For n EN we define the set Zn = {1, 2,... , n - 1} and we define modular product on this set as follows: for x, y, z E Zn: (x.y = z) e (x.y = z mod n). In other words, we get the number z by calculating the product of the numbers x and y as a common product of two of natural numbers and from this product we then calculate the remainder after dividing by the number n. Examples for n = 5 and different values of x and y: In Z3 :3.4 = 2, 2.3 = 1, 2.4 = 3. .. Assignment: We construct the graph G so that its vertices are elements of the set Z101and two vertices corresponding to the elements x and y will be joined by an edge just when the set Z101 holds: x.y = 1 in terms of the modular product defined above. a) Is the graph G regular? b) Is graph G continuous? c) Is graph G a tree? d) What will be the sum of all numbers in the adjacency matrix of graph G?
For n EN we define the set Zn = {1, 2,... , n - 1} and we define modular product on this set as follows: for x, y, z E Zn: (x.y = z) e (x.y = z mod n). In other words, we get the number z by calculating the product of the numbers x and y as a common product of two of natural numbers and from this product we then calculate the remainder after dividing by the number n. Examples for n = 5 and different values of x and y: In Z3 :3.4 = 2, 2.3 = 1, 2.4 = 3. .. Assignment: We construct the graph G so that its vertices are elements of the set Z101and two vertices corresponding to the elements x and y will be joined by an edge just when the set Z101 holds: x.y = 1 in terms of the modular product defined above. a) Is the graph G regular? b) Is graph G continuous? c) Is graph G a tree? d) What will be the sum of all numbers in the adjacency matrix of graph G?
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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Transcribed Image Text:For n EN we define the set Zn = {1, 2,... , n - 1} and we define modular product on this set
as follows:
for x, y, z E Zn: (x.y = z) (x.V = z mod n).
In other words, we get the number z by calculating the product of the numbers x and y as a common
product of two of natural numbers and from this product we then calculate the remainder after
dividing by the number n. Examples for n = 5 and different values of x and y:
In Z; : 3.4 = 2, 2.3 = 1, 2.4 = 3. ..
Assignment: We construct the graph G so that its vertices are elements of the set Z101and two
vertices corresponding to the elements x and y will be joined by an edge just when the set Z101
holds: x.y = 1 in terms of the modular product defined above.
a) Is the graph G regular?
b) Is graph G continuous?
c) Is graph G a tree?
d) What will be the sum of all numbers in the adjacency matrix of graph G?
The answers to all these questions must be duly substantiated, resp. proven.
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