Find the number of paths of length n between two different vertices in K4 if n is a) 2 .

step 1 So for this question, there are four parts. We're trying to determine the number of pads of leng

Find the number of paths of length n between two different...

Key Concepts

Complete Graph

A complete graph is a type of graph in which every pair of distinct vertices is connected by a unique edge. This rich connectivity means that there are direct connections between any two vertices, which simplifies many counting problems in graph theory as every two vertices have the possibility of being connected in one step.

Graph Paths and Walks

In graph theory, a path (or walk, when repeated vertices are allowed) is defined as a sequence of vertices where each consecutive pair is joined by an edge. The concept of path length refers to the number of edges traversed. Counting these paths is a common problem in combinatorics and helps in understanding the structure and connectivity of a graph.

Adjacency Matrix

An adjacency matrix is a representation of a graph as a square matrix, where the entry in the ith row and jth column indicates whether there is an edge from vertex i to vertex j. This matrix is a powerful tool in graph theory because its powers can be used to count the number of paths (or walks) of a given length between any two vertices.

Matrix Exponentiation in Graph Theory

Matrix exponentiation involves raising the adjacency matrix of a graph to a power, and the resulting matrix has entries that represent the number of paths (or walks) of that length between vertices. This method is particularly useful for counting paths in graphs, as seen in problems involving paths of specified lengths in complete graphs like K4.

Transcript

00:01 So for this question, there are four parts.

00:04 We're trying to determine the number of pads of length n between two different vertices in k4, for n equals 2, 3, 4, and 5.

00:14 So essentially, what we do is first we have to draw k4, right? k4 is the complete graph on 4 vertices, which means that there are 4 vertices.

00:25 So i just ranged them in a square.

00:27 And a complete graph means that there's an edge connecting, each pair of distinct vertices.

00:35 So we see that a is connected with b, c, d, b is connected with the other three, and so on, so forth.

00:41 And so the way we try to find the number of paths of length end between two different vertices is we, essentially, we have to find the adjacency matrix of k4 first, right? so, uh, in the order a, b, c, d, we see that there is no, like, for example, this corresponds with a, b, c, d, this column, b, c, and d.

01:12 All right, so a and a have zero edges.

01:16 B, right, we just simply do this.

01:19 We just fill out the adjacency matrix, and this is our adjacency matrix.

01:28 So it's basically all ones except for the top left, bottom right diagonal, which is zero.

01:36 And that makes sense, right? there's an edge between every distinct pair of vertices, and there is no edge connecting a vertex with itself, hence the zeros.

01:48 And so, now that we think about it, how do we find the number of paths of length two between two different vertices? well, we know from this section that all we have to do is square this matrix, right? we just multiply it by itself.

02:03 So if we let this matrix be a, right? then a squared will tell us details regarding the number of pads of length two between the vertices a cubed right will be the number of paths of length three if we just read off a four for four and a five for five so essentially right um for these four parts a b c and d or n is equal to two three four and five respectively we just need to find a two a squared a cubed, a to the fourth, and a to the fifth.

02:38 And then we will be able to get our answers.

02:43 So, first off, let's just do a squared.

02:52 And once we do a squared, i'm just going to give you the results of the matrix multiplication, because once you know how to do matrix multiplication, there's really no need for me to work out steps again.

03:09 So i'll just work it out for part a, and then for part b, c, and d, i'll just tell you what a cubed, a fourth, and a to the fifth are.

03:17 And you can check them on your own.

03:18 Obviously, if you're doing this for homework, you should show your work and, you know, make sure that you show that you understand how to do this.

03:28 But for this purpose, for a squared, right? so we just matrix multiply these.

03:38 And so this gives us for so for this we have for the top left right the top left we have zero times zero plus one times one plus one times one so we have three here we have here zero times one times one plus one times one plus one times one so that gives us two zeros and two one so we add them up and then we get two for this for the next uh next element zero times one plus one times one plus one times one plus one times one two again uh for the final one on the upper most column the uppermost row zero times one plus one times one plus one times zero gives us two and we just work through again right so so for this one, it's 1 times 0 plus 0 times 1 plus 0 times 1 plus 1 times 1 plus 1 times 1 gives us 2.

04:52 And as we work through, we see that we will see that this is our matrix...

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