How many graphs on $$n$$ labeled vertices exist which have at least $$\left( {{n^2} - 3n} \right)/2\,\,\,$$ edges?

In an M$$ \times $$N matrix such that all non-zero entries are covered in $$a$$ rows and $$b$$ columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is

If matrix $$X = \left[ {\matrix{ a & 1 \cr { - {a^2} + a - 1} & {1 - a} \cr } } \right]$$
and $${X^2} - X + 1 = 0$$
($${\rm I}$$ is the identity matrix and $$O$$ is the zero matrix), then the inverse of $$X$$ is

The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same colour, is