AbstractFor a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in $$L^p$$
L
p
, $$p\ge e(H)$$
p
≥
e
(
H
)
, denoted by t(H, W). One may then define corresponding functionals $$\Vert W\Vert _{H}\,{:}{=}\,|t(H,W)|^{1/e(H)}$$
‖
W
‖
H
:
=
|
t
(
H
,
W
)
|
1
/
e
(
H
)
and $$\Vert W\Vert _{r(H)}\,{:}{=}\,t(H,|W|)^{1/e(H)}$$
‖
W
‖
r
(
H
)
:
=
t
(
H
,
|
W
|
)
1
/
e
(
H
)
, and say that H is (semi-)norming if $$\Vert \,{\cdot }\,\Vert _{H}$$
‖
·
‖
H
is a (semi-)norm and that H is weakly norming if $$\Vert \,{\cdot }\,\Vert _{r(H)}$$
‖
·
‖
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 $$\Vert \,{\cdot }\,\Vert _{H}$$
‖
·
‖
H
, we prove that $$\Vert \,{\cdot }\,\Vert _{r(H)}$$
‖
·
‖
r
(
H
)
is neither 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 a negligence in the original statement of the aforementioned theorem by Hatami.