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Apart from being an academic, what can a Discrete Math PhD do?

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Short answer: Everything.

Long answer:
Mathematicians are often not (solely) hired for the graphs and algorithms they know, but for the skill to logically dissect a problem, come up with a solution, consider all possible cases, etc.

With discrete math, a career in software development would be possible for example. Look for R&D departments or positions and they might prefer a PhD with strong background and some coding skills over someone who has developed for years but is not able to turn a Wiki article on some algorithm into proper code because the math is lacking to understand it.

If you don't mind extending into statistics a little bit, data science and machine learning would open up, where also a lot of mathematicians are wanted currently.

The problem solving skills I mentioned above are also often liked by consultant agencies, where they care not so much about what you know (as you will need to learn and use their products either way mostly) but rather that you can take a client's problem and solve it. Also these guys really like PhDs, as that looks cool and they can bill you out higher than someone without one (even if your PhD is not that relevant to the client).

So, medium long answer: There are a lot of options, you simply need to get out there and look for them. You will need to do some convincing of the form "yes, I am missing these two skills you require, but I can more than make up for it by ...", but then you should be able to get into many fields.

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    There is a separate question however. You speak only of qualifications for various positions. The other consideration is whether there are actual jobs being offered.
    – Buffy
    Commented Feb 10, 2020 at 13:42
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    @Buffy Right, you need to also consider that of course, yes. There are a few jobs that specifically look for mathematicians (with PhD), but there are way more jobs that didn't even think about it, but where there is a chance to convince them with a nice cover letter why your math skills are more valuable than the degrees they put on the job ad.
    – Dirk
    Commented Feb 10, 2020 at 14:56
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    Indeed, the hard problem is to find those opportunities. My experience from industry is that everyone would love to hire the mathematicians for their problem solving skills, but only if their is no other appliciant who actually has some experience/knowledge in e.g. data science/machine learning..
    – user111388
    Commented Feb 10, 2020 at 20:22
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Logistics, routing, navigation systems would fit the bill (I've heard that TomTom has a lot of discrete math people, for example).

You may also move in the direction of discrete optimization and then you may sell yourself doing operations research.

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