# Edges and vertices relationship tips

### Vertices, Edges and Faces

Relation between the numbers of vertices, edges, faces and components I need to determine an equation relating vertices, edges, faces and components. Hints. If the separate components have v 1, v 2, , v k vertices. In the domain of mathematics and computer science, graph theory is the study of graphs that concerns with the relationship among edges and vertices. Different shapes have different numbers of edges and vertices. Some tips can help you tell the difference between them and to use them as.

Then you have this face right over here, also in the back. The only way we can see this is because they've drawn it so that it is transparent. So that is the second face. Now you have this triangular face on top. So let me color that in. So you have this triangular face on top. So that's going to be our third face, third face. And then you have this triangular face on the bottom.

That's gonna be our fourth face. That's going to be our fourth face.

## Graph Theory - Quick Guide

And then the key question is are we done? Looks like I've colored all the ones that I can see, but there's one a little bit tricky here. There's the one that we are actually seeing through.

**Euler's Formula - 3 Dimensional Shapes**

There is the face, there's, let me pick a color, there's the face out front that we can see through so that we can see faces one, two, and four. So that's actually going to be our fifth face.

The way they've drawn it, it's like it's made out of glass, so we can see faces one, two, and four. But that is our fifth face.

And so this thing has five faces. It is because if any two edges are adjacent, then the degree of the vertex which is joining those two edges will have a degree of 2 which violates the matching rule.

Example M1, M2, M3 from the above graph are the maximal matching of G. Maximum Matching It is also known as largest maximal matching. Maximum matching is defined as the maximal matching with maximum number of edges.

Hence by using the graph G, we can form only the subgraphs with only 2 edges maximum. Hence we have the matching number as two. Perfect Matching A matching M of graph G is said to be a perfect match, if every vertex of graph g G is incident to exactly one edge of the matching Mi. Example In the following graphs, M1 and M2 are examples of perfect matching of G.

### Graph Theory Quick Guide

A maximum matching of graph need not be perfect. If it is odd, then the last vertex pairs with the other vertex, and finally there remains a single vertex which cannot be paired with any other vertex for which the degree is zero.

It clearly violates the perfect matching principle. If G has even number of vertices, then M1 need not be perfect. Hamilton actually created this idea as a sort of puzzle, find the path to connect every vertex on a dodecahedron, visiting every vertex only once, and sold his dodecahedron puzzle to an Irish merchant who began to sell it.

It turns out that it is quite a difficult task to identify whether or not a solid has a Hamiltonian circuit or not, and it may require a proof by exhaustion to determine whether or not a path is possible. However, it has been proven that all Platonic solids, Archimedean solids, and planarconnected graphs have Hamiltonian circuits.

Therefore, by counting the number of edges that we shade red we can determine the number of vertices. To prove this, assume this is not the case. Then there would be a vertex that is not coloured, which means we were not done colouring our edges red. Next we will begin by examining faces.