The following traits affect the Admiral's ship, so he has to be there:
Analytical Mastermind
Defensive skill
Engineering Genius
Inspirational leader (the +1cq will work if he use a jump point bomb to exit)
Legendary Tactician
Linked targeting
Master Plan
Other traits:
Inspired Command (his ship must be on the table)
Instil fervour (he need not be there)
Lucky (he need not be there)
Master of deception (he need not be there)
Meticulous planner (he need not be there)
Operational control (he need not be there)
Revered Leader (he need not be there)
Top Gun (he need not be there)*
In a campaign, he has to be part of the fleet that is engaged in a battle for his traits to affect that battle, ie his ship has to have started in hyperspace (if allowed) or have withdrawn from the table.
* Matt ruled that Top Gun does not apply to fighters that are detached for a campaign battle, not fighters that are launched from the Admiral's ship in a battle that involves the carrier.