Radio transmitters are located at v1 = (0, 0), v1 = (2,0), v1 = (0, 4), and v4 = (5, 5). Assume that two transmitters must use different frequencies if and and only if the distance between them is less than or equal to 2r. Let D;(r) be a closed disk, of radius r, centered at vi, and let G, be the intersection graph with vertex set V, = {D1(r), D2(r), D3(r), D4(r)}. Two disks are adjacent if and only if their intersection is not empty. %3| v3 v4 v1 v2 (a) Shown above are the disks in V2 and the corresponding graph G2. Write down its adjacency matrix for G2, and determine the chromatic number (i.e. the number of frequencies needed).
Radio transmitters are located at v1 = (0, 0), v1 = (2,0), v1 = (0, 4), and v4 = (5, 5). Assume that two transmitters must use different frequencies if and and only if the distance between them is less than or equal to 2r. Let D;(r) be a closed disk, of radius r, centered at vi, and let G, be the intersection graph with vertex set V, = {D1(r), D2(r), D3(r), D4(r)}. Two disks are adjacent if and only if their intersection is not empty. %3| v3 v4 v1 v2 (a) Shown above are the disks in V2 and the corresponding graph G2. Write down its adjacency matrix for G2, and determine the chromatic number (i.e. the number of frequencies needed).
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:Radio transmitters are located at vl = (0,0), v1 = (2,0), v1
(5, 5). Assume that two transmitters must use different
frequencies if and and only if the distance between them is less than or
equal to 2r. Let D;(r) be a closed disk, of radius r, centered at vi, and let
G, be the intersection graph with vertex set V, = {D1(r), D2(r), D3(r), D4(r)}.
Two disks are adjacent if and only if their intersection is not empty.
(0, 4), and v4
v3
v4
v1
v2
(a) Shown above are the disks in V2 and the corresponding graph G2.
Write down its adjacency matrix for G2, and determine the chromatic
number (i.e. the number of frequencies needed).
(b) Now suppose the transmitters increase their power levels so that
r = 2.5. Draw the new graph G2.5 and determine the number of
frequencies needed.
(c) Optional extra credit: Use the LaTeX package tikz to plot the disks
and the graph.
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