  Random graphs undergo structural phase transitions that are crucial for
dynamical processes and cooperative behavior of models defined on graphs. In
this work we investigate the impact of a first-order structural transition on
the thermodynamics of the Ising model defined on Erd\"os-R\'enyi random graphs,
as well as on the eigenvalue distribution of the adjacency matrix of the same
graphical model. The structural transition in question yields graph samples
exhibiting condensation, characterized by a large number of nodes having
degrees in a narrow interval. We show that this condensation transition induces
distinct thermodynamic first-order transitions between the paramagnetic and the
ferromagnetic phases of the Ising model. The condensation transition also leads
to an abrupt change in the global eigenvalue statistics of the adjacency
matrix, which renders the second moment of the eigenvalue distribution
discontinuous. As a side result, we derive the critical line determining the
percolation transition in Erd\"os-R\'enyi graph samples that feature
condensation of degrees.
