A complete graph Gn on n vertices has one edge joining every distinct pair of vertices. (a) Draw

A complete graph Gn on n vertices has one edge joining every...

Key Concepts

Independent Circuits

Independent circuits refer to a set of cycles in a graph that form a basis for the cycle space, meaning every cycle in the graph can be expressed as a combination of these. For a connected graph, the number of independent circuits, also known as the cyclomatic number, is calculated by the formula m - n + 1, where m is the number of edges and n is the number of vertices. This concept is fundamental in network theory and graph algorithms, particularly in the analysis of potential redundancy and connectivity within the graph.

Counting Edges in Graphs

In a complete graph with n vertices, the number of edges is determined by the number of unique pairs of vertices. This is given by the combinatorial formula 'n choose 2', which simplifies to n(n - 1)/2. This count is important for understanding the density and complexity of connections in a complete graph.

Complete Graph

A complete graph is a graph in which every pair of distinct vertices is connected by a unique edge. This means that in a complete graph on n vertices, there is an edge between every possible pair of vertices. It represents the idea of maximum connectivity among a set of nodes and is often denoted by G?.

Graph Orientation

Graph orientation involves assigning a direction to each edge of an undirected graph, thereby converting it into a directed graph or digraph. This process is crucial when studying properties such as flow, connectivity, and incidence matrices in digraphs since the assigned direction can affect the analysis of paths and circuits.

Incidence Matrix

The incidence matrix of a digraph is a matrix that represents the relationship between vertices and edges. For each edge, the matrix indicates which vertex is the tail (entry -1) and which is the head (entry +1). This matrix is a useful algebraic tool for encoding the structure of a directed graph and for solving problems related to flows, connectivity, and network circuits.

Transcript

00:01 We are given a graph in each part, and we are asked to find how many vertices and how many edges these graphs have.

00:09 In part a, we're given the complete graph with n nodes, kn.

00:20 So now in the complete graph, we know that each vertex shares an edge with the n -minus -1 other vertices.

00:43 And so it follows that for all vertices, the degree of the vertices is n minus 1.

01:04 Also, for i get too far, there are n vertices by definition.

01:14 And since the degree of a vertex is n minus 1, we know by a theorem from the book called the handshaking lemma sometimes, but the handshaking theorem, as they call it, it tells us that the sum of the degrees of the vertices for all vertices in the vertex set is going to be equal to 2m for some integer m.

01:50 However, we have that this is the same as the sum for all vertices in v of n minus 1 is equal to 2m, and this tells us that we have n times n minus 1 equals to m so that m which is the number of edges in fact is equal to n times n minus 1 over 2 and so we have this complete graph is going to have n times n minus 1 over 2 edges in part b we are given the cycle with n vertices c n by definition it has n vertices and we have that it also has sort of easy to see.

03:01 Intuitively, you can prove it rigorously if you want, and edges.

03:07 And this is because each vertex has an edge coming off of it.

03:22 And so there's a one -to -one correspondence between the vertices and edges.

03:28 In part c, we are given the wheel with n vertices.

03:36 So this is going to have, well not n vertices, but the wheel n.

03:47 This is going to be the number of vertices in the cycle, cn, which is n vertices, plus one, for the extra vertex in the center.

03:59 So this is going to be n plus one vertices.

04:08 And we have that the number of edges is going to be, well, we have the number of edges from the cycle, which was n edges plus, and then we have that the central vertex is going to have n more edges, or not to each of the n outer vertices.

04:35 And so this graph has two n edges.

04:51 In part d, we are given the complete bipartite graph kmn for some integers m and n.

05:05 And by definition, since we have that v1 has m vertices and n has b2 as n vertices, the total number of vertices is the vertices in v1, m plus the vertices in v2, n, which is m plus n vertices.

05:35 And now thinking about, say, a vertex v1 from the vertex set v1, we know that the vertex set v1, we know that the so the degree of this vertex is going to be, well, there's n vertices in v2, which v1 has to have an edge, exactly one edge with each of them.

06:02 So the degree is going to be n.

06:04 And we have that for all v2s in the vertices set v2.

06:14 It follows that the degree of v2 is going to be, well, for each vertex in v1.

06:23 There has to be an edge to be 2, and so this is going to be m.

06:29 And so it follows that by the handshaking theorem, two times the number of edges, e, i'll call it, is the sum over all vertices in the vertex set v of the degree of the vertex.

07:00 And this is the same since v1 and v2 are disjoint, as the sum of all v1s in v1 of the degree of v v plus the sum for all vs in v2 of the degree of v and we have from earlier that this is going to be the sum for all v's in v1 of n plus the sum over all v's in v2 of m and we have that there are are n vertices in v1 or m vertices in v1 so this is going to be equal to m times n plus and there are n vertices in v2 so this is going to be n times m and so this is going to be two m and therefore we have the number of edges is m times n edges in part e we are given q of n.

08:41 So this is the n cube.

08:48 And we have that this is the graph which represents the two to the n ways to write a binary string of length n.

09:05 And so it follows that there's two to the n vertices...

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