I am working on a graphene patch antenna simulation. It works well at expected frequency. However I found there is a higher frequency resonance like fig below.
It is very hard to eliminate this - there will always be resonances in the patch at higher frequencies. It is possible you may be able to eliminate a particular resonance by choice of the feed method (position or combination of feeds). You can eliminate full-wave resonance by feeding at opposite edges out of phase, or slot-coupling under the centre of the patch. There may still be a resonance at close to 3 times the frequency.
You can move resonances by changing the shape of the patch, so that the ratio between the two frequencies is different, but removing higher frequency resonances entirely is probably not possible. To stop a receiver from picking up these higher frequencies there would usually be a low-pass or band-pass filter in the feed line somewhere.
Currents at the second resonance are probably in places where there are also currents in the first resonance, so slots will also disrupt the first resonance. This is assuming it is a simple rectangular patch, in which the first resonance has currents over the whole surface, falling to zero at two opposite edges, and the second resonance has currents in the same direction but with a zero current line across the middle as well as at the same two edges.
Arvind Kumar is correct. There is a line across a patch, somewhere near the middle, where the electric field is zero for the first mode. Vias can be placed here without disturbing that mode. They will short out the second mode.
There are some techniques defined in the paper entitled "A compact microstrip-fed patch antenna with enhanced bandwidth and harmonic suppression" publihed in IEEE Trans. Antennas Propag., vol. 64, no. 12, 2016 such as capacitive feeding and quarter wavelength resonators. The shorting pin can also be used.