Graph theory is an important part of mathematics, and the concept of connectivity comes up frequently in graph theory definitions. In this article, we’ll answer the question of what exactly a connected graph is.
A connected graph is one in which every pair of vertices is connected by a path. Two vertices are considered connected if there exists a path that connects them. For example, in a graph with vertices B, C, D, and E, vertices B and E are connected if there exists a path that goes from B to C, then from C to D, and finally from D to E. If there exists at least one such path connecting every pair of vertices in the graph, then it is considered a connected graph.
On the other hand, if a graph has a pair of vertices that are not connected, then it is considered a disconnected graph. For instance, if we add two new vertices, H and I, to the previous example graph, and there is no path connecting vertices E and I, the graph becomes disconnected.
A disconnected graph can be made up of connected components, which are the largest connected subgraphs. For example, in the disconnected graph we just created, there are two connected components: one containing vertices B, C, D, E, and F, and another containing vertices G, H, and I.
Sometimes, it is not immediately obvious whether a graph is connected or not, especially if the drawing of the graph is not clear. In such cases, it is important to check whether there exists a path connecting every pair of vertices.
In conclusion, a connected graph is one in which every pair of vertices is connected by a path. If there exists a pair of vertices that are not connected, then the graph is considered disconnected, and it may be made up of connected components. It is important to carefully examine the graph to determine whether it is connected or disconnected.