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

Key Concepts

Complete Bipartite Graphs

A complete bipartite graph is a type of graph that is divided into two disjoint sets of vertices such that every vertex in one set is connected to every vertex in the other set, and no vertices within the same set are adjacent. This structure plays a central role in many combinatorial problems and provides a well?structured framework for counting paths, as seen in K?,?, which has two sets of three vertices each.

Graph Walks and Paths

In graph theory, a path or walk refers to a sequence of vertices connected by edges. When counting paths of a given length, one often allows repetition of vertices and edges unless otherwise restricted, making the concept akin to counting walks. This concept is key when determining the number of sequences between two vertices over multiple steps.

Adjacency Matrix Powers

The adjacency matrix of a graph encodes the connections between vertices, and taking successive powers of this matrix reveals the number of walks of a certain length between vertices. Specifically, the (i, j) entry of the matrix raised to the n-th power gives the number of paths of length n from vertex i to vertex j. This algebraic method is a powerful tool for analyzing and counting paths in structured graphs.

Spectral Graph Theory

Spectral graph theory studies the properties of graphs through the eigenvalues and eigenvectors of matrices associated with the graph, such as the adjacency matrix. The approach allows one to derive closed-form expressions or recurrence relations for the number of paths of a given length between vertices, particularly in highly regular graphs like complete bipartite graphs.

Recurrence Relations in Combinatorial Path Counting

Recurrence relations are equations that express the number of paths of a certain length in terms of shorter paths. In structured graphs like complete bipartite graphs, these relations can often be derived due to the regular pattern of connections between vertices, and they provide a systematic way to compute or simplify the counting of paths for varying path lengths.

Transcript

00:01 We're asked to find the number of paths of length n between any two adjacent vertices in the bipartite graph k -33 for the values of n in exercise 19.

00:15 So in exercise 19, n took on the values of 2, 3, 4, and 5.

00:21 So in part a, let's assume that n is equal to 2.

00:25 So the number of pads of length 2 between any two adjacent vertices in k -3 -3.

00:32 We recall that k -3 -3 has two sets of 3 -3.

00:35 Each and there are edges between every vertex from one set and every vertex from the other set.

00:42 And no edges between vertices in the same set.

00:48 But first, to turn the adjacency matrix of k -3 -3, well, k -3 has six vertices, therefore the adjacency matrix is going to be a six -by -six matrix.

01:09 Now, if there is an edge between two vertices, the adjacency matrix contains a 1, or else the matrix contains a 0.

01:21 Therefore, the first three columns contain ones in the last three rows, and the first three rows contains ones in the last three columns.

01:34 So our adjacency matrix looks something like this.

01:40 0 -0 -1 -1 -0 -0 -0 -1 -1 -1 -1 -1 -1 -0 -1 -0 -1 -0 and 111 -0.

02:16 The adjacency matrix has the number of paths of length 1 between any pair of vertices.

02:22 The number of paths of length 2 between a pair of vertices is then given by the second power.

02:29 So if we call this adjacency matrix a, then a squared is going to be a times a.

02:38 And because this is a block matrix, so i'll call this 0 3 by 3, 1, 3 by 3, 1, 3 by 3, and 0 3 by 3.

02:52 When we do matrix multiplication, we can multiply block matrices together.

02:58 So we get 03 by 3 times the next matrix, which is another 03 by 3 matrix, plus a 1 3 by 3 times another 1 3 by 3.

03:20 And the next entry, this is going to be a 0 3 by 3 times a 1 3 by 3 plus a 1 3 by 3 times a 0 3 by 3.

03:33 In the next row, we have a 1 3x3 times 0 3 by 3 plus a 0 3 by 3 times a 1 3 by 3.

03:49 And the last entry is a 1 3 by 3 times a 1 3 by 3 plus a 0 3 by 3 times a 0 3 by 3.

04:04 So this reduces 2.

04:06 Well, the 0 3 by 3 is just give us another 0 3 by 3.

04:09 Add to this, the product of one of two one three by three is just another, well, it's a little more complicated, actually.

04:19 So this is a zero three by three plus, well, one three by three times a one three by three gives us a three three three by three.

04:35 A zero three by three times a one three by three is another zero three by three.

04:38 And then we have the last entry is a 3 3 by 3 plus a 0 3 by 3.

04:58 So this simplifies to a 3 3 by 3, a 0 3 by 3, a 0 3 by 3, and a 3 3 by 3.

05:22 Alternatively, this is the matrix 333, 33, 333, 333, 333, 333, 3 3 3, 3.

05:30 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -3 -33 -33 -33 -33 -33.

05:54 Now, all pairs of adjacent vertices correspond with the elements which are all zero.

06:15 This is because, notice that in a, all the pairs of adjacent vertices have a 1.

06:37 This is important because we see that now all of those entries have 1.

06:41 Have a 0 in a squared.

07:08 Thus, there are no paths of length 2 between adjacent vertices.

07:25 So the answer to part a is a big 0.

07:34 And in part b, we have to find the number of pads of length 3 between a pair of vertices that are adjacent.

07:47 Well, now let's find a cubed.

07:54 This will contain the number of paths of length 3 between a pair of vertices.

07:57 And so this is same as a squared times a...

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