Question

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.

Answer #1

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.

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.

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

Given a tree with p vertices, how many edges must you add
without adding vertices to obtain a maximal planar graph?

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.

Prove that if a graph has 1000 vertices and 4000 edges then it
must have a cycle of length at most 20.

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.)

Show that if G is connected with n ≥ 2 vertices and n − 1 edges
that G contains a vertex of degree 1.
Hint: use the fact that deg(v1) + ... + deg(vn) = 2e

A graph is called planar if it can be
drawn in the plane without any edges crossing. The Euler’s formula
states that v − e + r = 2,
where v,e, and r are the numbers of vertices,
edges, and regions in a planar graph, respectively. For the
following problems, let G be a planar simple graph with 8
vertices.
Find the maximum number of edges in G.
Find the maximum number of edges in G, if G
has no...

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