<p>I originally wanted to pose the question, but then I started investigating and found a few very helpful interpretation that haven't been collected anywhere (to my knowledge). Hence, I will write my results here for future reference:</p>
<p>Interpretation of the entries of <span class="math-container">$L^{-1}$</span>:</p>
<ul>
<li><p>The i'th row (or column, it's symmetric) of <span class="math-container">$L^{-1}$</span> 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 <span class="math-container">$1/|V|$</span> units of flow.</p>
</li>
<li><p>Let the energy of a flow <span class="math-container">$f \in \mathbb{R}^E$</span> be <span class="math-container">$\sum_{e \in E} f_i^2$</span>. Let the effective resistance between two vertices <span class="math-container">$u, v$</span> be the minimum energy requred to send a unit of flow from <span class="math-container">$u$</span> to <span class="math-container">$v$</span>. The effective resistance between <span class="math-container">$u, v \in V$</span> is <span class="math-container">$(1_u - 1_v)^T L^{-1} (1_u - 1_v)$</span> where <span class="math-container">$1_i$</span> is the i'th canonical basis vector.</p>
</li>
<li><p>(Assume the graph is connected, degree regular, and not bipartite.) Let <span class="math-container">$s, t \in V$</span>. Start with +1 units of &quot;supply&quot; in <span class="math-container">$s$</span> and -1 of &quot;supply&quot; in <span class="math-container">$t$</span>. 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 <span class="math-container">$M_{sj} - M_{tj}$</span> where <span class="math-container">$M := L^{-1}$</span> (I had to introduce M due to some MathJax+SE formatting issue).</p>
</li>
</ul>

CC BY-SA 4.0
2023-02-05T21:48:01.640Z
2023-02-05T21:48:01.640Z
Zuza
<p>Let <span class="math-container">$L$</span> be a <a href="https://en.wikipedia.org/wiki/Laplacian_matrix" rel="nofollow noreferrer">graph Laplacian</a>. What is the meaning of the entries of its (pseudo)inverse <span class="math-container">$L^{-1}$</span>? In other words, are there any interpretations which might help with understanding the entries of <span class="math-container">$L^{-1}$</span>?</p>

CC BY-SA 4.0
2023-02-06T02:14:17.947Z
There is no graph - your link is missing.
joseph h
CC BY-SA 4.0
2023-02-06T09:50:49.513Z
I fixed it, thank you.
Zuza
<graph-theory><spectral-graph-theory><random-walks><flow-problems>
Entries of the Inverse Laplacian
Zuza
# Entries of the Inverse Laplacian

(Asked by: username_0 on 2023-02-05)

Let $L$ be a [graph Laplacian](https://en.wikipedia.org/wiki/Laplacian_matrix). 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 $L^{-1}$?

**Tags:** graph-theory, spectral-graph-theory, random-walks, flow-problems

> **Comments:**
> **username_1**: There is no graph - your link is missing.  
> **username_0**: I fixed it, thank you.  

---

## 1 answers

### Answer by username_0 on 2023-02-05. Score: 1

I originally wanted to pose the question, but then I started investigating and found a few very helpful interpretation that haven't been collected anywhere (to my knowledge). Hence, I will write my results here for future reference:

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).


---

