INC command in assemblxy x86 : does the INC command used on registers increments by 1 byte or by 4 bytes?

For example, first I set mov ecx,0. and then inc ecx.
what does ecx hold?

example number 2: esi holds an address in memory.
What happens in this case?

I think that in the first case is 4 bytes and in the second its 1 byte (because memory), am I right?