E suppose that g is a graph of order 25 and size 99

(e) Suppose that G is a graph of order 25 and size 99. From
this information, we know that any path in G can be no longer
than what number l? Provide the best upper bound.

TEST1/MAD3305 Page 2 of 4
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2. (20 pts.) Provide mathematical definitions for each of the
following terms.
(a) A graph G:
(b) Subgraph:
(c) Spanning Subgraph:
(d) Bipartite Graph:
(e) Diameter:
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3. (5 pts.) List the r-regular graphs of order 5 for all
possible values of r. They are all old friends.

TEST1/MAD3305 Page 3 of 4
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4. (10 pts.) Use the Havel-Hakimi Theorem to construct a graph
with degree sequence
s: 7,5,4,4,4,3,2,1
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5. (10 pts.) Use the ideas from the proof of Theorem 2.7, to
construct a 3-regular graph G that contains K
3
as an induced
subgraph. Show each stage of the construction.
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6. (5 pts.) Sketch a graph G that has the following adjacency
matrix:
A
G




















0 1 0 1 1 0
1 0 1 0 0 1
0 1 0 1 0 0
1 0 1 0 1 0
1 0 0 1 0 1
0 1 0 0 1 0

TEST1/MAD3305 Page 4 of 4
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7. (5 pts.) Construct a 3-regular graph G of minimum order that
contains C
4
as an induced subgraph. [Use the ideas of Paul Erdos
and Paul J. Kelly.]
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8. (10 pts.) Prove exactly one of the following propositions.
Indicate clearly which you are demonstrating.
(a) If G is a non-trivial graph, then there are distinct
vertices u and v in G with deg(u) = deg(v).
(b) If G is a graph of order n and deg(u) + deg(v) ≥ n - 1 for
each pair of non-adjacent vertices u and v, then G is connected.
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9. (10 pts.) (a) Suppose G is a bipartite graph of order at
least 5. Prove that the complement of G is not bipartite. [Hint:
At least one partite set has three elements. Connect the dots?]
(b) Display a bipartite graph G of order 4 and its bipartite
complement. Label each appropriately and give partite sets for
each bipartite graph.