Radio transmitters are located at vl = (0,0), v1 = (2,0), v1(5, 5). Assume that two transmitters must use differentfrequencies if and and only if the distance between them is less than orequal to 2r. Let D;(r) be a closed disk, of radius r, centered at vi, and letG, 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 v4v3v4v1v2(a) Shown above are the disks in V2 and the corresponding graph G2.Write down its adjacency matrix for G2, and determine the chromaticnumber (i.e. the number of frequencies needed).(b) Now suppose the transmitters increase their power levels so thatr = 2.5. Draw the new graph G2.5 and determine the number offrequencies needed.(c) Optional extra credit: Use the LaTeX package tikz to plot the disksand the graph.

(Solving the first question as per our guidelines) Solution: The graph is

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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Question

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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