Context :

Aims : Investigation of relationships between dust and gas , and study of the star formation law in M 31 .

Methods : We derive distributions of dust temperature and dust opacity across M 31 at 45 \arcsec resolution using the Spitzer data .
With the opacity map and a standard dust model we de-redden the H \alpha emission yielding the first de-reddened H \alpha map of M 31 .
We compare the emissions from dust , H \alpha , HI and H _ { 2 } by means of radial distributions , pixel-to-pixel correlations and wavelet cross-correlations .
We calculate the star formation rate and star formation efficiency from the de-reddened H \alpha emission .

Results : The dust temperature steeply decreases from 30 K near the center to 15 K at large radii .
The mean dust optical depth at the H \alpha wavelength along the line of sight is about 0.7 .
The radial decrease of the dust-to-gas ratio is similar to that of the oxygen abundance .
Extinction is about linearly correlated with the total gas surface density within limited radial intervals .
On scales < 2 kpc , cold dust emission is best correlated with that of neutral gas and warm dust emission with that of ionized gas .
H \alpha emission is slightly better correlated with emission at 70 \mu m than at 24 \mu m. The star formation rate in M 31 is low .
In the area 6 kpc < R < 17 kpc , the total SFR is \simeq 0.3 { M } _ { \odot } { yr } ^ { -1 } .
A linear relationship exists between surface densities of SFR and H _ { 2 } .
The Kennicutt-Schmidt law between SFR and total gas has a power-law index of 1.30 \pm 0.05 in the radial range of R = 7-11 kpc increasing by about 0.3 for R = 11-13 kpc .

Conclusions : The better 70 \mu m–H \alpha than 24 \mu m–H \alpha correlation plus an excess in the 24 \mu m/70 \mu m intensity ratio indicates that other sources than dust grains , e.g .
of stellar origin , contribute to the 24 \mu m emission .
The lack of H _ { 2 } in the central region could be related to the lack of HI and the low opacity/high temperature of the dust .
Since neither SFR nor SFE is well correlated with the surface density of H _ { 2 } or total gas , other factors than gas density must play an important role in the formation of massive stars in M 31 .
The molecular depletion time scale of 1.1 Gyr indicates that M 31 is about three times less efficient in forming young massive stars than M 33 .