T is a tree graph on 10 vertices, each is labled with an integer from 1-10....

T is a tree graph on 10 vertices, each is labled with an integer from 1-10. There are four leaves: 1, 2, 3, and 4.

1. What is the number of possible trees, such that the vertices labeled 1, 2, 3, and 4 are leaves, and the other labeled vertices can be either a leaf or not.

1. What is the number of possible trees, such that only 1, 2, 3, and 4 are leaves.

Expert Solution

Colby Messinger answered 2 years ago

Show that a graph T is a tree if and only if for every two vertices...

Show that a graph T is a tree if and only if for every two vertices x, y ∈ V (T), there exists exactly one path from x to y.

2. Let G be a bipartite graph with 10^7 left vertices and 20 right vertices. Two...

2. Let G be a bipartite graph with 10^7 left vertices and 20 right vertices. Two vertices u, v are called twins if the set of neighbors of u equals the set of neighbors of v (triplets, quadruplets etc are defined similarly). Show that G has twins. Show that G has triplets. What about quadruplets, etc.? 3. Show that there exists a bipartite graph with 10^5 left vertices and 20 right vertices without any twins. 4. Show that any graph...

Let T be a graph. Suppose there is a unique path between every pair of vertices...

Let T be a graph. Suppose there is a unique path between every pair of vertices in T. Prove that T is a tree. Can I do this using the contraposative? Like Let u,v be in T and since I am assuming T to not be a tree this allows cycles to occur thus the paths must not be unique. Am I on the right track?

Is it possible for a planar graph to have 6 vertices, 10 edges and 5 faces?...

Is it possible for a planar graph to have 6 vertices, 10 edges and 5 faces? Explain.

Let G be a bipartite graph with 107 left vertices and 20 right vertices. Two vertices...

Let G be a bipartite graph with 107 left vertices and 20 right vertices. Two vertices u, v are called twins if the set of neighbors of u equals the set of neighbors of v (triplets, quadruplets etc are defined similarly). Show that G has twins. Bonus: Show that G has triplets. What about quadruplets, etc.?

Write the code to manage a Binary Tree. Each node in the binary tree includes an integer value and string.

Programming CWrite the code to manage a Binary Tree. Each node in the binary tree includes an integer value and string. The binary tree is sorted by the integer value. The functions include:• Insert into the binary tree. This function will take in as parameters: the root of the tree, the integer value, and the string. Note that this function requires you to create the node.• Find a node by integer value: This function takes in two parameters: the root...

Question 1 a) Prove that if u and v are distinct vertices of a graph G,...

Question 1 a) Prove that if u and v are distinct vertices of a graph G, there exists a walk from u to v if and only if there exists a path (a walk with distinct vertices) from u to v. b) Prove that a graph is bipartite if and only if it contains no cycles of odd length. Please write legibly with step by step details. Many thanks!

Consider the m by n grid graph: n vertices in each of m rows, and m...

Consider the m by n grid graph: n vertices in each of m rows, and m vertices in each of n columns arranged as a grid, and edges between neighboring vertices on rows and columns (excluding the wrap-around edges in the toric mesh). There are m n vertices in total. a)What is the diameter of this graph? b) From the top left vertex to the bottom right vertex, how many shortest paths are there? Please explain.

An m × n grid graph has m rows of n vertices with vertices closest to...

An m × n grid graph has m rows of n vertices with vertices closest to each other connected by an edge. Find the greatest length of any path in such a graph, and provide a brief explanation as to why it is maximum. You can assume m, n ≥ 2. Please provide an explanation without using Hamilton Graph Theory.

Suppose T is a minimal spanning tree found by Prim’s algorithm in a connected graph G....

Suppose T is a minimal spanning tree found by Prim’s algorithm in a connected graph G. (i) Show that either T contains the n-1 smallest edges, or the n-1 smallest edges form a subgraph which contains a cycle. (ii) Show that if an edge (a, b) is not in T and if P is the unique path in T from a to b, then for each edge e in P the weight of e is less than the weight of...

Question