Let D, = (r,f\r* = e = f², rf = fr¯!). Verify that6 * [3 * (r, 0), (f, 0), 3 * (r, 0), (e, 1)]is a Hamiltonian circuit inCay({(r, 0), (f, 0), (e, 1)}:D, OZ).

Answered: Let D, = (r,f\r* = e = f², rf = fr¯!).…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let D, = (r,f\r* = e = f², rf = fr¯!). Verify that
6 * [3 * (r, 0), (f, 0), 3 * (r, 0), (e, 1)]
is a Hamiltonian circuit in
Cay({(r, 0), (f, 0), (e, 1)}:D, OZ).

Expert Solution

Step 1

Given,

$D_{4} = <r, f | r^{4} = e = f^{2}, r f = f r^{- 1}> .$

To prove that

$6 * [3 * (r, 0), (f, 0), 3 * (r, 0), (e, 1)],$

is a Hamiltonian circuit in

Cay $(\{(r, 0), (f, 0), (e, 1)\} : D_{4} \oplus Z_{6}) .$

Step 2

Now we have

$D_{4} = <r, f | r^{4} = e = f^{2}, r f = f r^{- 1}>,$

Then,

$D_{4} = \{e, r, r^{2}, r^{3}, f, r f, r^{2} f, r^{3} f\} .$

Now we draw the $C a y (\{r, f\} : D_{4})$ i.e.

By using the graph, then the path $6 * [3 * (r, 0), (f, 0), 3 * (r, 0), (e, 1)] .$

Therefore the path starts from e with the sequence

$2 * [3 * (r, 0), (f, 0), 3 * (r, 0), (e, 1)]$

is given by

$(e, 0) \to (r, 0) \to (r^{2}, 0) \to (r^{3}, 0) \to (r^{3} f, 0) \to (r^{2} f, 0) \to (r f, 0) \to (f, 0) \to (f, 1)$ .

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Let D, = (r,f\r* = e = f², rf = fr¯!). Verify that 6 * [3 * (r, 0), (f, 0), 3 * (r, 0), (e, 1)] is a Hamiltonian circuit in Cay({(r, 0), (f, 0), (e, 1)}:D, OZ).