The graph with correct feasible region of L.P.P. for the constraints $2 x+\mathrm{y} \leqslant 10, \mathrm{y} \leqslant x, \mathrm{y} \leqslant 2, x, \mathrm{y} \geqslant 0$ is $\ldots$

The feasible region is the intersection of all constraint inequalities: $2x + y \le 10$, $y \le x$, $y \le 2$, $x \ge 0$, and $y \ge 0$.

The boundary lines include $2x + y = 10$ through $(0,10)$ and $(5,0)$, $y = x$ through the origin, and $y = 2$ horizontally.

The origin $(0,0)$ satisfies all constraints, confirming the region lies below or on each line in the first quadrant.

The vertices occur at intersections: $(0,0)$, $(5,0)$ from $y=0$ and $2x+y=10$, $(4,2)$ from $y=2$ and $2x+y=10$, and $(2,2)$ from $y=2$ and $y=x$.

These four points form a quadrilateral entirely within the first quadrant, bounded by the axes and the constraint lines.

Option A shows this quadrilateral with vertices at $(0,0)$, $(5,0)$, $(4,2)$, and $(2,2)$, matching the derived feasible region.

\boxed{A}