Question

Let \(G=(V, E)\) be a connected graph with at least two vertices. Prove that if \(|V|>\) \(|E|,\) then \(G\) has a vertex of degree one.

Short answer

If \\(|V| > |E|\\), then \\ G \\ has a vertex with degree 1.

Step-by-step solution

Understanding Graph Connectivity and Degree

In a connected graph, every vertex is part of a path or connection path to another vertex. The degree of a vertex is defined as the number of edges incident to it. Furthermore, if a graph is connected, no vertex is completely isolated, meaning that each vertex has at least degree one.

Establish the Relationship between Vertices and Edges

Given \(|V| > |E|\), where \(|V|\) is the number of vertices and \(|E|\) is the number of edges, ensure the validity of the inequality. It implies there are more vertices than there are edges, which suggests that the graph cannot be a complete graph because in a complete graph with \(n\) vertices, \(|E|\) would be \(|V|(|V|-1)/2\).

Apply Handshaking Lemma

The Handshaking Lemma states that \( \sum v_i \text{ degree }(v_i) = 2|E| \). This lemma is derived from the fact that each edge contributes two ends, thus counting each edge twice. If the sum of all vertex degrees is less than double the number of vertices, we cannot equally distribute these degrees among the number of vertices.

Conclusion from Handshaking Lemma and Given Inequality

Given \[|V| > |E|\], consider the implications of redistribution of degrees. The Handshaking Lemma tells us \(|E| \<\) total degrees sum, which if adequately distributed, would bring at least one vertex having degree 1 since \(rac{2|E|}{|V|}\) would have to be less than 2, leading to not all vertices being degree 2 or more.

Verification

Verify with simple examples: if there are 3 vertices with 2 edges only possible when one vertex at least is having degree 1 to maintain the connectiveness, e.g., a linear or a chain graph. This supports that one vertex has to have degree 1 when \(|V| > |E|\).

Key concepts

Connected Graph

In the world of graph theory, the concept of a connected graph is fundamental. A connected graph in simple terms is a graph in which there exists a path between every pair of vertices. This essentially means that you can travel from any vertex to any other vertex in the graph without ever "lifting your pen"—or breaking the path.

Being connected implies that no vertex stands alone, ensuring that all vertices are linked directly or indirectly. The graph is "connected" if there are no isolated subgraphs or singular vertices. This characteristic is pivotal in studying and understanding complex networks and is often a prerequisite in many graph-theoretical theorems and problems.

For instance, in a connected graph with a condition like \(\vert V\vert > \vert E\vert\), it is intriguing because it guides us to expect a combination of vertices and edges that prevent the graph from being a complete graph, which requires more edges proportionally.

Vertex Degree

The degree of a vertex in a graph is a key attribute that describes the number of edges connected to that vertex. In more technical terms, the degree is the count of edges that terminate at the vertex. Notably, in a directed graph, one might differentiate between indegree and outdegree, yet in an undirected graph—like the one in this exercise—such distinction isn't necessary.

Understanding vertex degree helps in analyzing the structure of the graph. A graph with a vertex of degree one signifies a certain simplicity, like a leaf in a tree, which connects to the graph via exactly one edge. This concept comes in handy, especially when working with problems concerning network flow, circuit design, or even social networks.

In our context, knowing that \(\vert V\vert > \vert E\vert\), implies certain vertices cannot have very high degrees, because having every vertex connect to others would demand more edges than available. Understanding these degree distributions allows us to hypothesize about the presence of vertices with minimum connections, often leading to some vertices having a degree of one.

Handshaking Lemma

The Handshaking Lemma is a straightforward yet profound concept in graph theory. It essentially states that in any undirected graph, the sum of the vertex degrees is twice the number of edges. This is because each edge contributes to the degree count of two vertices—its endpoints.

Mathematically, this can be expressed as \( \sum_{v\in V} \text{degree}(v) = 2\vert E\vert\). This lemma is helpful in deducing properties about certain kinds of graphs. Given the rule \(\vert V\vert > \vert E\vert\), the application of the Handshaking Lemma leads us to investigate whether all vertices can share a higher degree.

In these cases, the total degrees must be more than \(2\vert E\vert\), and there aren't enough edges available to distribute equally among vertices, which supports the likely conclusion that at least one vertex will have a minimal degree, or degree of one. Such insights showcase the balance between \(\vert V\vert\) and \(\vert E\vert\) and the implications it holds on the graph's architecture.

Vertex-Edge Relationship

The relationship between vertices and edges in a graph offers insights into its overall structure and complexity. Unlike many aspects of graph theory, which can become complex, the vertex-edge relationship can often tell a more straightforward story. By comparing the counts of vertices \(\vert V\vert\) and edges \(\vert E\vert\), we derive crucial characteristics about the graph.

When \(\vert V\vert > \vert E\vert\), it often implies a "sparse" graph; that is, not every vertex is connected to many others. In such graphs, this relationship suggests the absence of high connectivity and indicates that some vertices may only connect to a single edge—resulting in a degree of one.

This particular relationship is crucial when analyzing whether a graph possesses a tree-like structure or several components with minimal connections. Understanding the delicate balance between vertices and edges not only helps in visualizing graphs but also aids in designing efficient algorithms for network flow, data structures, and more encompassing connected structures in graph theory.