  Galaxy clusters provide an excellent probe in various research fields in
astrophysics and cosmology. However, the number of galaxy clusters detected so
far in the $AKARI$ North Ecliptic Pole (NEP) field is limited. In this work, we
provide galaxy cluster candidates in the $AKARI$ NEP field with the minimum
requisites based only on coordinates and photometric redshift (photo-$z$) of
galaxies. We used galaxies detected in 5 optical bands ($g$, $r$, $i$, $z$, and
$Y$) by the Subaru Hyper Suprime-Cam (HSC), assisted with $u$-band from
Canada-France-Hawaii Telescope (CFHT) MegaPrime/MegaCam, and IRAC1 and IRAC2
bands from the $Spitzer$ space telescope for photo-$z$ estimation. We
calculated the local density around every galaxy using the 10$^{th}$-nearest
neighbourhood. Cluster candidates were determined by applying the
friends-of-friends algorithm to over-densities. 88 cluster candidates
containing 4390 member galaxies below redshift 1.1 in 5.4 deg$^2$ have been
detected. The reliability of our method was examined through false detection
tests, redshift uncertainty tests, and applications on the COSMOS data, giving
false detection rates of 0.01 to 0.05 and recovery rate of 0.9 at high
richness. 3 X-ray clusters previously observed by $ROSAT$ and $Chandra$ were
recovered. The cluster galaxies show higher stellar mass and lower star
formation rate (SFR) compared to the field galaxies in two-sample Z-tests.
These cluster candidates are useful for environmental studies of galaxy
evolution and future astronomical surveys in the NEP, where $AKARI$ has
performed unique 9-band mid-infrared photometry for tens of thousands of
galaxies.
