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.
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