If G is a connected graph, a centre of G is a vertex v with...

If $G$ is a connected graph, a centre of $G$ is a vertex $v$ with the property that the maximum of the distances between $v$ and the other vertices of $G$ is as small as possible. By successively removing all the end-vertices, prove that every tree has either one centre or two adjacent centres. Give an example of a tree of each type with seven vertices.

Key Concepts

Leaf-Removal Process

The process of successively removing all end-vertices (or leaves) of a tree is a method used to determine its center. By repeatedly eliminating vertices with degree one, the tree shrinks until what remains is either a single vertex or two adjacent vertices. This process underpins the proof that every tree has either one center or two adjacent centers.

Eccentricity and Radius

The eccentricity of a vertex is the greatest distance from that vertex to any other vertex in the graph. The radius of the graph is the minimum eccentricity among all vertices. These concepts are fundamental in identifying the center of a graph, as the vertex or vertices with eccentricity equal to the radius form the center.

Center of a Graph

The center of a graph is defined as the vertex (or vertices) that minimizes the maximum distance (or eccentricity) to all other vertices in the graph. In other words, it is the most 'central' part of the graph, ensuring that from a center vertex, the farthest vertex is as close as possible compared to any other vertex in the graph.

Tree

A tree is a type of connected graph that contains no cycles. Trees have the property that there is exactly one unique path between any pair of vertices, which simplifies many analyses and proofs in graph theory. This unique structure also implies that trees have a specific relationship between the number of vertices and edges and often supports inductive strategies in proofs.

Transcript

00:02 We are given a bipartite simple graph, and we are asked to prove an inequality between the number of edges and the number of vertices of this graph.

00:15 So let g be a simple bipartite graph, let v be the number of vertices of this graph, and e be the number of edges.

00:25 Now, since g is bipartite, it follows that the n vertices, not n vertices, but v vertices, partitioned.

00:41 Into two sets v1 and v2.

00:52 And so if we have that, suppose the set v1 contains n vertices, then it follows that the set v2 will contain total number of vertices v minus the number of vertices in v1, which is n, so v minus n.

01:17 And we have that every vertex in the set v1 can only connect with vertices in v2, and vice versa.

01:25 So it follows that if v1 lies in set v1, then it follows that the degree of v1 must be less than or equal to v minus n.

01:51 And there are n vertices in v1, so the max number of edges, and this is going to be a function that depends on that value n, that we chose for, the size of v1, this is going to be the n vertices, so the number of vertices in v1, n times the maximum number of edges, which is going to be v minus n per vertex.

02:29 So we get n times v minus n.

02:32 And just to simplify this, i'll write this as a function f of n.

02:38 And we want to find an upper bound on the number of edges.

02:42 And to do this, you want to find the maximum number of edges...

Please give Ace some feedback