How many cycles does a connected graph G with n vertices and n edges have?
Table of Contents
- 1 How many cycles does a connected graph G with n vertices and n edges have?
- 2 How many edges has a complete graph G with n vertices?
- 3 Which of the following is correct about a graph of V vertices and P pendent vertices?
- 4 How many vertices are there in a complete graph with n vertices?
- 5 How many graphs are possible with n vertices and m edges?
- 6 When can a graph G v E with N vertices have chromatic number 1?
- 7 How many vertices can a graph have without a cycle?
- 8 Can a graph with n vertices and (n) edges have a circuit?
- 9 What is the minimum degree of a graph with a cycle?
How many cycles does a connected graph G with n vertices and n edges have?
one cycle
Show that G has exactly one cycle. Let G have n vertices and n edges. Since G is a connected graph, it has a spanning tree T with n vertices and n − 1 edges.
How many edges has a complete graph G with n vertices?
A complete graph has an edge between any two vertices. You can get an edge by picking any two vertices. So if there are n vertices, there are n choose 2 = (n2)=n(n−1)/2 edges.
How do you show that a graph is complete?
To be a complete graph:
- The number of edges in the graph must be N(N-1)/2.
- Each vertice must be connected to exactly N-1 other vertices.
Which of the following is correct about a graph of V vertices and P pendent vertices?
Question 87
| Graph | Vertices | Faces |
|---|---|---|
| A tree on n vertices | n | 1 |
| Cn | n | 2 |
| K4 | 4 | 4 |
| K2 | n+2 | n |
How many vertices are there in a complete graph with n vertices?
Definition: A complete graph is a graph with N vertices and an edge between every two vertices.
What is the edge connectivity of a complete graph with n vertices?
The complete graph on n vertices has edge-connectivity equal to n − 1. Every other simple graph on n vertices has strictly smaller edge-connectivity. In a tree, the local edge-connectivity between every pair of vertices is 1.
How many graphs are possible with n vertices and m edges?
What is the maximum number of simple graphs possible with n vertices and m edges? The number of edges possible in a simple graph with n vertices would be (n2). So the total number of possible graphs would involve the total number of subsets possible out of this which would be 2(n2).
When can a graph G v E with N vertices have chromatic number 1?
Edgeless graphs: If a graph G has no edges, its chromatic number is 1; just color every vertex the same color. These are also the only graphs with chromatic number 1; any graph with an edge needs at least two colors to properly color it, as both endpoints of that edge cannot be the same color.
How many Hamilton circuits are in a graph with 4 vertices?
The complete graph above has four vertices, so the number of Hamilton circuits is: (N – 1)! = (4 – 1)! = 3!
How many vertices can a graph have without a cycle?
Thus, a graph with ’n’ vertices can have maximum n-1 edges (as the first edge requires 2 new vertices and after that at least 1 vertex each edge) without having a cycle (which will be a tree of n vertices).
Can a graph with n vertices and (n) edges have a circuit?
Thus a connected graph of n vertices and (n-1) edges cannot have a circuit. Hence a graph G is a tree. Theorem 5: Prove that a graph with n vertices, (n-1) edges and no circuit is a connected graph.
How do you find the number of edges in a graph?
Hence the number of edges in a graph without cycles is n − k, where k is the number of connected components. Let G be a graph with n vertices and n edges. Keep removing vertices of degree 1 from G until no such removal is possible, and let G ′ denote the resulting graph.
What is the minimum degree of a graph with a cycle?
Note that in each removal, we’re removing exactly 1 vertex and 1 edge, so G ′ cannot be empty, otherwise before the last removal we’d have a graph with 1 vertex and 1 edge, and G ′ has the same number of vertices and edges. Therefore the minimum degree in G ′ is at least 2, which implies that G ′ has a cycle.