A matrix M and its 6th power are given below.
Let G be the graph with vertex set {Vi, V2, V3, V4, V5, V6} whose adjacency matrix is M, where vertex Vi corresponds to the ith row and column of M for each i € {1,2, 3, 4, 5,6}.
(i) Draw a picture of G (with the vertices labelled).
(ii) How many walks of length 6 that start and end at V5 are there in G? (iii) How many walks of length 6 that end at V are there in G?
(iv) Is G bipartite? If so, divide the vertices of G into sets A and B such that each edge of G joins a vertex in A to a vertex in B. If not, give an example of an odd length cycle in G.
(v) How many walks of length 660 from V3 to V6 are there in G? Explain your answer.
0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0
36 0 0 0 48 12 0 42 30 42 0 0 0 30 24 30 0 0 0 42 30 42 0 0 48 0 0 0 66 18 12 0 0 0 18 6
M =
M6
