As I understand it, the Treaty of Verdun partitioned Francia into Western Francia, Middle Francia, and Eastern Francia. The capitals were Paris and Aachen for the first two, and I'm not clear where there was a fixed capital for Eastern Francia.
Aachen was an important city for a very long time, although Middle Francia itself faded in importance within a relatively short amount of time.
There were periods when the Kingdom of France based in Paris were not as strong as other regional powers: the Duchy of Aquitaine, Brittany, Normandy, were quite strong.
Were attempts to usurp Paris as the historic capital of Western Francia?
It is important to note that the historical importance of the treaty of Verdun is often greatly overblown.
Here is a great and easy to understand blog by a Medievalist about this problem.
The Treaty of Verdun (843) was the first of nearly a dozen post-Charlemange partitions of the Frankish Empire that happened over the following 50+ years. And it did not even mark the final partition of the Empire, as the whole empire would be united again under the rule of Charles the Fat in 884.
The suggestion is often made that West and East Francia, and therefore France and Germany, were created at Verdun, but as per above that is really a bit too simple when you go into what as actually going on over those centuries.
For your question it's probably mostly tangential, but I thought it should be noted.
If West Francia became France, and East Francia became the Kingdom of Germany (later Holy Roman Empire), how did "Middle Francia", become the Duchies of Burgundy and Savoy (not to mention the city-states of the Low Countries)?