My daughter graduated college in May 2016, lived with us until August 2017 when she moved to another state to attend graduate school. Can I claim her as a dependent?

Probably yes, if she was still under 24 at year end. 

Graduation year

If he/she was a student (under age 24) for at least 5 months and lived with you for more than half the year, and did not provide more than 1/2 his own support for the whole year, you can still claim him. Be sure he knows you're claiming him, so he doesn't claim himself. He can only be claimed once. But, he can "file taxes" without claiming his own exemption.

The real question is who should be claiming him in this "transition" year to adulthood. You two have to agree on who is going to claim his exemption. Each should do their taxes both ways and see which way the family comes out best.  Even then, you have to meet the rules. The rule is that a child of a taxpayer can still be a “Qualifying Child” dependent, regardless of  his income, if:

1. he is a full time student under 24 for at least 5 calendar months of the year (graduating in May usually means you meet the 5 month rule)

2. he did not provide more than 1/2 his own support  (scholarships are considered 3rd party support and not support provided by the student). 

3. lived with the parent (including time away at school) for more than half the year

So, it usually hinges on  "Did he provide more than 1/2 his own support in 2016.

The support value of the home you provided is the fair market rental value of the home plus utilities & other expenses divided by the number of occupants. IRS Publication 501 on page 20 has a worksheet that can be used to help with the support calculation. See: http://www.irs.gov/pub/irs-pdf/p501.pdf

But,  you don't mention any income for her. So, she may still be (but less likely) your dependent even if she was over age 23. 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 $4,050 (2016)

3. The taxpayer must have provided more than 1/2 his support. Note that this rule is different from the support rule for a QC. Now, scholarships count as support not provided by you.

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