Pendent Vertex, Isolated Vertex and Adjacency of a graph

In graph theory, vertices are classified based on their degree (the number of edges connected to them). Two special types are pendent vertices (degree 1) and isolated vertices (degree 0). Adjacency describes the relationship between vertices or edges that share a common connection.

Pendent Vertex

A vertex with degree one is called a pendent vertex (also known as a leaf vertex). It has exactly one edge connected to it.

Example

Here, vertex 'a' and vertex 'b' are connected by edge 'ab'. Each vertex has only one edge, so both have degree 1. Both 'a' and 'b' are pendent vertices.

Isolated Vertex

A vertex with degree zero is called an isolated vertex. It has no edges connected to it.

Example

Here, vertices 'a' and 'b' have no connectivity between each other or to any other vertices. The degree of both is zero, making them isolated vertices.

Adjacency

Adjacency describes how vertices and edges are connected to each other −

• Adjacent vertices − Two vertices are adjacent if there is an edge directly connecting them.
• Adjacent edges − Two edges are adjacent if they share a common vertex (endpoint).

Example 1

In the above graph −

• 'a' and 'b' are adjacent vertices, as there is a common edge 'ab' between them.
• 'a' and 'd' are adjacent vertices, as there is a common edge 'ad' between them.
• 'ab' and 'be' are adjacent edges, as there is a common vertex 'b' between them.
• 'be' and 'de' are adjacent edges, as there is a common vertex 'e' between them.

Example 2

In the above graph −

• 'a' and 'd' are not adjacent − there is no direct edge between them.
• 'c' and 'b' are not adjacent − there is no direct edge between them.
• 'ac' and 'cd' are adjacent edges, as there is a common vertex 'c' between them.
• 'ab' and 'bd' are adjacent edges, as there is a common vertex 'b' between them.

Conclusion

A pendent vertex has degree 1 (connected by exactly one edge), an isolated vertex has degree 0 (no edges), and adjacency describes direct connections between vertices (sharing an edge) or edges (sharing a vertex).