In a statement, the Turkish armed forces said the car bomb went off at 8:45 am and targeted on-leave military personnel from the Kayseri Commando Brigade.
The army said 48 troops were wounded and said that civilian citizens may have also been hurt in the "treacherous act."
Speaking in Kayseri shortly after the blast, Interior Minister Suleyman Soylu said 55 people had been wounded in total and that six were in critical condition.
Speaking alongside him at a joint press conference, Turkish Chief of Staff Hulusi Akar vowed to continue the "fight against terrorists inside and outside the country" until "every last terrorist is neutralized."
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There was no immediate claim of responsibility for the attack.
The state-run Anadolu Agency said the car bomb went off at an entrance gate of Erciyes University, hitting a public transportation bus that included on-leave soldiers among its passengers.
Images taken moments after the explosion showed a smoking public bus, still in flames, with its windows blown open and its interior blackened.
Turkey is facing a wide range of security challenges including renewed conflict in the predominantly Kurdish southeast. Turkey is a member of NATO and partner in the US-led coalition against the Islamic State group, which has been blamed for multiple attacks in Turkey.
A state of emergency was declared following a botched July 15 coup attempt.
Turkey's prime ministry office imposed a temporary blackout on coverage of the explosion and urged media to refrain from publishing anything that may cause "fear in the public, panic and disorder and which may serve the aims of terrorist organisations.