  Temperature fluctuations of a finite system follows the Landau bound $\delta
T^2 = T^2/C(T)$ where $C(T)$ is the heat capacity of the system. In turn, the
same bound sets a limit to the precision of temperature estimation when the
system itself is used as a thermometer. In this paper, we employ graph theory
and the concept of Fisher information to assess the role of topology on the
thermometric performance of a given system. We find that low connectivity is a
resource to build precise thermometers working at low temperatures, whereas
highly connected systems are suitable for higher temperatures. Upon modelling
the thermometer as a set of vertices for the quantum walk of an excitation, we
compare the precision achievable by position measurement to the optimal one,
which itself corresponds to energy measurement.
