The maximum sum of the size of all intersections within intersecting families and crossing-intersecting families

Let ω(F)=∑_,B\|A∩ B| and ω(A,B)=∑(A,B)∈ A× B|A∩ B|. A family F is intersecting if F₁∩ F₂≠ ∅ for any F₁,F₂ and two family A and B are crossing-intersecting if A∩ B≠ ∅ for any (A,B)∈ A. For an intersecting family F, Erd{o}s, Ko and Rado determined the upper bound of |F|, consequently yielding an upper bound of |F|2=∑_,B\1. If we replace $1$ with |A∩ B| in the summation ∑_,B\1, then this summation transforms into ω(F). In this paper, for an intersecting family F, we determine the upper bound of ω(F), which is a generalization of Erd{o}s-Ko-Rado Theorem. Further, for crossing-intersecting families A and B, we determine the upper bound of ω(A,B).