How do you prove a graph is connected?

Given a graph with n vertices, prove that if the degree of each vertex is at least (n−1)/2 then the graph is connected. The distance between two vertices in a graph is the length of the shortest path between them. The diameter of a graph is the distance between the two vertices that are farthest apart.

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.

Why is a graph connected?

A graph is said to be connected if there is a path between every pair of vertex. From every vertex to any other vertex, there should be some path to traverse. That is called the connectivity of a graph. A graph with multiple disconnected vertices and edges is said to be disconnected.

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.

Can you show that a graph consisting of simply one circuit with n ≥ 3 vertices is 2 Chromatic if’n is even and is 3 Chromatic if’n is odd?

Therefore we see that a graph containing a complete graph of r vertices is at least r-chromatic. For example, every graph containing a triangle is at least 3-chromatic. 6. A cycle of length n ≥ 3 is 2-chromatic if n is even and 3-chromatic if n is odd.

How do you find the number of vertices?

How to Figure How Many Vertices a Shape Has.

How do you know if a graph is connected?

A graph G is connected if there is a path in G between any given pair of vertices, otherwise it is disconnected. Every disconnected graph can be split up into a number of connected subgraphs, called components. Subgraph Let Gbe a graph with vertex set V(G) and edge-list E(G).

How many cut vertices does a connected graph have?

A connected graph ‘G’ may have at most (n–2) cut vertices. In the following graph, vertices ‘e’ and ‘c’ are the cut vertices. By removing ‘e’ or ‘c’, the graph will become a disconnected graph. Without ‘g’, there is no path between vertex ‘c’ and vertex ‘h’ and many other. Hence it is a disconnected graph with cut vertex as ‘e’.

How do you know if a graph is disconnected?

For example, one can traverse from vertex ‘a’ to vertex ‘e’ using the path ‘a-b-e’. In the following example, traversing from vertex ‘a’ to vertex ‘f’ is not possible because there is no path between them directly or indirectly. Hence it is a disconnected graph. Let ‘G’ be a connected graph.

What is meant by connectivity in graph theory?

Graph Theory – Connectivity. Whether it is possible to traverse a graph from one vertex to another is determined by how a graph is connected. Connectivity is a basic concept in Graph Theory. Connectivity defines whether a graph is connected or disconnected.