5. Suppose that A is the adjacency matrix for K|B|,|C|,...

5. Suppose that A is the adjacency matrix for K|B|,|C|, where B = {v1, v2, ..., v20} and C = {v21, v22, ..., v50}. (a) What is the value of A3,4? (b) What is the value of A15,35? 6. An adjacency matrix for a simple graph G has exactly 20 ones. How many edges does G have? 7. Suppose that G is a simple graph with vertices v1, v2, v3, v4, v5, v6, v7. Further suppose that d(v1) = d(v3) = d(v5) = d(v7) = 6. Lastly, suppose that |E(G)| = 18. Determine the degree of v4. Definition: An adjacency matrix A for a graph G is block diagonal if A = (A1 01 02 A2), where A1 and A2 are adjacency matrices for subgraphs of G and 01, 02 are matrices consisting of all zeros. Definition: A graph G is disconnected if G has at least two subgraphs G1 and G2 such that there is no way to get from a vertex of G1 to a vertex of G2 using the edges of G. Theorem: A graph G is disconnected <=> There is a way to label the vertices of G so that the resulting adjacency matrix is block diagonal. 8. Suppose that the graph G has adjacency matrix A = (0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0) Fill in the circle for the correct answer. There is a way to label the vertices of G so that the resulting adjacency matrix is block diagonal. There is not a way to label the vertices of G so that the resulting adjacency matrix is block diagonal.

Transcript

00:01 Okay, so let's first look at question five.

00:04 So we have a is the adjacency matrix for k, b, and the c, and we have b equals v1 to v20, and c equals v21 to v50.

00:26 So what this means is we have a complete bipartite graph.

00:31 So here we have one, all the way to 20.

00:34 And here we have 21 all the way to 50.

00:41 So every single vertex in b is connected to c is adjacent to the vertices in c, and every vertices in c is adjacent to every vertex in b.

00:57 So a asks, what is the value of a3 comma 4? well, vertex 3, b3 is going to be right there.

01:09 In this set and then v4 is also going to be in that set.

01:13 Since it's a bipartite graph, there's no edges inside the part b, so we know it's going to be zero, because we only put one if there is an edge between these two vertices.

01:26 So b asks, what about a15 -35? well, a15 is going to be, b -15 is going to be around there.

01:36 B -35 is going to be somewhere in here.

01:39 And since it's going to be going to be, complete bipartite graph, we know their existing edge between v15 and v35.

01:46 So this is going to equal 1.

01:51 All right, so let's clear off the screen.

01:54 Question 6 asks, if we have an adjacency matrix for a simple graph g that has exactly 20 ones, how many edges does g have? well, if it's a simple graph, we know there are no loops, which means the main diagonal are all zeros.

02:15 So we know our adjacency matrix looks something like this.

02:24 Now what happens if we have a vertex here? let's call it vi.

02:29 And then we have some vertex over here, vj.

02:32 What happens if there's an edge between them? well, it gets a 1.

02:38 Well, vj is also represented over here and vi is also represented up here.

02:43 And we know there's an edge between them, so it gets a 1.

02:46 So each edge is represented twice in our.

02:51 And since it's simple we know they can only be one and there's no loops so we know that the number of ones corresponds to the sum of the degrees so we know the sum of the degrees so we know the sum of the degrees of all the vertices is equal to 20 and every degree corresponds to one end of an edge and edge has two ends so we know that half of 20 is going to equal the size of the edges of g, which is equal to 10.

03:33 So g has 10 edges, because each edge is represented twice in our adjacency matrix.

03:45 All right, so we'll move on to number seven.

03:48 This one's a little trickier.

03:51 So we'll suppose the g is a simple graph with seven vertices, and we'll suppose that the degree of the i, or, sorry, v1, v3, v5, and v7 are all equal to six.

04:11 Well, what do you notice about this number six? it's one less than a number of vertices in our graph.

04:19 So that means v1, v3, v5, and b7 are adjacent to every vertex in our graph.

04:27 So that means vi is also adjacent to v2, v4, and v6.

04:36 V3 is also adjacent to v2, v4, and v6.

04:41 V5 is also adjacent to v2, v4, and v6...

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