Graphs and Trees

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Graphs and Trees

• This handout
• Terminology of Graphs
• Applications of Graphs

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The process of mathematical reasoning

• We considered numbers and their properties
• Next Graphs, their properties and applications

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Terminology of Graphs

• A graph (or network) consists of
• a set of points
• a set of lines connecting certain pairs of the
  points.
• The points are called nodes (or vertices).
• The lines are called arcs (or edges or links).
• Example

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Graphs in our daily lives

• Transportation
• Telephone
• Computer
• Electrical (power)
• Pipelines
• Molecular structures in biochemistry

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Terminology of Graphs

• Each edge is associated with a set of two nodes,
  called its endpoints.
• Ex a and b are the two endpoints of edge e
• An edge is said to connect its endpoints.
• Ex Edge e connects nodes a and b.
• Two nodes that are connected by an edge are
  called adjacent.
• Ex Nodes a and b are adjacent.
• An edge is said to be incident on each of its
  endpoints.
• Two edges incident on the same endpoint are
  called adjacent.
• Ex. Edges e and f are adjacent.

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Terminology of Graph degrees

• The degree of node v, denoted deg(v), equals the
  number of edges that are incident on v.
• Ex deg(a) deg(c) 1, deg(b) 2
• The total degree of a graph is the sum of the
  degrees of all its nodes.
• Ex The total degree of the graph below
• is 121 4 .

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Graph properties

• Theorem If G is any graph, then the total degree
  of G equals twice the number of edges of G
• the total degree of G 2 (the number of edges
  of G)
• Corollary 1 The total degree of a graph is even.
• Corollary 2 In any graph there are an even
  number of vertices of odd degree.
• Application to an Acquaintance Graph
• Is it possible in a group of five people
• for each to be friends with exactly three
  others?

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Terminology of Graph Paths

• A path between two nodes is a sequence of
  distinct nodes and edges connecting these nodes.
• Example
• Two nodes are called connected if there is a path
  between them.
• Fact For any two nodes a and b of a graph, there
  is an efficient way to determine whether a and b
  are connected or not.

a
b
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An application of graphs in solving a puzzle

• From an initial position on the left bank of a
  river,
• a ferryman wants to transport
• a wolf, a goat, and a cabbage to the right
  bank.
• Ferrymans boat is only big enough
• to transport one object at a time, other than
  himself.
• For obvious reasons,
• the wolf cannot be left alone with the goat
• the goat cannot be left alone with the cabbage.
• How should the ferryman proceed?

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An application of graphs in solving a puzzle

• To solve the puzzle, create the following graph
• Create a node for each allowable arrangement.
• E.g., ( fg wc ) is an allowable arrangement
• since the ferryman and the goat are on the left
  bank,
• and the wolf and the cabbage are on the right
  bank.
• Create an edge between two nodes if it is
  possible to go from the arrangement of one node
  to the arrangement of the other node by a single
  ferry trip.
• E.g., there is an arc between nodes ( fgw c )
  and ( w fgc ) because the transition from the
  first node to the second node can be realized
• by a single trip of the ferryman with the goat
• from the left bank to the right bank.

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An application of graphs in solving a puzzle

• The resulting graph is
• To transport everything from the left bank to the
  right bank, we need to find a path from node (
  fwgc ) to node ( fwgc ) in the graph.
• There are two this kind of paths. One of them
• (fwgc ) ? (wc fg) ? (fwc g) ? (w fgc) ?
  (fwg c) ?
• (g fwc) ? (fg wc) ? ( fwgc)

fwgc
fwg c
fwc g
fgc w
fg wc
c fwg
fwgc
wc fg
w fgc
g fwc