CMPSCI 250 Discussion #9: Paths and Matrices

David Mix Barrington

24/26 April 2006

We put a directed multigraph on the board with five nodes {1,2,3,4,5} and eight edges {a from 1 to 2, b from 1 to 2, c from 2 to 4, d from 3 to 5, e from 4 to 3, f from 4 to 5, g from 5 to 1, and h from 5 to 4}.

• Question 1: Find all four-step paths out of vertex 1: aced, acfg, acfh, bced, bcfg, and bcfh. Find all four-step paths from vertex 5 to itself: gach, gbch, hfhf.
• Question 2: Write the matrix A of naturals for this, compute A⁴, and compare the answer to Question 1.

  A = 0 2 0 0 0      A^2 = 0 0 0 2 0   A^4 = 2 0 0 2 2
      0 0 0 1 0            0 0 1 0 1         1 2 1 1 1
      0 0 0 0 1            1 0 0 1 0         1 0 0 3 1
      0 0 1 0 1            1 0 0 1 1         1 2 1 3 2
      1 0 0 1 0            0 2 1 0 1         1 2 3 1 3
  

  The first row of A⁴ adds to 6, which matches the six paths out of vertex 1 found in Question 1. The lower-right corner of A⁴, the entry A_5,5, is 3, which matches the three paths from vertex 5 to itself found in Question 1.

  How many total four-step paths are in the graph? The sum of all 25 entries in the matrix A⁴, which is 36.

• Question 3: Determine the path relation for G, that is, for which pairs of vertices i and j is there a path (of any length) from i to j? Can you justify your conclusion without listing all the paths?

  There is a path from i to j for every pair i and j. There are several ways to see this:

  • The graph has cycles that visit all vertices, such as the paths acedg or bcdeg. You can get from any vertex to any other vertex simply by following this cycle for long enough.
  • The matrix A⁴ has only four zero entries. This means that there is a four-step path from i to j unless the pair (i,j) is (1,2), (1,3), (3,2), or (3,3). So we can just find paths for these four pairs: a, ace, dga, and λ, and we know paths for all 25 pairs exist.
  • It turns out that A⁵ has no nonzero entries. So not only is there a path of some length from every i to every j, there is always a path of length 5.
  • The calculation in Question 4 also shows that paths with all possible start and finish vertices exist.
• Question 4: Let B be the boolean adjacency matrix of G, where the entry B_i,j is 1 if there is at least one edge from i to j and is 0 if there is not (except that we set B(i,i) to 1 for each i -- thus "B" is the matrix called "B+I" in lecture). Write down B and compute B⁴ using boolean matrix multiplication, where 1 + 1 = 1. How does your answer relate to Question 3?

  B = 1 1 0 0 0      B^2 = 1 1 0 1 0   B^3 = B^4 = 1 1 1 1 1
      0 1 0 1 0            0 1 1 1 1               1 1 1 1 1
      0 0 1 0 1            1 0 1 1 1               1 1 1 1 1
      0 0 1 1 1            1 0 1 1 1               1 1 1 1 1
      1 0 0 1 1            1 1 1 1 1               1 1 1 1 1
  

  We can think of the matrix B^k as indicating whether there is a path of length exactly k from i to j in the graph made from G by adding a loop at each vertex. This is true if and only if there is a path in G of length k or less. Thus our matrix of all ones matches our conclusion in Question 3 that the path relation is always true. The fact that B³ = B⁴ for this graph means that here, there is always a path of length at most three from i to j. In any graph of n vertices, as we discussed in lecture, there is a path from i to j at all if and only if there is a path of length at most n-1.

  Last modified 27 April 2006