Can we claim matthew as a dependent, he attended college and graduated may 2023. Parents paid all expenses upto august 2023. He was not a full time student.

Q. Can we claim Matthew as a dependent, he attended college and graduated may 2023?

A. No, unless he had less than $4700 of income for all of 2023. 

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, full time student status, a relationship test and residence test.

Since he was not a full time student, during his last term, ending with the May graduation, he cannot be a QC. In order to qualify as a Qualifying Relative dependent, his income must be under $4700. 

A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:

1. Closely Related OR live with the taxpayer ALL year
2. His/her gross taxable income for the year must be less than $4700 (2023) ($4400 for 2022).
3. The taxpayer must have provided more than 1/2 his support

In either case:

4. He must be a US citizen or resident of the US, Canada or Mexico
5. He must not file a joint return with his spouse or be claiming a dependent of his own
6. He must not be the qualifying child of another taxpayer

See full dependent rules at: https://turbotax.intuit.com/tax-tools/tax-tips/Family/Rules-for-Claiming-a-Dependent-on-Your-Tax-Return/INF12139.html