  If $K$ is a field with involution and $E$ an arbitrary graph, the involution
from $K$ naturally induces an involution of the Leavitt path algebra $L_K(E).$
We show that the involution on $L_K(E)$ is proper if the involution on $K$ is
positive definite, even in the case when the graph $E$ is not necessarily
finite or row-finite.
  It has been shown that the Leavitt path algebra $L_K(E)$ is regular if and
only if $E$ is acyclic. We give necessary and sufficient conditions for
$L_{K}(E)$ to be $^\ast$-regular (i.e. regular with proper involution). This
characterization of $^\ast$-regularity of a Leavitt path algebra is given in
terms of an algebraic property of $K,$ not just a graph-theoretic property of
$E.$ This differs from the known characterizations of various other algebraic
properties of a Leavitt path algebra in terms of graph-theoretic properties of
$E$ alone.
  As a corollary, we show that Handelman's conjecture (stating that every
$^\ast$-regular ring is unit-regular) holds for Leavitt path algebras.
Moreover, its generalized version for rings with local units also continues to
hold for Leavitt path algebras over arbitrary graphs.
