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Chapter 7 The Mathematics of Networks
 Network
Another name for a connected graph
 Tree
A network with no circuits
 Spanning Tree
A subgraph that connects all the vertices of the network and has no circuits
 Minimum Spanning Tree (MST)
Among all spanning trees of a weighted network, one with the least total weight
Property 1
• In a tree, there is one and only one path joining any two vertices.
• If there is one and only one path joining any two vertices of a graph, then the graph must
be a tree.
Property 2
• In a tree, every edge is a bridge.
• If every edge of a graph is a bridge, then the graph must be a tree.
Property 3
• A tree with N vertices has N – 1 edges.
• If a network has N vertices and N – 1 edges, then it must be a tree.
Property 4
• If a network has N vertices and M edges, then M  N – 1. [R = M – (N – 1) as the
redundancy of the network.]
• If M = N – 1, the network is a tree; if M  N – 1, the network has circuits and is not a
tree. (In other words, a tree is a network with zero redundancy and a network with
positive redundancy is not a tree.
Kruskal’s algorithm
 Pick the cheapest edge available. (In case of a tie, pick one at random.) Mark it.
 Pick the next cheapest link available and mark it.
 Steps 3, 4, …, N – 1. Continue picking and marking the cheapest unmarked link
available that does not create a circuit.
The Shortest Network Rule
 a minimum spanning tree (no interior junction points) or
 A Steiner tree. [A Steiner tree is a network with no circuits (a tree) such that all
interior junction points are Steiner points. A Steiner point of a network is an
interior junction point consisting of three line segments coming together forming
equal 120˚ angles.]
There are three important terms that we will use in connection with junction points:
• In a network connecting a set of vertices, a junction point is said to be native junction
point if it is located at one of the vertices.
• A nonnative junction point located somewhere other than at one of the original vertices is
called an interior junction point of the network.
• An interior junction point consisting of three line segments coming together forming
equal 120˚ angles (a “perfect” Y- junction if you will) is called a Steiner point of the
network.
The Shortest Network Connecting Three Points
•
•
If one of the angles of the triangle is 120˚ or more, the shortest network linking the three
vertices consists of the two shortest sides of the triangle (a).
If all three angles of the triangle are less than 120˚, the shortest network is obtained by
finding a Steiner point S inside the triangle and joining S to each of the vertices (b).
•
The only possible interior junction points in a shortest network are Steiner points. For
convenience, we will call this the interior junction rule for shortest networks.