△，-free→lEl≤n2/4 Second Proof. ∑(d+d)=∑品 uw∈E w∈V △-free→d,+d≤n→ ∑(d.+d)≤mE uU∈E Cauchy-Schwarz 4E2 (handshaking) m nE≥ 4E2 n2 > E≤ m 4 -free 㱺 |E| ≤ n2/4 Second Proof. ￾ uv￾E (du + dv) = ￾ v￾V d2 v Cauchy-Schwarz ⇤ v￾V d2 v ￾ 1 n ￾⇤ v￾V dv ⇥2 = 4|E| 2 n -free ⇥ du + dv ￾ n ⇥ ￾ uv￾E (du + dv) ￾ n|E| n|E| ￾ 4|E| 2 n (handshaking) |E| ￾ n2 4 u v (du + dv)