Do the wands in Harry Potter have names?

There is a unique wand for every wizard in the Harry Potter series, by JK Rowling. In her books, each kind of wand consists of a four major components...

1.) Size- each wand has a different length and circumference.

2.) Wood- there are many different kinds. Some examples are birch and vine wood. (I believe there are many others too.)

3.) A core- gives a wand its magical properties. There are three kinds. Phoenix feather, unicorn hair, and dragon heartstring (whatever that is...)

4.) and finally, the wizard who wields it! After all, "the wand choices the wizard!"