We face the problem of devising optimal unification operators for
domains of abstract substitutions. In particular, we are interested
in abstract domains for sharing and linerity properties, which are
widely used in logic program analysis. We propose a new (infinite)
domain $ShLin^\omega$ which can be thought of as a general framework from
which other domains can be easily derived by abstraction. The
advantage is that abstract unification in $ShLin^\omega$ is simple and
elegant, and it is based on a new concept of sharing graph which
plays the same role of alternating paths for pair sharing
analysis. We also provide an alternative, purely algebraic
description of sharing graphs.  Starting from the results for
$ShLin^\omega$, we derive optimal abstract operators for two well-known
domains which combine sharing and linearity: a domain proposed by
Andy King and the classic Sharing X Lin.