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The Laplacian equals $B B^{T}$ for every oriented incidence matrix $B$

Statement

Let $G$ be a finite simple graph, let $B$ be any oriented incidence matrix of $G$ , and let $L (G)$ be its Laplacian matrix. Then

$L (G) = B B^{T} .$

Facts & Assumptions

Given: A finite simple graph $G$ with oriented incidence matrix $B = (b_{i j})$ and Laplacian matrix $L = L (G)$ .

[F1]

Each column of $B$ has one entry $1$ , one entry $- 1$ , and all other entries $0$ (An oriented incidence matrix of a finite simple graph).

[F2]

The Laplacian has diagonal entries equal to vertex degrees and off-diagonal entries $- 1$ on adjacent pairs and $0$ otherwise (The Laplacian matrix of a finite simple graph).

Proof

technique · direct

1.1

The $(i, i)$ entry of $B B^{T}$ is $\sum_{j} b_{i j}^{2}$ . By [F1], the term $b_{i j}^{2}$ is $1$ exactly when the edge $e_{j}$ is incident with $v_{i}$ , and is $0$ otherwise. Therefore $(B B^{T})_{i i}$ is the number of edges incident with $v_{i}$ , namely $\deg (v_{i}) = L_{i i}$ by [F2].

F1F2

1.2

For $i \neq r$ , the entry $(B B^{T})_{i r} = \sum_{j} b_{i j} b_{r j}$ receives a nonzero contribution only from edges incident with both $v_{i}$ and $v_{r}$ . In a simple graph there is at most one such edge. If $v_{i}$ and $v_{r}$ are not adjacent, the sum is $0$ . If they are adjacent, the unique common edge contributes $(1) (- 1)$ or $(- 1) (1)$ , hence $- 1$ . So $(B B^{T})_{i r} = L_{i r}$ by [F2].

F1F2

2.1

Steps 1.1 and 1.2 show that every entry of $B B^{T}$ matches the corresponding entry of $L (G)$ , so the matrices are equal.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard P. Stanley, MIT 18.314 handout, Lemma 1.6(a) (standard reference, not scraped)

The Laplacian equals BBTBBT for every oriented incidence matrix BB