Let T:V 6V be the adjacency operator of the Petersen graph as illustrated in the enclosed file. Here V is the vector spa

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Let T V 6v Be The Adjacency Operator Of The Petersen Graph As Illustrated In The Enclosed File Here V Is The Vector Spa 1 (86.93 KiB) Viewed 26 times

Let T:V 6V be the adjacency operator of the Petersen graph as illustrated in the enclosed file. Here V is the vector space of all formal real linear combinations of the vertices V1 , ..., V10 of the Petersen graph. 1. Show that the subspace spanned by u = V1 + V2 + V3 +34 + Vs and w = vg + % + V8 + vg + v10 is stable under T, by calculating T(u) and T(w) explicitly. 2. Use part 1 to calculate (by hand) an eigenvector for T satisfying T(U) = 3v, and an eigenvector satisfying T(v) = v. 3. (Extra credit). Use Pari to compute the spectrum of the adjacency operator T of the Petersen graph. What do you observe? 6 1 17 to UTC 8