Apologies for not having replied sooner - it has been hectic here!
Which veggies you can have on a low carb diet depends on which low carb plan you are following, as there are many of them!
If you are following a specific low carb plan (eg the Atkins Diet) the only sure way to know what to do to follow it correctly is to get the diet author's book. Having said that, no diet author can list every possible vegetable and it may be that a non-listed veggie is perfectly OK for that low carb diet.
Where things start to get a little tricky is that different versions of a low carb diet have different carb allowances (whether or not they say so directly or simply list what you should eat without discussing the carb values). So whichever diet you are on, the issue of whether a particular vegetable is OK or not really comes down to one thing - does the amount of it you are likely to eat as a portion keep you within your carb allowance for the meal or for the day? If it is not listed in the diet author's book, it is a simple matter of looking up the carb count.
Also to be factored in is that many low carb diets such as the Atkins Diet start strictly and gradually increase your carbs after a certain period, up to the level where you stop losing weight. Since everyone's carb tolerance is different, a veggie that's OK for one person may not be acceptable for the next, and a veggie that's not OK for you at the start of the diet might be OK at a later stage.
So the bottom line is to know which specific plan you are following and the carb allowance you are aiming for at this particular stage. And if you want to have a vegetable that is not universally listed as low carb, look its carb count up and check whether it will fit into your current carb allowance.
I hope this helps, and good luck with your weight loss journey.