Liouville property of an operator on graphs

Liouville property of an operator on graphs --- свойство Лиувиля оператора на графе. Given an operator [math]\displaystyle{ {\mathcal L} }[/math] on a graph and a class [math]\displaystyle{ {\mathcal F} }[/math] of solutions of [math]\displaystyle{ {\mathcal L} }[/math], by the Liouville property of the pair [math]\displaystyle{ ({\mathcal L},{\mathcal F}) }[/math] we mean that the dimension of the space of all solutions of the operator [math]\displaystyle{ {\mathcal L} }[/math] in [math]\displaystyle{ {\mathcal F} }[/math] is at most one.