Saturated configuration and new large construction of equiangular lines

A set of lines through the origin in Euclidean space is called equiangular when any pair of lines from the set intersects with each other at a common angle. We study the maximum size of equiangular lines in Euclidean space and use a graph theoretic approach to prove that all the currently known constructions for maximum equiangular lines in R^d cannot be added by any more lines to form a larger equiangular set of lines when d=14,16,17,18,19, and 20. We give new constructions of large equiangular lines which are 248 equiangular lines in R⁴², 200 equiangular lines in R⁴¹, 168 equiangular lines in R⁴⁰, 152 equiangular lines in R³⁹ with angle 1/7, and 56 equiangular lines in R¹⁸ with angle 1/5.