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Graphs, relations and matrices


... it becomes important to use a computer to analyze graph properties, so we need ... More definitions and terms. A walk in a directed graph is a sequence v1e1v2e2. ... – PowerPoint PPT presentation

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Title: Graphs, relations and matrices

Graphs, relations and matrices
  • Section 7.4

  • 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

One representation for a graph
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?

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.

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
  • 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.

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
  • 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.

  • 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

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

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.

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.

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

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?

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.

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.

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!!

Binary Relations, Directed Graphs, and Adjacency
  • 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.

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

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

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)!

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.
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