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A. Recall that an undirected graph is connected when it has at least one vertex and there is a path between every pair of vertices. Give a simple rule for determining based on m and k, whether or not Gm.k is connected.

A. Recall that an undirected graph is connected when it has at least one vertex and there is a path between every pair of vertices. Give a simple rule for determining based on m and k, whether or not Gm.k is connected.

Advanced Engineering Mathematics
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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For each pair of natural numbers m and k with m ≥ 2 and 0 <k < m, let Gm.k be the undirected graph whose vertices are the m elements of the ring Nm where N is the set of natural numbers, i.e., {0, 1, 2, 3, .....m}. The edges in the graph consist of all pairs (i, j), where i ≤ {0, 1, 2, ....m} and j= (i+k) mod m. Example: We show two graphs below for G3,1 and G4,2. 2 G 3,1 _i= 0 (initially) 1 Next i = (i+k) mod m, where k = 1 and m = 3 3 G 4,2 2 i = 0 (initially) 1 Next i = (i+k) mod m, where k = 2 and m = 4 A. Recall that an undirected graph is connected when it has at least one vertex and there is a path between every pair of vertices. Give a simple rule for determining based on m and k, whether or not Gm.k is connected. B. Prove that your rule from (D) works. If you use properties seen in class, you must clearly indicate what property it is.
Transcribed Image Text:For each pair of natural numbers m and k with m ≥ 2 and 0 <k < m, let Gm.k be the undirected graph whose vertices are the m elements of the ring Nm where N is the set of natural numbers, i.e., {0, 1, 2, 3, .....m}. The edges in the graph consist of all pairs (i, j), where i ≤ {0, 1, 2, ....m} and j= (i+k) mod m. Example: We show two graphs below for G3,1 and G4,2. 2 G 3,1 _i= 0 (initially) 1 Next i = (i+k) mod m, where k = 1 and m = 3 3 G 4,2 2 i = 0 (initially) 1 Next i = (i+k) mod m, where k = 2 and m = 4 A. Recall that an undirected graph is connected when it has at least one vertex and there is a path between every pair of vertices. Give a simple rule for determining based on m and k, whether or not Gm.k is connected. B. Prove that your rule from (D) works. If you use properties seen in class, you must clearly indicate what property it is.
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