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From: ljdickey@watmath.UUCP (Lee Dickey)
Newsgroups: net.math
Subject: A graph theory question
Message-ID: <5448@watmath.UUCP>
Date: Sun, 26-Jun-83 19:51:09 EDT
Article-I.D.: watmath.5448
Posted: Sun Jun 26 19:51:09 1983
Date-Received: Mon, 27-Jun-83 00:26:39 EDT
Sender: ljdickey@watmath.UUCP
Organization: U of Waterloo, Ontario
Lines: 39

About a week or so ago, I passed along an example of two graphs that had the 
same set of leaf distances, but which were not isomorphic.  Here is another pair
of graphs, discovered to Prof. J. Michael Robinson, smaller than the previous
example, which have that same property:


                    o                  o
                    |                  |
       o------------o                  o------------o
       |           / \                 |           / \
       o          o   o                o          o   o
      / \         |   |               / \         |   |
     o   o        o   o              o   o        o   o

Each of these graphs has:
      1 pair  of leaves with distance 2
      2 pairs of leaves with distance 3
      3 pairs of leaves with distance 4
      4 pairs of leaves with distance 5.

Prof. Ronald C. Read has discovered the smallest (counting the number of
vertices) example of a pair of graphs that have the same
distance sequences, but which are not isomorphic:

           o   o                      o o o
            \ /                        \|/
          o--o--o                       o
             |                          |
             o                          o
            / \                         |
           o   o                        o
          /     \                      /|\
         o       o                    o o o

Each of these graphs has:
      6 pairs of leaves with distance 2
      9 pairs of leaves with distance 4.

It is interesting to note that they have different numbers of vertices.