Question

Graph Theory

.

While it has been proved that any tree with n vertices must have n − 1 edges. Here, you will prove the converse of this statement. Prove that if G = (V, E) is a connected graph such that |E| = |V | − 1, then G is a tree.

Answer #1

To show that graph G is a tree

In lecture, we proved that any tree with n vertices must have n
− 1 edges. Here, you will prove the converse of this statement.
Prove that if G = (V, E) is a connected graph such that |E| =
|V| − 1, then G is a tree.

Let G = (V,E) be a graph with n vertices and e edges. Show that
the following statements are equivalent:
1. G is a tree
2. G is connected and n = e + 1
3. G has no cycles and n = e + 1
4. If u and v are vertices in G, then there exists a unique path
connecting u and v.

A spanning tree of connected graph G = (V, E) is an acyclic
connected subgraph (V, E0 ) with the same vertices as G. Show that
every connected graph G = (V, E) contains a spanning tree. (It is
the connected subgraph (V, E0 ) with the smallest number of
edges.)

Let G be an undirected graph with n vertices and m edges. Use a
contradiction argument to prove that if m<n−1, then G is not
connected

Suppose G is a simple, nonconnected graph with n vertices that
is maximal with respect to these properties. That is, if you tried
to make a larger graph in which G is a subgraph, this larger graph
will lose at least one of the properties (a) simple, (b)
nonconnected, or (c) has n vertices.
What does being maximal with respect to these properties imply
about G?G? That is, what further properties must GG possess because
of this assumption?
In this...

please solve it step by step. thanks
Prove that every connected graph with n vertices has at least
n-1 edges. (HINT: use induction on the number of vertices
n)

Prove that a bipartite simple graph with n vertices must have at
most n2/4 edges. (Here’s a hint. A bipartite graph would have to be
contained in Kx,n−x, for some x.)

Let G be a connected simple graph with n vertices and m edges.
Prove that G contains at least m−n+ 1 different subgraphs
which are polygons (=circuits). Note: Different polygons
can have edges in common. For instance, a square with a diagonal
edge has three different polygons (the square and two different
triangles) even though every pair of polygons have at least one
edge in common.

Call a graph on n vertices dendroid if it has n edges and is
connected. Characterize degree sequences of dendroids.

Graph Theory.
A simple graph G with 7 vertices and 10 edges has the
following properties: G has six vertices of degree
a and one vertex of degree b. Find a and
b, and draw the graph.
Show all work.

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