Inspired by the linear Schr\"odinger operator, we consider a generalized $p$-Laplacian operator on discrete graphs and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous $p$-Laplacian, we prove that the variational spectrum of the discrete generalized $p$-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl's inequalities for the Laplacian operator to the nonlinear case and prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized $p$-Laplacian on generic graphs, including variational eigenpairs. In particular, when applied to the linear case $p=2$, in addition to recovering well-known features, the new results provide novel properties of the linear Schr\"odinger operator.

翻译：在线性Schr\'odinger操作器的启发下,我们考虑一个通用的美元-拉普拉西亚操作员在离线图形上的操作员,并提出新的结果,说明该操作员的若干光谱特性,特别注意其电子元件的节点域计数。就像单维连续的美元-拉普拉西亚一样,我们证明,在森林上离的通用美元-拉普拉西亚的变异频谱是整个光谱。此外,我们展示了如何将Weyl的拉普拉西亚操作员的不平等转移到非线性案例,并证明普通图形上通用的美元-拉普拉加西亚每个节点的节点域数,包括变异的egenpair。特别是,在应用线性案例时,除了收回众所周知的特征外,新的结果提供了线性Schr\'oder操作员的新特性。