In the 1970 movie TORA! TORA! TORA!
, which was about the Japanese attack on Pearl Harbor, the movie depicts (what starts out as) a peacetime U.S. Navy with Vice Admiral William F. "Bull" Halsey
(portrayed by James Whitmore) in command of one battle group centered around the aircraft carrier U.S.S. Enterprise, and one of his colleagues being Rear Adm. John H. Newton (portrayed by Ken Lynch, who also portrayed Chief Vanderberg in "Devil in the Dark") commanded the battlegroup centered around the carrier U.S.S. Lexington, while Admiral Husband Kimmel
(Martin Balsam) served as CINC-Pac at Pearl.
Has anyone tracked the names and ranks of cannon Federation starship-of-the-line C.O.s seen in TOS? There seemed to be quite a few commodores assigned to Constitution-class vessels.
Off the top of my head, I can think of Wesley and Decker. It could be argued that Wesley was a taskforce CO, although he sat in the Lexington's center seat. I always supposed that Connies were commanded by captains and commodores because they were the Federation's most elite ships. (At least in TOS.)
Whether this arrangement had something to do with the commodore's Connie serving as the immediate flagship of a larger squardon or wing never occurred to me.
The term "commodore" as a U.S. naval rank was abolished in the 1980's. The O-7 naval rank for one-star admirals has since been re-labled "Rear Admiral, Lower Half". O-8 two -star admirals, originally called "Rear Admiral", are now called "Rear Admiral, Upper Half". All of which is silly, of course, just for rare one-star admirals to avoid be addressed with the name of a defunct brand of computer from that same era.
You can review U.S. military ranks and insignia here
FWIW, I prefer to regard the TOS arrowhead badges as signifying the First Fleet, the largest and most central unit in the Federation. Other badges, such as Decker's pretzel-shaped insignia, could be other units, such as the Fourth Fleet or whatever.