  For a graph $H$, its homomorphism density in graphs naturally extends to the
space of two-variable symmetric functions $W$ in $L^p$, $p\geq e(H)$, denoted
by $t(H,W)$. One may then define corresponding functionals
$\|W\|_{H}:=|t(H,W)|^{1/e(H)}$ and $\|W\|_{r(H)}:=t(H,|W|)^{1/e(H)}$ and say
that $H$ is (semi-)norming if $\|.\|_{H}$ is a (semi-)norm and that $H$ is
weakly norming if $\|.\|_{r(H)}$ is a norm. We obtain two results that
contribute to the theory of (weakly) norming graphs. Firstly, answering a
question of Hatami, who estimated the modulus of convexity and smoothness of
$\|.\|_{H}$, we prove that $\|.\|_{r(H)}$ is not uniformly convex nor uniformly
smooth, provided that $H$ is weakly norming. Secondly, we prove that every
graph $H$ without isolated vertices is (weakly) norming if and only if each
component is an isomorphic copy of a (weakly) norming graph. This strong
factorisation result allows us to assume connectivity of $H$ when studying
graph norms. In particular, we correct an error in the original statement of
the aforementioned theorem by Hatami.
