Math 575
Problem Set 5
1. Suppose that G is a graph with V(G) = {v1, ,v2 ,…,vn }. For each i = 1,2, …, n, let ti
denote the number of triangles in G ! vi . Express t, the number of triangles of G,
in terms of n and the ti ' s . (Note: A triangle in a graph G is a set of 3 mutually
adjacent vertices.
! n$
So, for example, Kn contains # & triangles.
" 3%
Hint: How often does a given triangle get counted in the sum of the ti ' s ? Try
your answer on some point-deleted graphs of your choice.
Answer: For the triangle K consisting of the vertices v, u, w, will get counted in
every vertex-deleted subgraph except for G – v, G – u and G – w.
n
n
So K gets counted exactly n – 3 times in the sum, ! t k . So we get, t =
k =1
!t
k
k =1
n" 3
.
2. Prove: If G is a regular graph, then G is completely determined by its pointdeleted subgraphs.
Solution:
Proof. Since We can always determine the degrees of the vertices from the pointdeleted subgraphs, we can determine that G is regular and what the common
degree of the vertices is. Suppose that we determine that G is regular of degree r.
Suppose that G – v is any one of the vertex-deleted subgraphs. Then it must be
that v is adjacent to precisely the vertices of degree r – 1 in G – v.
3. Show that if G is a connected graph that is not complete, then there exists a set S
of vertices of G such that G – S is not connected.
Solution: If G is not complete, then there exist two vertices v and u that are not
adjacent. So, now just let S = V (G) ! {v,u} .
4. Show that if G is a non-trivial, connected graph with no induced path on four
vertices, then the complement of G is not connected.
Hint: This is clearly true for complete graphs. So suppose that G is not complete
and consider a minimal set S such that G – S is not connected.
5. Prove or disprove:
(a). If two graphs have the same degree sequence, then they are isomorphic.
(b). If two trees have the same degree sequence, then they are isomorphic.
6. Show that the two graphs below are not isomorphic.
Answer: The left hand graph is bipartite but the right hand graph is not.
7. What can you say about a graph that is bipartite and its complement is also
bipartite? In particular, what is the maximum number of vertices that G could
have? Solution: G can have at most 4 vertices.
8. (a). How many different isomorphisms are there from one path on n vertices to
another path on n vertices? Answer: 2
(b). How many different isomorphisms are there from one cycle on n vertices to
another cycle on n vertices? Answer: 2n
9. (a). How many spanning trees are there for the graph below?
Answer: We must remove exactly one of the 4 edges in the lower 4-cycle. For
the upper portion, consider the case where the edge fg is in the tree and the case
where it is not. There are 27 trees that span the upper portion and so there are
27 ! 4 = 108 spanning trees altogether.
(b). How many spanning trees are there for the graph below?
Hint: Consider several cases, perhaps involving the middle two edges.
Answer: Consider the cases where neither of cd and ef are in the tree, exactly
one of cd and ef is in the tree, and where both of cd and ef are in the tree. There
are 56 spanning trees.
10. How many distinct trees T on {1, 2, 3, …, n} n ≥ 5 have degT (1) = 2, and deg T (2) = 3 .
" n ! 3%
Answer: $
( n ! 2 )n ! 4
'
# 2 &