  Lower bounds on Hilbert-Samuel multiplicity are known for several types of
commutative noetherian local rings, and rings with multiplicities which achieve
these lower bounds are said to have minimal multiplicity. The first part of
this paper gives characterizations of rings of minimal multiplicity in terms of
the Ext-algebra of the ring; in particular, we show that minimal multiplicity
can be detected via an Ext-algebra which is Gorenstein or Koszul AS-regular.
The second part of this paper characterizes rings of minimal multiplicity via a
numerical homological invariant introduced by J. Herzog and S. B. Iyengar
called linearity defect. Our characterizations allow us to answer in two
special cases a question raised by Herzog and Iyengar.
