  We undertake a systematic investigation of the maxima and minima of the
eigenfunctions associated with the first nontrivial eigenvalue of the Laplacian
on a metric graph equipped with standard (continuity--Kirchhoff) vertex
conditions. This is inspired by the famous hot spots conjecture for the
Laplacian on a Euclidean domain, and the points on the graph where maxima and
minima are achieved represent the generically "hottest" and "coldest" spots of
the graph. We prove results on both the number and location of the hot spots of
a metric graph, and also present a large number of examples, many of which run
contrary to what one might na\"ively expect. Amongst other results we prove the
following: (i) generically, up to arbitrarily small perturbations of the graph,
the points where minimum and maximum, respectively, are attained are unique;
(ii) the minima and maxima can only be located at the vertices of degree one or
inside the doubly connected part of the metric graph; and (iii) for any fixed
graph topology, for some choices of edge lengths all minima and maxima will
occur only at degree-one vertices, while for others they will only occur in the
doubly connected part of the graph.
