Mathematics 502
Homework (due Apr 25)
A. Hulpke
42) A graph X = (V, E) is called bipartite if one can partition V = V1 ∪ V2 (V1 ∩ V2 = ∅)
so that the subgraphs induced on V1 or on V2 have no edges.
Show that a graph is bipartite if and only if all cycles have even length.
43) Let Ω = {0, . . . , 7} and V be the set of the 35 partitions of Ω into two sets of size 4.
We consider two elements of V adjacent if and only if the intersection of two 4-sets has size
2. (For example {{0, 1, 2, 3}, {4, 5, 6, 7}} is adjacent to {{0, 1, 4, 5}, {2, 3, 6, 7}}.
a) What is the valency of this graph?
b) Show that the automorphism group of the graph contains a subgroup isomorphic to S8 .
c) Show that the graph is isomorphic to J(7, 3, 1).
44) The diameter of a graph is the maximum distance of two distinct vertices. Determine
the diameter of J(v, k, k − 1) when v > 2k.
45) For a Graph X = (V, E) with vertex set V = {1, . . . , n} we define the adjacency
matrix A = (ai,j) ) by setting ai,j = 1 iff {i, j} ∈ E (Note that the set notation implies that
this is the case iff {j, i} ∈ E, thus A is symmetric.)
Show that the entry (i, j) of Ad counts the number of walks (i.e. following edges but
permitted to go backwards on the same edge you came) from vertex i to vertex j.
47)
Show that J(2k + 1, k, 0) is at least 2-arc transitive.