Question

Draw all non-isomorphic trees on four nodes. Let T be a tree with average degree α. Determine the number of vertices in T. Let T=(V,E) be a tree. Show that T has a vertex v such that for all e∈E, the component of T−e containing v has at least ⌈∣V∣/2⌉ vertices. Prove that either v is unique or there are two adjacent vertices. Suppose that G be a connected graph with at least one cycle and let C be a cycle in G. Show that the complement of any spanning tree of G includes at least one edge from C. A singles badminton tournament involves 62 people. Using a tree model, determine how many matches will be there. Draw the tree represented by the prüfer code (2,2,4,5,6,6,6) Give a prüfer code representation of the tree given below.

Answer 1

A tree is a type of graph that consists of vertices connected by edges, with no cycles or loops. We can draw all non-isomorphic trees on four nodes, which include a linear tree and a triangular tree. The number of vertices in a tree with average degree α can be calculated using the formula |V| = (2α)/(α+1). Additionally, in a singles badminton tournament with 62 people, a tree model can be used to determine that there will be 61 matches in total.

Step-by-step explanation:

A tree is a type of graph that consists of vertices connected by edges, with no cycles or loops. In this question, we are asked to draw all non-isomorphic trees on four nodes. Non-isomorphic trees are trees that cannot be transformed into each other by rearranging the vertices. There are two non-isomorphic trees on four nodes:

1. Tree 1: Vertices are connected in the shape of a line (linear tree).
2. Tree 2: Vertices are connected in the shape of a triangle (triangular tree).

To determine the number of vertices in a tree T with average degree α, we can use the formula: |V| = (2α)/(α+1), where |V| is the number of vertices. This formula gives the number of vertices based on the average degree of the tree.

To prove that a tree T has a vertex v such that for all e∈E, the component of T−e containing v has at least ⌈∣V∣/2⌉ vertices, we can use the concept of connectivity in a tree. By removing an edge e from the tree T, we divide the tree into two components. One component will contain the vertex v, and the other component will contain the remaining vertices. Since a tree is connected, the component containing v will have at least ⌈∣V∣/2⌉ vertices.

In a singles badminton tournament involving 62 people, a tree model can be used to determine the number of matches. A tree is a useful model in this case because it ensures that every player plays against every other player exactly once. Since there are 62 players, the tree will have 61 edges, and therefore there will be 61 matches in total.