I am confused if this is a valid DFA or not.
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Yes. That one state must be the initial state. If it is also an acceptor state, the DFA accepts the language $\{\epsilon\}$; if not, the DFA accepts the language $\varnothing$. Both of these are languages over any alphabet, including the empty alphabet.
Brian M. Scott
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Thank you! Can a NFA have one state and an empty alphabet as well? – newarsenic Oct 03 '20 at 01:14
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@newarsenic: Yes: DFAs are just a special case of NFAs, so every DFA is technically also an NFA. You’re welcome! – Brian M. Scott Oct 03 '20 at 01:41