Find the number of paths from a to e in the directed graph in Exercise 2 of length a) 2 .

Find the number of paths from a to e in the directed graph...

Key Concepts

Directed Graph

A directed graph is a set of vertices connected by edges that have a specific direction, indicating a one-way relationship between vertices. In counting problems, this structure is crucial because it restricts the sequence in which vertices can be visited, making the direction of traversal fundamental to the problem.

Path in Graphs

A path in a graph is a sequence of vertices where each adjacent pair is connected by an edge. For directed graphs, the order and direction of edges matter, and a valid path must respect these directions.

Path Length

Path length refers to the number of edges used to traverse from the starting vertex to the destination vertex in a graph. It is a critical factor when specifying or counting paths of a particular length, as it sets the exact number of steps or transitions required.

Combinatorial Path Counting

Combinatorial path counting involves determining the number of distinct paths that satisfy given constraints such as starting and ending points, and path length. Techniques often include recursive reasoning, dynamic programming, or matrix exponentiation to methodically count all possible paths.

Transcript

00:01 So for this question, we're trying to find the number of paths of length like two all the way to seven between vertices a and e on in a graph.

00:22 In a graph that was given an exercise two.

00:26 And so if we look at the graph, which is directed, we can create an adjacency matrix for the graph, right? you know that adjacency matrix a of the graph will be as follows.

00:46 And we, the way we create this is by, for each entry, we see, we give, um, the entry here gives the number of paths between two vertices.

01:00 So for example, this first column corresponds with a, first top row corresponds to a, second column b, second row b, etc.

01:11 So if we want to find the number of paths from vertex a to vertex e, then we look at the, then we look at this element of the adjacency matrix.

01:25 So for now, for one, paths of length one, there are no paths of length one between a and e because this is simply, this is the element we look at.

01:37 So the way i'm going to do this is not very elegant, but, but if we want to really find the number of pads of length 2, 3, 4, 5, 6, and 7 between vertices a and e in this graph, one way to do it is to calculate a squared, a cubed, a4, 8 to the 5th, 8 to the 6th, and a to the 7th, and look at this element in each of these matrices.

02:08 Because by theorem 2, we know that that will give us the number of paths of length 2, 3, 4, 5, 6, and 7, respectively between the two desired vertices.

02:18 And so i'm just going to give you the matrices.

02:22 I'm assuming that we all know how to do matrix multiplication here.

02:26 So when you're doing this problem and if you choose this solution path, then just matrix multiply it out and just check with the matrices that i list below to make sure that you're done correct matrix multiplication but here are the matrices as follows and i realize that this is not a very elegant solution but this is um of like this is a an algorithmic way of determining determining the number of paths which is what we want in this in this problem right so a squared will be this matrix here which means that for two for part a for a our answer is one a cubed is this matrix, 01120, 2 ,0 ,0 ,0 ,0 ,0 ,0 ,0, 1, 0 ,1.

03:36 So for part b, if we look at the top right element, see that there are zero paths of length, 3 between a and e, and we'll go down...

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