We complete the understanding of the question of the essential self-adjoitness and non-essential self-adjointness of the discrete Laplacian acting on 1-forms. We also discuss the notion of completeness. Moreover, we study the relationship between the adjacency matrix of the line graph and the discrete Laplacian acting on 1-forms. Thanks to it, we exhibit a condition that ensures that the adjacency matrix on line graph is bounded from below and not essentially self-adjoint.

中文翻译：

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邻接矩阵和离散拉普拉斯算子作用于形式

我们完成了对作用于 1-形式的离散拉普拉斯算子的本质自伴性和非本质自伴性问题的理解。我们还讨论了完整性的概念。此外，我们研究了线图的邻接矩阵和作用于 1-形式的离散拉普拉斯算子之间的关系。多亏了它，我们展示了一个条件，确保线图上的邻接矩阵从下方有界，而不是本质上自伴随的。