Deck 10: Graphs and Trees

Question 1

Draw a directed graph with the following adjacency matrix:

Answer 1

Question 2

A certain graph has 19 vertices, 16 edges, and no circuits. Is it connected? Explain.

Answer 2

No. The graph is not connected because if it were connected, then, since it is circuit-free, it
would be a tree and would have 18 edges instead of 19 edges.

Question 3

a. Prove that having n vertices, where n is a positive integer, is an invariant for graph
isomorphism.b. Prove that having a vertex of degree 3 is an invariant for graph isomorphism.

Answer 3

Note: An alternative proof for this problem could consider two cases separately. In one case
the vertex v0 in G of degree 3 would have three distinct edges incident on it, and in the other
case there would be one edge and one loop incident on it.

Question 4

Determine whether each of the following graphs has an Euler circuit. If it does have an Euler
circuit, find such a circuit. If it does not have an Euler circuit, explain why you can be 100%
sure that it does not.

Question 5

Find the following matrix product:

Question 6

Question 7

For each of (a)-(c) below, either draw a graph with the specified properties or else explain why
no such graph exists.
(a) Graph with six vertices of degrees 1, 1, 2, 2, 2, and 3.
(b) Graph with four vertices of degrees 1, 2, 2, and 5.
(c) Simple graph with four vertices of degrees 1, 1, 1, and 5.

Question 8

Either draw a graph with the given specification or explain why no such graph exists.
(a) full binary tree with 16 vertices of which 6 are internal vertices
(b) binary tree, height 3, 9 vertices
(c) binary tree, height 4, 18 terminal vertices

Question 9

Consider the adjacency matrix for a graph that is shown below. Answer the following questions
by examining the matrix and its powers only, not by drawing the graph. Show your work in a
way that makes your reasoning clear.

Question 10

Determine whether each of the following graphs has a Hamiltonian circuit. If it does have an
Hamiltonian circuit, find such a circuit. If it does not have an Hamiltonian circuit, explain
why you can be 100% sure that it does not.

Question 11

A certain connected graph has 68 vertices and 72 edges. Does it have a circuit? Explain.

Question 12

Question 13

If a graph has vertices of degrees 1, 1, 2, 3, and 3, how many edges does it have? Why?

Question 14

(a) Use Kruskal's algorithm to find a minimum spanning tree for the graph, and indicate the
order in which edges are added to form the tree.
(b) Use Prim's algorithm starting with vertex a to find a minimum spanning tree for the
graph, and indicate the order in which edges are added to form the tree.
(c) Use Dijkstra's algorithm to find the shortest path from a to