We are mainly concerned with sequences of graphs having a grid geometry,
with a uniform local structure in a bounded domain $${\Omega} {\subset} \mathbb{R}^{d}, d \geq 1$$
Ω
⊂
R
d
,
d
≥
1
. When $$\Omega = [0, 1]$$
Ω
=
[
0
,
1
]
, such graphs include the standard Toeplitz graphs and, for $$\Omega = [0, 1]^{d}$$
Ω
=
[
0
,
1
]
d
,
the considered class includes d-level Toeplitz graphs. In the general case, the underlying
sequence of adjacency matrices has a canonical eigenvalue distribution, in
the Weyl sense, and we show that we can associate to it a symbol $$\mathfrak{f}$$
f
. The knowledge
of the symbol and of its basic analytical features provides many information on
the eigenvalue structure, of localization, spectral gap, clustering, and distribution
type.
Few generalizations are also considered in connection with the notion of generalized
locally Toeplitz sequences and applications are discussed, stemming e.g.
from the approximation of differential operators via numerical schemes. Nevertheless,
more applications can be taken into account, since the results presented
here can be applied as well to study the spectral properties of adjacency matrices
and Laplacian operators of general large graphs and networks.
We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger innerouter curvature growth than a model graph, then it has faster volume growth too. We also study the relationhips of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.
In this paper we investigate the $$ L^1 $$
L
1
-Liouville property, underlining its connection with stochastic completeness and other structural features of the graph. We give a characterization of the $$ L^1 $$
L
1
-Liouville property in terms of the Green function of the graph and use it to prove its equivalence with stochastic completeness on model graphs. Moreover, we show that there exist stochastically incomplete graphs which satisfy the $$ L^1 $$
L
1
-Liouville property and prove some comparison theorems for general graphs based on inner–outer curvatures. We also introduce the Dirichlet $$L^1$$
L
1
-Liouville property of subgraphs and prove that if a graph has a Dirichlet $$L^1$$
L
1
-Liouville subgraph, then it is $$L^1$$
L
1
-Liouville itself. As a consequence, we obtain that the $$ L^1$$
L
1
-Liouville property is not affected by a finite perturbation of the graph and, just as in the continuous setting, a graph is $$ L^1$$
L
1
-Liouville provided that at least one of its ends is Dirichlet $$ L^1$$
L
1
-Liouville.
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