a31=2' only that there are two pathways from node 1 to node 3. You can find how many paths go from nod1 to node2 in 2 steps by squaring the matrix, in 3 by cubing, etc. The total number of possible paths from noden to nodem is the "mn" element of A+ A2+ A3+ ...
 
  • #3
Thanks!

Thanks a lot for the very appreciated explanation. :approve:
 

FAQ: Understanding Adjacency Matrices for Digraphs?

1. What is an adjacency matrix for digraphs?

An adjacency matrix for digraphs is a mathematical representation of a directed graph, where each row and column corresponds to a vertex in the graph. The values in the matrix indicate whether there is an edge between two vertices, with a 1 representing a connection and a 0 representing no connection.

2. How is an adjacency matrix different from an adjacency list?

An adjacency list is a data structure that lists all the vertices adjacent to a particular vertex, whereas an adjacency matrix stores this information in a matrix format. The choice between using an adjacency matrix or list depends on the type of graph and the operations that need to be performed on it.

3. What are the advantages of using an adjacency matrix for digraphs?

One advantage of using an adjacency matrix is that it allows for efficient lookup of edges between vertices. It also requires less storage space compared to an adjacency list for sparse graphs. Additionally, many graph algorithms and operations can be easily implemented using an adjacency matrix.

4. Are there any limitations of using an adjacency matrix for digraphs?

One limitation of using an adjacency matrix is that it requires a fixed amount of memory, regardless of the number of vertices in the graph. This can be an issue for large graphs with many vertices. Additionally, adding or removing edges in an adjacency matrix can be more time-consuming compared to an adjacency list.

5. Can an adjacency matrix be used for weighted digraphs?

Yes, an adjacency matrix can be used for weighted digraphs by assigning the weight of each edge as the corresponding value in the matrix. This allows for efficient calculation of the shortest path between two vertices using algorithms like Dijkstra's algorithm.

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