Officers said they believed the attack on Christina Edkins during Thursday morning's rush hour in Birmingham appeared to be "random and senseless".
A 22-year-old man has been arrested on suspicion of murdering the 16-year-old who was attacked on the bus's top deck.
Police said they were still searching for the murder weapon.
Officers have until 01:00 GMT on Saturday either to charge or release the man they have in custody.
They said the suspect, a British national with links to Walsall, was not known to mental health units in the area and has no known link to the teenager.
Christina was stabbed once in the chest on the bus on Hagley Road while on her way to school in Halesowen.
A woman claiming to have witnessed the attack on the bus said she waited for the girl's attacker to get off the vehicle before performing CPR on the teenager.
Writing on Facebook, Stephanie Bent said: "I gave CPR until medics arrived and am deeply traumatised I could not save her. I certainly wasn't going to confront a knifeman knowing he had just stabbed someone I thought he may stab me too, all I cared about was getting up to the girl to help."
Police have said they "urgently" need to speak to all of the 16 people who were on the bus at the time of the attack, at about 07:30 GMT.
In a press conference Det Supt Richard Baker said police have been "overwhelmed" by the public response, which had provided the investigation with "crucial information".
He said there had been three people on the top deck of the bus at the time of the attack - Christina, her attacker and another man, who has been identified.
Mr Baker said the suspect "calmly waited to get off the bus" after Christina was attacked.
Detectives called for anyone who might have seen the suspect travelling on buses around the Birmingham area after the stabbing to get in touch.
Counsellors have been brought into Leasowes High School, where Christina was a pupil, to support pupils and staff.
Throughout the day, pupils and people living nearby have been leaving flowers, soft toys and balloons outside the school gates.
The school said it thanked "everyone for many offers of support" and extended heartfelt sympathy to Christina's family.
Friends of the teenager also gathered to pay their respects at Hagley Road, where a growing number of floral tributes have also been placed.
A book of remembrance has been opened at nearby St Peter's Church, in Lapal, Halesowen, where people can go to light a candle.
Several supermarkets in Halesowen have started collections for Christina's family.
Passengers on the lower deck of a bus where a girl was stabbed to death heard screaming but thought it was teenagers messing about, police have said.