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.

Expert Solution

Colby Messinger answered 2 years ago

Show that a graph with at least 2k vertices is k-connected if and only if for...

Show that a graph with at least 2k vertices is k-connected if and only if for any two unrelated subsets X, Y of V, such that | X | = k = | Y |, there are k foreign paths between X and Y.

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.

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?

Show that a graph without isolated vertices has an Eulerian walk if and only if it...

Show that a graph without isolated vertices has an Eulerian walk if and only if it is connected and all vertices except at most two have even degree.

An eulerian walk is a sequence of vertices in a graph such that every edge is...

An eulerian walk is a sequence of vertices in a graph such that every edge is traversed exactly once. It differs from an eulerian circuit in that the starting and ending vertex don’t have to be the same. Prove that if a graph is connected and has at most two vertices of odd degree, then it has an eulerian walk.

(a) Prove the following claim: in every simple graph G on at least two vertices, we...

(a) Prove the following claim: in every simple graph G on at least two vertices, we can always ﬁnd two distinct vertices v,w such that deg(v) = deg(w). (b) Prove the following claim: if G is a simple connected graph in which the degree of every vertex is even, then we can delete any edge from G and it will still be connected.

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.?

let G be a simple graph. show that the relation R on the set of vertices...

let G be a simple graph. show that the relation R on the set of vertices of G such that URV if and only if there is an edge associated with (u,v) is a symmetric irreflexive relation on G

Use the fact that every planar graph with fewer than 12 vertices has a vertex of...

Use the fact that every planar graph with fewer than 12 vertices has a vertex of degree <= 4 to prove that every planar graph with than 12 vertices can be 4-colored.

1. What if only 5 people were at the conference? a. Draw a graph (vertices and...

1. What if only 5 people were at the conference? a. Draw a graph (vertices and edges: pg. 3, example #1) that pictures this situation. b. Explain the total number of handshakes for the 5 people based on the graph. Explain how you use the graph to find the number of handshakes. c. Draw a tree to picture this situation (pg. 4, example #2) d. Explain the total number of handshakes for the 5 people based on the tree. Explain...

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