论文标题
签名图上的$ p $ -laplacians的节点域定理
Nodal domain theorems for $p$-Laplacians on signed graphs
论文作者
论文摘要
我们在签名图上为$ p $ laplacians建立了各种节点域定理，该图将大多数现有结果在图形$ p $ laplacians和任意对称矩阵上的大部分结果统一。根据我们的节点域的估计，我们获得了更高级的cheeger不等式，该不平等与$ p $ laplacians的变异特征值和Atay-liu在签名图上的多路cheeger常数有关。在$ p = 1 $的特定情况下，这导致了几种与变异特征值和多路cheeger常数有关的身份。有趣的是，我们的方法还会在通常的图表上取得新的结果，包括Sturm的振荡定理的薄弱版本，用于图形$ 1 $ laplacians，以及在$ P $ -Laplacians的最大和第二大变量特征值之间具有$ P $ -Laplacians的特征值，并在连接的BipArtite上$ P> 1 $。
We establish various nodal domain theorems for $p$-Laplacians on signed graphs, which unify most of the existing results on nodal domains of graph $p$-Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we obtain a higher order Cheeger inequality that relates the variational eigenvalues of $p$-Laplacians and Atay-Liu's multi-way Cheeger constants on signed graphs. In the particular case of $p=1$, this leads to several identities relating variational eigenvalues and multi-way Cheeger constants. Intriguingly, our approach also leads to new results on usual graphs, including a weak version of Sturm's oscillation theorem for graph $1$-Laplacians and nonexistence of eigenvalues between the largest and second largest variational eigenvalues of $p$-Laplacians with $p>1$ on connected bipartite graphs.
