D

Step 1: Calculate the total distance covered by the particle from point A to point E on the hexagon. The length of each side of the regular hexagon can be denoted as 's'. Hence, the total distance from A to E (which covers 3 sides) is 3s.
Step 2: The average velocity is defined as the total displacement divided by the total time taken.
The displacement from A to E can be calculated; since A to E is a straight line through the center of the hexagon, the displacement is equal to the distance between A and E, which simplifies to 's * sqrt{3}'.
Step 3: The time taken can be calculated using the speed 'v', as time is distance divided by speed. Thus, time = (3s) / v.
Step 4: Average velocity can therefore be computed as:
\( \text{Average Velocity} = \frac{\text{Displacement}}{\text{Time}} = \frac{s\sqrt{3}}{(3s/v)} = \frac{v\sqrt{3}}{3}. \)
Upon comparing it to the given options, the average velocity corresponds to Option D: Both.