C we have a natural weight function deg on v gn find a

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 3 (or vice-versa), or • u ≡ 2 and v ≡ 1 (or vice-versa). We can therefore use the bipartition A = {u ∈ V (Gn ) | u is even}, B = {v ∈ V (Gn ) | v is odd}. (c) We have a natural weight function deg on V (Gn ). Find a formula for ΦV (Gn ) (x). Be careful, your answer depends on the class of n mod 4. SOLUTION. The solution will of course depend on class of n mod 4. From the previous solution, we see that |{v ∈ {1, . . . , n} | v ≡ 3 mod 4}|, if u ≡ 0 mod 4, |{v ∈ {1, . . . , n} | v ≡ 2 mod 4}|, if u ≡ 1 mod 4, deg(u) = |{v ∈ {1, . . . , n} | v ≡ 1 mod 4}|, if u ≡ 2 mod 4, |{v ∈ {1, . . . , n} | v ≡ 0 mod 4}|, if u ≡ 3 mod 4. Let’s check the various cases. Suppose first that n = 4m. Then all the classes [0], [1], [2], [3] have the same number of elements, m, since [0] = {4, . . . , 4m} [1] = {1, . . . , 4(m − 1) + 1}, [2] = {2, . . . , 4(m − 1) + 2}, [3] = {3, . . . , 4(m − 1) + 3}. Hence all the vertices have degree m, and ΦV (G4m ) (x) = 4mxm . If n = 4m + 1, the class [1]...
View Full Document

This document was uploaded on 03/07/2014.

Ask a homework question - tutors are online