Table of Contents

## How do you prove a graph is connected?

A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected.

**What is the order of graph G?**

The order of a graph G is the cardinality of its vertex set, and the size of a graph is the cardinality of its edge set. Given two vertices u and v, if uv ∈ E, then u and v are said to be adjacent.

### 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.

**Can you have a simple graph with n vertices with the degrees of vertices 0 1 n − 1 if yes construct such a graph for N 5 If not argue why not?**

Since there are n vertices, if they all have different degrees, they must be 0,1,2,…,(n-1). But then we have that the vertex of degree (n-1) must have an edge to all other vertices, and the vertex of degree 0 has no edges. This is a contradiction so no such graph can exist.

#### How do you prove a simple graph does not exist?

If any of the vertices is connected to vertices, then it is connected to all the vertices except from itself. If this were true, no vertex would be disconnected (being connected with 0 other). Therefore, a graph with vertices with a 0 degree vertex and another of degree cannot exist.

**How do you prove that a graph is 2 connected?**

A graph is connected if for any two vertices x, y ∈ V (G), there is a path whose endpoints are x and y. A connected graph G is called 2-connected, if for every vertex x ∈ V (G), G − x is connected.

## How do you determine the order of a graph?

First order, would be natural log of concentration A versus time. If you get a straight line with a negative slope, then that would be first order. For second order, if you graph the inverse of the concentration A versus time, you get a positive straight line with a positive slope, then you know it’s second order.

**How do you determine graph order and size graph?**

Order of a graph is the number of vertices in the graph. Size of a graph is the number of edges in the graph.

### How do you prove a graph is a cycle?

Proof: Let G be a graph with n vertices. If G is connected then by theorem 3 it is not a tree, so it contains a cycle. If G is not connected, one of its connected components has at least as many edges as vertices so this component is not a tree and must contain a cycle, hence G contains a cycle.

**How do you show that a graph is complete?**

In the graph, a vertex should have edges with all other vertices, then it called a complete graph. In other words, if a vertex is connected to all other vertices in a graph, then it is called a complete graph.

#### How do you find the sum of degrees of vertices?

The number of edges connected to a single vertex v is the degree of v. Thus, the sum of all the degrees of vertices in the graph equals the total number of incident pairs (v, e) we wanted to count. For the second way of counting the incident pairs, notice that each edge is attached to two vertices.

**How many edges can a simple graph G on n have if it is not connected?**

Alternate solution There are exactly k(n−k) edges between vertices in the two pieces. If you add them to your graph, you get a simple graph, which by handshaking lemma, has at most n(n−1)2 edges.