Graphs, relations and matrices

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Graphs, relations and matrices

• Section 7.4

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Overview

• Graph structures are valuable because they can
  represent relationships among pairs of objects,
  and they remain simple in structure even when the
  number of objects is large. In application
  problems, it becomes important to use a computer
  to analyze graph properties, so we need a
  representation of a graph that a computer can
  understand.

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One representation for a graph
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Example
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Practice Problem 1

• Write the adjacency matrices of each of the
  following graphs. Make clear which vertex
  corresponds to which row/column. In each graph,
  what is the value of M12? Of M34? Of M21?

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Directed Graphs

• Note that all of our examples involve symmetric
  matrices because the number in Mab and Mba must
  be the same for any graph. However, in some
  applications, this is not the case.

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Directed Graphs

• A directed graph, like a graph, consists of a set
  V of vertices and a set E of edges. Each edge is
  associated with an ordered pair of vertices
  called its endpoints. In other words, a directed
  graph is the same as a graph, but the edges are
  described as ordered pairs rather than unordered
  pairs.
• If the endpoints for edge e are a and b in that
  order, we say e is an edge from a to b, and in
  the diagram we draw the edge as a straight or
  curved arrow from a to b.
• For a directed graph, we use (a, b) rather than
  a, b to indicate an edge from a to b. This
  emphasizes that the edge is an ordered pair, by
  using the usual notation for ordered pairs.

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More definitions and terms

• A walk in a directed graph is a sequence v1e1v2e2
  . . . vnenvn1 of alternating vertices and edges
  that begins and ends with a vertex, and where
  each edge in the list lies between its endpoints
  in the proper order. If there is no chance of
  confusion, we omit the edges when we describe a
  walk.
• The adjacency matrix for a directed graph with
  vertices v1, v2, . . . , vn is the n n matrix
  where Mij (the entry in row i , column j) is the
  number of edges from vertex vi to vertex vj.

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Example

• Consider a two-player game where there is a
  single pile of 10 stones and each player may
  remove one or two stones at a time on his or her
  turn. If we use a node for each state of the
  game and an edge to denote a move, we get the
  directed graph below

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Questions

1. Why is it important to use a directed graph in
   modeling this game?
2. Find the adjacency matrix for this directed graph.

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Matrix arithmetic

• Matrices can be multiplied and added using some
  standard mathematical rules. The surprising thing
  is that these standard operations, when applied
  to the adjacency matrix of a graph, have a real
  interpretation in terms of the graph properties
  we already know.

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What does M2 represent?

• Given the adjacency matrix M for a directed graph
  G, the arithmetic operation M M has an
  interpretation in G. Lets see if we can figure
  out what it is in this example.

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Matrix multiplication

• To compute the (2,3)-entry in M2, we multiply Row
  2 of M times Column 3 of M as follows

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Matrix multiplication

• If we complete the multiplication (every row
  times every column), we get the result at the
  right. What do these numbers mean in terms of
  the original graph?

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Interpretation of Mk

• Theorem. Let M be the adjacency matrix of a
  directed graph G with vertex set 1, 2, 3, . . .
  , n. The row i, column j entry of Mk counts the
  number of k-step walks from node i to node j in
  the graph G.

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Interpretation of the sum M M2 Mk

• We find the sum of two matrices by adding entries
  in the same position. This gives us the
  following extension of our theorem.
• Corollary. Let M be the adjacency matrix of a
  directed graph G with vertex set 1, 2, 3, . . .
  , n. The row i, column j entry of M M2
  Mk counts the number of walks from node i to node
  j in the graph G of length k or less.

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Example
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Connection to Relations

• Note that a directed graph looks exactly the same
  as a one-set arrow diagram for a relation R on a
  set A. This is no coincidence!!

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Binary Relations, Directed Graphs, and Adjacency
Matrices

• For a relation R on the set A 1, 2, 3, . . .
  , n, the following statements are equivalent for
  all a, b ? A
• (a, b) ? R (which we write sometimes as aRb).
• There is a directed edge from node a to node b in
  the graph of R.
• There is a 1 in the row a, column b entry of the
  adjacency matrix for R.

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Examples
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Solutions
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Boolean Operations and Composition of Relations

• We can find a connection between matrix
  arithmetic and composition of relations as long
  as we use Boolean arithmetic in our
  computations

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Boolean matrix multiplication

• In this example, we multiply A A in the same
  way as before, except we use the Boolean addition
  and multiplication among the entries. To
  distinguish this from A2, we denote this A(2).
  Try it before checking the next slide for the
  answer!

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Boolean matrix multiplication

• Take a moment to find the composition R ? R, and
  write the adjacency matrix for this new relation.
  Compare to A(2)!

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Composition and Boolean matrix multiplication

• Theorem. If R is a binary relation on a set A
  with adjacency matrix M, then the matrix M(2) is
  the adjacency matrix for the relation R ? R on
  the set A.