Advanced Mathematics
Name: ___________________
Graph Theory
5/11/17
A graph is non-empty set of points (called vertices/nodes) and (possible empty) set of edges. Each edge connects two different vertices
or one to itself (called a loop). An edge is called incident to the vertices it connects. Two vertices are adjacent if there is an edge
connecting them. The degree of a vertex is the number of edges incident with it (a loop contributes twice to the degree). A vertex
that has a degree of 0 is an isolated vertex. Two edges are parallel if they connect the same vertices. A graph is called simple if it
contains no loops or parallel edges. A graph is called complete if there is an edge between every pair of vertices, but contains no loops
or parallel edges. A path from vertex v to a not necessarily different vertex w is a sequence of adjacent edges where the first edge is
incident to vertex v and the last edge is incident to vertex w. If a path consists of n edges, then it has length n. A path is called simple
if it does not contain and edge more than once. A simple path is called a cycle or circuit if it begins and ends at the same vertex. A
graph is called connected if there exists a path between any two distinct vertices.
1.
Given each graph.
a.
How many edges does the graph have? Vertices?
b.
What is the degree of each vertex?
c.
Does the graph have any parallel edges? Loops? Isolated vertices?
d.
Is the graph simple?
e.
Is the graph complete?
f.
Is the graph connected?
2.
Let
denote a complete graph with n vertices. How many edges does a
have? How many edges does a
have?
?
3.
If possible, draw a graph to meet each description. Along the way, look for any conjectures.
a.
A graph with 3 vertices of degrees 1, 2, and 3.
b.
A graph with 4 vertices of degrees 1, 2, 3, and 4.
c.
A graph with 4 vertices of degrees 1, 1, 2, and 2.
d.
A graph with 2 vertices of degrees 1 and 2.
e.
A graph with 2 vertices of degrees 1 and 3.
f.
A graph with 3 vertices each of degree 3.
g.
A graph with an odd number of vertices or odd degree.
h.
A graph with 5 edges and the sum of the degrees of the vertices is 8.
i.
A graph with 5 edges and the sum of the degrees of the vertices is 12.
j.
A graph with any number of edges where the sum of the degrees of the vertices is 7.
k.
A complete graph with 7 edges.
l.
A complete graph with 6 edges.
m.
A graph with 2 edges and 1 vertex.
n.
A graph with 2 edges and 3 vertices.
o.
A graph with 2 vertices and 6 edges, exactly 3 are loops.
4.
Draw all possible graphs with 2 vertices and 3 edges.
5.
A simple graph is called bipartite if its vertices can be partitioned into two disjoint non-empty sets, S and T, such that every
edge in the graph connects a vertex from S to a vertex in T. Which of the following graphs are bipartite? If a graph is
bipartite, partition the vertices into two disjoint sets S and T. If the graph is not bipartite, explain why.
6.
Suppose that G is a bipartite graph whose vertices are portioned into two sets S and T where S has m elements and T has n
elements. The G is called a complete bipartite graph if each element of S is connected to each element of T. The graph is
denoted by
. How many edges does
have? How many edges does
have?
7.
What is the maximum number of edges in a simple bipartite graph with n vertices?
8.
If every vertex of a simple graph G has degree n, then G is called an n-regular graph.
a.
Draw two different 2-regular graphs.
b.
Draw three different 3-regular graphs with the same number of vertices.
c.
What is the minimum number of vertices needed to draw a 4-regular graph?
d.
How many vertices does a 4-regular graph with 20 edges have?
9.
Suppose that is G is a connected graph. The distance between vertices v and w is the length of the shortest path from v to
w. The diameter of G = max{distance(v, w): v, w are distinct vertices of G).
a.
Find the diameter of each of the graphs from Exercise 5.
b.
What is the diameter of
?
?