as an accompanying spouse/family to employees/self-employed persons residing in Denmark, you are allowed to work full-time in Denmark.
And if you complete an education with a duration of at least 18 months (PhD for instance) then you are eligible for Unemployment benefits according to the special rule for graduates.
Be aware that this requires that you were living in Denmark not only when graduating, but also the day before you commenced you education (PhD) here in Denmark.
For using the graduate rule it is not nessasary to have been member with A-kasse for 1 year, but it is an advantage, because this will mean that you can avoid the 1 month quarantine, which graduates without prior membership must face.
However, under all circumstances, even if you already are member of an A-kasse, you must contact your A-kasse at latest 2 weeks after graduation and tell them that you now want to have your status changed from student to graduate.
You don't tell us what kind of membership you have currently in your A-kasse. We assume its a free study-membership.
If you have paid for membership like a normal paid employee (PhD) and you have had an income for at least DKK 228.348 during the last 3 years (only periods with A-kasse membership counts here), then you will be eligible for Unemployment benefits as a normal/salaried person. This means a higher benefit rate.