For n ∈ N we define the set Zn = {1, 2, . . , n - 1} and on this set we define the modular product as follows: For x, y, x ∈ Zn : (x.y = z) ⇔ (x.y ≡ z mod n). In other words, we get the number z by computing the product of x and y as the ordinary product of two natural numbers and then calculating the remainder of this product after dividing by n. Examples for n = 5 and different values of x and y: in Z5 : 3.4 = 2 , 2.3 = 1 , 2.4 = 3 . . . Problem: We construct a graph G such that its vertices are elements of the set Z101 and the two vertices corresponding to the elements x and y are connected by an edge if and only if on the set Z101 : x.y = 1 in the sense of the modular product defined above. a) Is the graph G ordinary? b) Is the graph G continuous? c) Is the graph of G a tree? d) What will be the sum of all the numbers in the adjacency matrix of graph G?
For n ∈ N we define the set Zn = {1, 2, . . , n - 1} and on this set we define the modular product as follows:
For x, y, x ∈ Zn : (x.y = z) ⇔ (x.y ≡ z mod n). In other words, we get the number z by computing the product of x and y as the ordinary product of two natural numbers and then calculating the remainder of this product after dividing by n.
Examples for n = 5 and different values of x and y: in Z5 : 3.4 = 2 , 2.3 = 1 , 2.4 = 3 . . .
Problem: We construct a graph G such that its vertices are elements of the set Z101 and the two vertices corresponding to the elements x and y are connected by an edge if and only if on the set Z101 : x.y = 1 in the sense of the modular product defined above.
a) Is the graph G ordinary?
b) Is the graph G continuous?
c) Is the graph of G a tree?
d) What will be the sum of all the numbers in the adjacency matrix of graph G?
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