  Let $G$ be a group and $\ell$ a commutative unital $\ast$-ring with an
element $\lambda \in \ell$ such that $\lambda + \lambda^\ast = 1$. We introduce
variants of hermitian bivariant $K$-theory for $\ast$-algebras equipped with a
$G$-action or a $G$-grading. For any graph $E$ with finitely many vertices and
any weight function $\omega \colon E^1 \to G$, a distinguished triangle for
$L(E)=L_\ell(E)$ in the hermitian $G$-graded bivariant $K$-theory category
$kk^h_{G_{\mathrm{gr}}}$ is obtained, describing $L(E)$ as a cone of a matrix
with coefficients in $\mathbb{Z}[G]$ associated to the incidence matrix of $E$
and the weight $\omega$. In the particular case of the standard
$\mathbb{Z}$-grading, and under mild assumptions on $\ell$, we show that the
isomorphism class of $L(E)$ in $kk^h_{\mathbb{Z}_{\mathrm{gr}}}$ is determined
by the graded Bowen-Franks module of $E$. We also obtain results for the graded
and hermitian graded $K$-theory of $\ast$-algebras in general and Leavitt path
algebras in particular which are of independent interest, including hermitian
and bivariant versions of Dade's theorem and of Van den Bergh's exact sequence
relating graded and ungraded $K$-theory.
