Find the number of paths between c and d in the graph in Figure 1 of length a) 2 . b)

Find the number of paths between c and d in the graph in...

Key Concepts

Adjacency Matrix Method

The adjacency matrix is a matrix representation of a graph where each element indicates the presence or absence of an edge between a pair of vertices. One important property of the adjacency matrix is that when it is raised to the power n, the resulting matrix gives the number of paths of length n between each pair of vertices. This algebraic method provides a systematic way to count paths in graphs.

Path Length

Path length refers to the number of edges in a path. It is a critical parameter when analyzing graph properties because many problems impose restrictions on the number of steps or hops between two vertices. Evaluating the path length helps in optimizing routes, understanding graph metrics, and ensuring constraints are met in traversal problems.

Counting Paths

Counting paths involves determining the number of distinct sequences (or routes) between two vertices that satisfy given constraints, such as a fixed path length. This process requires combinatorial analysis and sometimes recursive or algebraic techniques to systematically enumerate all possible valid paths without repetition.

Graph Theory

Graph theory is a branch of mathematics that deals with the study of graphs, which are abstract structures used to model pairwise relations between objects. In this context, vertices (or nodes) represent entities and edges (or links) represent the relationships between these entities. This field provides tools and techniques for analyzing connectivity, network structure, and traversal properties.

Path

A path in graph theory is a sequence of vertices connected by edges, representing a way to move from one vertex to another within the network. Paths are fundamental in understanding connectivity in graphs, and the concept is employed in various applications such as network routing, social network analysis, and algorithm design.

Transcript

00:02 We're given a length, and we're asked to find the number of paths between points c and d in the graph in figure 1 of this length.

00:13 So the graph of figure 1 is the beginning of this section, and in part a, we're asked to find the number of paths between c and d of length 2.

00:27 In order to find the number of paths of length 2, first we'll find the adjacency matrix of this graph.

00:35 We'll identify the entry in this adjacency matrix, which corresponds to the number of paths of length 1 between c and d.

00:48 Then we'll square the adjacency matrix, hence the 2, and then we'll look at the same entry, and this will be the number of paths of length 2 between c and d.

01:01 So first let's find the adjacency matrix.

01:04 Well, i'm not going to write the whole thing out here for you, but find adjacency matrix.

01:13 First of all, notice that our graph has six vertices, and therefore the adjacency matrix is going to be a six -by -six matrix.

01:27 So i'm going to make an adjacency matrix where the entries correspond to the vertices as follows.

01:38 So we'll have six rows, a row for a, b, c, d, e, and f, and six columns.

01:52 One for a, one for b, one for c, one for d, one for e, and one for f.

02:04 And now we simply write down the entries.

02:09 We know automatically that the entries in the main diagonal are going to be zero since we have a simple graph.

02:20 The rest of the entries, simply look at the graph and see how it's connected.

02:27 So this is our adjacency matrix.

02:31 I'll call this matrix a.

02:35 Now again, because we're looking for the number of path between c and d of length 2, well first of all, the corresponding entry, which i'll circle in red, since this is an undirected graph, could be either this entry or this entry.

02:53 If you look at both of those, but to find the number of pads of length 2, we need to find a squared.

03:03 For the purposes of this exercise, we really need this entry.

03:10 We see that it's clearly zero, and therefore the number of pads of length 2 between c and d is zero.

03:20 For later, parts of this exercise, though, you're going to need to know what a squared is to calculate higher powers of a...

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