The vertex in a graph with degree one is known as ______.
pendant vertex
Understanding Vertex Degree in Graphs
In graph theory, a graph consists of vertices (or nodes) and edges that connect these vertices. A fundamental property of a vertex is its degree. The degree of a vertex is a count of the number of edges connected to it. Understanding vertex degrees is crucial for analyzing graph structures.
What is Vertex Degree?
The degree of a vertex \(v\), denoted as \(\deg(v)\), is the number of edges incident to it. In a simple graph (which contains no loops and no multiple edges between the same pair of vertices), each edge adds exactly one to the degree of each of the two vertices it connects. If a vertex has a loop (an edge connecting a vertex to itself), the loop usually contributes 2 to the degree of that vertex.
Analyzing Vertices by Degree
Vertices in a graph can be classified based on their degree:
- Degree Zero: A vertex with a degree of zero is not connected to any other vertex by an edge.
- Degree One: A vertex with a degree of one is connected to exactly one other vertex.
- Degree Greater Than One: Vertices with degrees greater than one are connected to two or more other vertices.
Identifying the Vertex with Degree One
The question asks for the specific name given to a vertex in a graph that has a degree of one. Let's look at the options provided and their definitions in graph theory:
- pendant vertex: A pendant vertex (also known as a leaf vertex) is a vertex with degree one. It is connected to only one other vertex.
- loop vertex: This term is not standard. A vertex with a loop edge might be involved, but a loop increases the degree by 2 (in simple graphs) or 1 (in multigraphs depending on definition), and doesn't define a vertex by having degree one.
- external vertex: This term is not a standard definition based on vertex degree in general graph theory. It might relate to specific types of graphs like trees or planar graphs, but not universally based on degree one.
- isolated vertex: An isolated vertex is a vertex that has a degree of zero. It is not connected to any other vertex.
Based on these definitions, the term specifically used for a vertex with degree one is a pendant vertex.
Comparison of Vertex Types by Degree
| Vertex Type | Degree | Description |
|---|---|---|
| Isolated Vertex | \(\deg(v) = 0\) | No edges connected to it. |
| Pendant Vertex (Leaf Vertex) | \(\deg(v) = 1\) | Connected by exactly one edge. |
| Non-Pendant Vertex | \(\deg(v) > 1\) | Connected by two or more edges. |
Therefore, a vertex in a graph with degree one is known as a pendant vertex.
Revision Table: Key Graph Terms
| Term | Definition | Relevant Degree |
|---|---|---|
| Vertex | A node in a graph. | N/A (is a component) |
| Edge | A connection between two vertices. | Impacts vertex degree |
| Degree of a Vertex | Number of edges connected to a vertex. | \(\deg(v)\) |
| Isolated Vertex | Vertex with degree 0. | \(\deg(v)=0\) |
| Pendant Vertex | Vertex with degree 1. | \(\deg(v)=1\) |
| Loop | An edge connecting a vertex to itself. | Adds to degree |
Additional Information: Graph Theory Concepts
Graph theory is a major area in discrete mathematics with many practical applications in computer science, engineering, and social sciences. Understanding basic concepts like vertices, edges, and degrees is fundamental to studying more complex graph properties and algorithms. The sum of the degrees of all vertices in a graph is equal to twice the number of edges (Handshaking Lemma).
Pendant vertices are particularly important in certain graph structures, such as trees, where they are also referred to as leaves. In trees (connected acyclic graphs), every tree with more than one vertex has at least two pendant vertices.
Important Questions from Graph Theory
Which of the following statement(s) is/are correct regarding about the undirected graph?
I. Number of odd degree vertices is even.
II. Sum of degrees of all vertices is even.
A principal node is a
Let G be a simple undirected planar graph on 10 vertices with 15 edges. If G is a connected graph, then the number of bounded faces in any embedding of G on plane is equal to_________.
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Let $G(V, E)$ be an undirected and unweighted graph with 100 vertices. Let $d(u, v)$ denote the number of edges in a shortest path between vertices $u$ and $v$ in $V$. Let the maximum value of $d(u, v)$, $u, v \in V$ such that $u \neq v$, be 30. Let T be any breadth-first-search tree of G. Which ONE of the given options is CORRECT for every such graph G?
