# Entries of the Inverse Laplacian

Let $$L$$ be a graph Laplacian. What is the meaning of the entries of its (pseudo)inverse $$L^{-1}$$? In other words, are there any interpretations which might help with understanding the entries of $$L^{-1}$$?

• There is no graph - your link is missing. Feb 6, 2023 at 2:14
• I fixed it, thank you.
– Zuza
Feb 6, 2023 at 9:50

Interpretation of the entries of $$L^{-1}$$:
• The i'th row (or column, it's symmetric) of $$L^{-1}$$ represents the voltage at vertices required such that the unique electrical flow sends a total of 1 unit of flow from the i'th vertex in a way that each vertex receives $$1/|V|$$ units of flow.
• Let the energy of a flow $$f \in \mathbb{R}^E$$ be $$\sum_{e \in E} f_i^2$$. Let the effective resistance between two vertices $$u, v$$ be the minimum energy requred to send a unit of flow from $$u$$ to $$v$$. The effective resistance between $$u, v \in V$$ is $$(1_u - 1_v)^T L^{-1} (1_u - 1_v)$$ where $$1_i$$ is the i'th canonical basis vector.
• (Assume the graph is connected, degree regular, and not bipartite.) Let $$s, t \in V$$. Start with +1 units of "supply" in $$s$$ and -1 of "supply" in $$t$$. Do a random walk (each vertex chooses a random incident edge and pushes all of its supply to the other node). The expected amount of supply at node j (summed up over all steps) is exactly $$M_{sj} - M_{tj}$$ where $$M := L^{-1}$$ (I had to introduce M due to some MathJax+SE formatting issue).