Graph Laplacians with Higher Accuracy
Mary Yoon
TL;DR
The paper introduces the $m$‑Laplacian $L^{(m)}_G$, a family of higher‑order graph operators inspired by finite‑difference methods, with $L^{(1)}_G$ recovering the ordinary graph Laplacian. It defines the discrete operator $\Delta^{(m)}$ on a grid and constructs $L^{(m)}_G$ as the Laplacian of a weighted graph $G_m$ with adjacency $A_{G_m}=\sum_{k=1}^{m} a_{k,m}P_{G,k}$, where the weights satisfy $a_{k,m}=(-1)^{k+1}\frac{2\binom{2m}{m-k}}{k^2\binom{2m}{m}}$, yielding $O(h^{2m})$ accuracy. The authors derive explicit spectra for key graph families (cycles, complete graphs, stars) and establish an existence result for weighted graphs with a given Laplacian spectrum; they also show that the $2$‑Laplacian $L^{(2)}_G$ can be PSD up to $k\le 8$ for $k$‑regular graphs and exhibits markedly fewer cospectral mates than other Laplacian variants, suggesting greater discriminative power in graph analysis. Overall, the work extends spectral graph theory with a tunable, higher‑order Laplacian framework that accommodates signed weights and opens avenues for refined graph characterization and spectrum design.
Abstract
Motivated by discrete Laplacian differential operators with various accuracy orders in numerical analysis, we introduce new matrices attached to a simple graph that can be considered graph Laplacians with higher accuracy. In particular, we show that the number of graphs having cospectral mates with these matrices is significantly less than the ones with other known matrices. We also investigate their spectral properties and explicitly compute their eigenvalues and eigenvectors for some graphs. Along the line, we also prove the existence of a weighted signed graph with given Laplacian eigenvalues.