Let G be an undirected graph with a loop at every vertex....

Let $G$ be an undirected graph with a loop at every vertex. Show that the relation $R$ on the set of vertices of $G$ such that $u R v$ if and only if there is an edge associated to $\{u, v\}$ is a symmetric, reflexive relation on $G .$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Symmetric Relation

A symmetric relation requires that if one element is related to another, then the second element is also related to the first. In undirected graphs, the inherent nature of the edges ensures that if there is an edge between two vertices, the relation is automatically symmetric.

Reflexive Relation

A reflexive relation is one where every element in the set is related to itself. In graph terms, this means every vertex has a self-loop, guaranteeing that the relation holds for every vertex with itself.

Undirected Graph

An undirected graph is a collection of vertices connected by edges that do not have a direction. This means that an edge connecting two vertices indicates a mutual relation between them, which is fundamental when defining symmetric relations in graph theory.

Graph with Loops

A graph with loops includes an edge that connects a vertex to itself. Such loops are critical when considering reflexive properties in a relation, ensuring that every vertex is related to itself.

Relation

A relation on a set is a rule that assigns pairs of elements from the set. In the context of graphs, the relation is typically defined based on the existence of an edge connecting two vertices.

Transcript

Please give Ace some feedback