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Can I do an arbitrage trade between the two exchanges?

This can be explained with an example: 

Assume Reliance stock is trading on NSE at 1000 and on BSE at 1004, you can do an arbitrage trade for intraday, so you can buy on NSE at 1000 and short on BSE at 1004, when the price difference comes lower than 4 (1004-1000) sell on NSE and buy back the shares on BSE. When you cover, if the difference between NSE and BSE prices is lesser than 4 you make a profit, if not, losses. 

This is an arbitrage trade because it doesn’t matter to you if reliance goes up or down, all you are bothered is about the price difference between Reliance on NSE and BSE.

You can also do such a trade if you already have stocks in your DEMAT. So for example, if you held Reliance in your DEMAT, you could sell it at 1004 on BSE and immediately buy it on NSE for 1000, this way to reduce the cost of your holding.

What you cannot do is - delivery based arbitrage, i.e buying on NSE and Selling on BSE without holding the stock in your DEMAT account.  (CNC orders)

Assume you buy reliance stock on NSE at Rs 1000, the delivery of this stock will come to you only on Wednesday evening (T+2 day), whereas the stock you sold on BSE at 1004, you will have to give delivery of that to the exchange latest by Wednesday morning (T+2). 

But until evening you will not have reliance shares, so on Wednesday morning you will end up with a short delivery and will be taken to auction which can have huge implications in terms of penalty, where you can lose much more than the Rs 4 arbitrage profit you were trying to put the trade for.