△-free→lEl≤n2/4 Second Proof. (d+d)=∑d品 uv∈E w∈V △-free→du+d,≤n→(d+dw)≤nlE uu∈E Cauchy-Schwarz 4E2 (handshaking） m n网≥d+d)=上e区a- 4E2 m uU∈E m-free 㱺 |E| ≤ n2/4 Second Proof. ￾ uv￾E (du + dv) = ￾ v￾V d2 v u v (du + dv) 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| ￾ ⌅ uv￾E (du + dv) = ⌅ v￾V d2 v ￾ ￾⇤ v￾V dv ⇥2 n = 4|E| 2 n (handshaking)