  In this paper we first give a lower bound on multiplicities for Buchsbaum
homogeneous $k$-algebras $A$ in terms of the dimension $d$, the codimension
$c$, the initial degree $q$, and the length of the local cohomology modules of
$A$. Next, we introduce the notion of Buchsbaum $k$-algebras with minimal
multiplicity of degree $q$, and give several characterizations for those rings.
In particular, we will show that those algebras have linear free resolutions.
Further, we will give many examples of those algebras.
