Mihawk sees and recognizes Luffy's potential. During the battle at Marineford, he expresses to himself that Luffy has the most dangerous ability on the seas, which is turning anyone around him into his allies, including former enemies.
Mihawk said that the true power of Luffy was not his physical strength or his Devil Fruit or Haki, but his power to make people his allies so easily.
Mihawk doesn't seem interested in having a crew and he is probably the only Shichibukai, who was happy that the organization was dissolved. Even though he respects Luffy's determination, he probably won't join the Grand Fleet.
Another possibility, Mihawk wanted to see Luffy's strength firsthand who managed to defeat strong fighters such as Crocodile, Luci, and Moria. Mihawk might not be surprised if Luffy managed to beat them in a one-on-one fight, but Crocodile has Baroque Works, Lucci has the Marines, and Moria has an army of zombies.
Mihawk actually genuinely wasn't interested in fighting Shanks and was pretty much pitying him because he was handicapped and knows that he would win.
Such an example is when Zoro challenged him to fight at Baratie, and even though he lost, Mihawk was impressed enough to spare him and declare his respect (although indirectly). Despite that, however, he does not seem to have anyone whom he is particularly fond of.
Dracule Mihawk
Though antagonistic towards Luffy during the Summit War of Marineford, he agreed to train Zoro in swordsmanship during the two-year timeskip.
Mihawk is said to be close to the same level as Shanks and is even dubbed as the greatest swordsman, so to say Luffy would defeat Mihawk is still debatable in the eyes of many fans.
Mihawk was purely neutral. He did not fight for right or wrong, good or evil, but merely acting in his own good side. Mihawk also shows curiosity for the people he decides to attack. He attempted to strike Whitebeard just to see what the power distance between himself and Whitebeard was.
It is one of few One Piece pairings where there has been a substantial amount of time spent together, as Mihawk and Perona have been by each other's side now for at least two years. Their moments can be seen closely as husband and wife, with how Perona tries to care for Mihawk and how he has shown worry for her safety.
He is the World's Greatest Swordsmen, meaning that he's famously outclassed several of One Piece's infamously overpowered sword fighters. In addition to delicate skill, Mihawk also has the devastating power necessary for taking Boa Hancock out with one slice.
Mihawk's strength is so great that Don Krieg accuses him of having the power of a Devil Fruit. While it has not yet been officially confirmed that he does not, it has only been indicated that his abilities are from pure skill and power.
Kozuki Oden and Roronoa Zoro both gave him permanent scars with their swords attacks. Keep in mind that neighter fighters have unlocked the ability to forge "black blades." Mihawk should be strong enough to scar Kaido as well. It should also be noted that Kaido considers Shanks to be his equal.
Luffy frees himself from the rubble of the collapsed wall and begins to attack Mihawk with a Gomu Gomu no Jet Bazooka, but he foresees both his arms getting cut off by Mihawk's sword, prompting him to rapidly redirect his arms.
Because he recognizes and decides to reward Zoro's Loyalty to Luffy. What Zoro did, Begging Mihawk to train him isn't something Zoro would have done for himself. Zoro chooses to forsake his pride as a swordsmen to get stronger for Luffy's sake and Mihawk admires that.
Mihawk is Rayleigh's son | Fandom.
Over the years, Shanks has grown into one of the most powerful people in One Piece. Even if he has one arm less, it hasn't hindered his progress at all. Shanks has managed to surpass Mihawk, and he is perfectly capable of beating him.
Abilities and Powers
Perona is physically strong, as she was seen carrying Kumashi's body, which is bigger than she is, while levitating with her Devil Fruit powers. Perona is able to cook, having cooked several times during the two years while she was living with Mihawk.
The enigma of the ringed eyes
So far, Mihawk and Imu are the only One Piece characters to bear these peculiar "spiral eyes" with red irises and two black circles around the pupils. In the anime, Mihawk's eyes are yellow to look more like those of a hawk, but in the manga's original coloring they are red like Imu's.
I don't think you can simply say "he can beat an admiral" because even within the Admirals, there's such a huge difference in power. I personally think that Luffy could probably just barely beat Aokoji now (or in the near future), and he could fight but not beat Kizaru, and has no chance of beating Akainu.
Mihawk outclassed Zoro when the two fought over Baratie, and even now, he is superior to the straw hat swordsman in terms of skill. On the Sabaody archipelago, Rayleigh fought Kizaru and hit him in battle and even wounded him during the battle. Rayleigh is much stronger than Zoro right now.
Throughout his journey Luffy has made strong bonds with numerous women. At least two have fallen in love with him, Alvida and Hancock, whilst another two have become enamoured with him, Rebecca and Shirahoshi. Nami, Robin and Vivi all share close and deep bonds with Luffy and have become some of his closest friends.
Bartolomeo is Luffy's biggest fan and also the other Straw Hat members who display their posters on his ship. Bartolomeo also became part of the Straw Hat Grand Fleet after the incident at Dressrosa.