# math algebra questions

Anonymous

### Question Description

Exercise11.1

7. a) How many paths of length 5 are there in the complete

bipartite graph K3,7? (Remember that a path such

as v1 v2 v3 v4 v5 v6 is considered to be the

same as the path v6 v5 v4 v3 v2 v1.)

How many paths of length p are there in the complete bipartite

graph Km,n?

8. LetX _ {1, 2, 3, . . . , n}, where n 2. Construct the loopfree

undirected graph G _ (V , E) as follows:

(V ): Each two-element subset of X determines a vertex

of G.

(E): If v1, v2 V correspond to subsets {a, b} and {c, d},

respectively, of X, draw the edge {v1, v2} in G when

{a, b} ∩ {c, d} _ .

a) Show that G is an isolated vertex when n _ 2 and that

G is disconnected for n _ 3, 4.

b) Show that for n 5, G is connected. (In fact, for all

v1, v2 V , either {v1, v2} E or there is a path of length 2

connecting v1 and v2.)

6. Find all (loop-free) nonisomorphic undirected graphs with

four vertices. How many of these graphs are connected?

Exercise 11.2

13. Let G be a cycle on n vertices. Prove that G is selfcomplementary

if and only if n _ 5.

Exercise 11.3

21. Determine the value(s) of n for which the complete graph

Kn has an Euler circuit. For which n doesKn have an Euler trail

but not an Euler circuit?

Exercise 11.4

14. Determine which of the graphs in Fig. 11.69 are planar. If

a graph is planar, redraw it with no edges overlapping. If it is

nonplanar, find a subgraph homeomorphic to either K5 or K3,3.

Exercise 11.5

7. a) For n 3, how many different Hamilton cycles are there

in the complete graph Kn?

b) How many edge-disjoint Hamilton cycles are there in

K21

c) Nineteen students in a nursery school play a game each

day where they hold hands to form a circle. For how many

days can they do this with no student holding hands with

the same playmate twice?

Exercise  12.1

7. Give an example of an undirected graphG _ (V , E) where

|V | _ |E| + 1 but G is not a tree.

This question has not been answered.

Create a free account to get help with this and any other question!

Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors