Given the spectral graph clustering optimization problemFind y that minimizes y Lysubject toandy¹y = n1¹y = 0,which of the following optimization problems produce a vector y that leads to the same sweep cut as the optimizationproblem above? M is a diagonal mass matrix with different masses on the diagonal.Minimize y™ LyA: subject toandMinimize yO B: subject toandy¹y = 11¹y = 0LyVi, y₁ = 1 or y¡ = -11¹y = 0Minimize y¹ Ly/(yy)subject to1¹y = 0Minimize y¹ LyOD: subject toandy My = 11¹ My = 0

Let's break down the given spectral graph clustering optimization problem: Objective: minimizes…

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Let Tmn be a BTTB matrix with a generating function f(x, y) ∈C2π×2π. Let λmin(Tmn) and λmax(Tmn) denote the smallest and largest eigenvaluesof Tmn, respectively. Then we havefmin ≤ λmin(Tmn) ≤ λmax(Tmn) ≤ fmax,where fmin and fmax denote the minimum and maximum values of f(x, y), respectively. In particular, if fmin > 0, then Tmn is positive definite.
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Let Tmn be a BTTB matrix with a generating function f(x, y) ∈C2π×2π. Let λmin(Tmn) and λmax(Tmn) denote the smallest and largest eigenvaluesof Tmn, respectively. Then we havefmin ≤ λmin(Tmn) ≤ λmax(Tmn) ≤ fmax,where fmin and fmax denote the minimum and maximum values of f(x, y), respectively. In particular, if fmin > 0, then Tmn is positive definite.
Problem 2. The half-clique problem is to determine if a graph of n vertices has a clique of at least n/2 vertices. Prove that half-clique problem is NP-complete via a polynomial time reduction from Clique problem to it.
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