Graph laplacian — where it appears
Named by 18 essays across 3 fields — each of them below, with the objects they name alongside it.
A matrix with no numbers in it
A graph arrives as vertices and edges. Two different matrices can be built from it, they answer different questions, and one of them has a null vector that is exact — the only object in these essays whose kernel is known before anything runs.
Two Laplacians of one graph
The combinatorial Laplacian D − A and the normalised one, which conjugates it by the inverse square roots of the degrees, are built from the same object, are not similar to each other, and answer different questions. On a graph whose degrees are equal they coincide. On one whose degrees span an order of magnitude their second eigenvalues are sixteen times apart.
The vertex nobody solves for
A Laplacian is singular, so every solve with one has to remove its kernel first. There are three ways, they agree to fourteen digits, and the one everybody uses carries a free parameter that no account of the method mentions and that moves the condition number by nine hundred.
Spread resistances make the loops easy
Scaled to a unit diagonal, the loop equations on the least-resistance tree get easier as a network's resistances spread — from 120 to 5.44 over six decades — and stop depending on the grid's size, while the node equations of the same flow get harder, from 538 to 4.6·10⁴. The spread that ruins the range-space formulation rescues the null-space one, though the loops' density means the work saved is a factor of two, not the factor of nine the iteration counts suggest.
A distance computed by a solve
Effective resistance is the one quantity in this field with no combinatorial route to it — it is defined by a linear system. On a small unweighted graph the answer is a ratio of two integers, so for once the error is known rather than estimated, and every resistance in a graph has to add up to a number fixed in advance.
The tree the resistances choose
On a network every basic set is a spanning tree and every null-space basis is a set of loops with entries 0 and ±1, so no tree can make Z badly conditioned. The tree with the best-conditioned Z still gives loop equations 4.8 times worse than the tree of least resistance: the basis has to be chosen against the Hessian, and pivoting finds it only when it pivots on the resistances too.
A graph with a tenth of the edges
Keeping 344 of 1,225 edges, sampled by effective resistance and reweighted, preserves every eigenvalue of the Laplacian to within a factor of 1.7. It preserves no degree — half of them are wrong by more than a third — and it takes the diameter from one to three.
A hierarchy with no grid behind it
On a graph Laplacian the algebraic V-cycle converges at 0.199 a cycle, its grid complexity is an unremarkable 3.05, and its operator complexity is 17.7 — one level of forty-one unknowns is entirely dense. The number people quote is the one that does not measure the work.
Eliminating a vertex is a graph operation
Gaussian elimination on a Laplacian deletes a vertex and joins its neighbours into a clique with conductances wᵢwⱼ over Σw. The matrix that remains is still a graph — symmetric, zero row sums, nonpositive off the diagonal — and the ordering decides whether the fill is thirty-one edges or four hundred and sixty-five.
A preconditioner that is a tree
Every eigenvalue of a tree-preconditioned Laplacian is at least one and at most the total stretch — a combinatorial integer with no arithmetic in it. Measured, the bound is two to four times loose, and on a grid the preconditioner makes the conditioning worse by a factor of 1.85 at every size.
A count that comes out of a determinant
The number of spanning trees of a graph is the determinant of its grounded Laplacian, so it is a whole number known in advance. The elimination that computes it is backward stable at every size — and from sixteen vertices the answer is wrong, because the count has seventeen digits and a binary64 has sixteen.
The spectrum is not the graph
Two graphs on six vertices with the same Laplacian characteristic polynomial — as integer polynomials, not to fourteen digits. One contains a triangle; the other is bipartite. Every method in this field that reads only the spectrum is answering about the class.
A Laplacian that is not symmetric
Point the edges and the matrix stops being symmetric. Its row sums are still exactly zero, so the null vector survives; everything built on the quadratic form does not, and the eigenvalues need a plane rather than a line. Asymmetry permits that and does not force it, which the smallest case here demonstrates by being asymmetric and real.
A conductance the arcs do not measure
Symmetrising a directed Laplacian with respect to its walk recovers everything the arrows took — a real spectrum, a sweep cut, a Cheeger inequality. What it does not recover is the quantity: the inequality bounds the probability that a step of the walk crosses the cut, which on one graph here is three times the weight of the arcs that do.
A finer invariant that hears less
The Laplacian spectrum fixes a graph's number of spanning trees and not the group those trees form, so the Smith normal form of the grounded Laplacian is a strictly finer integer invariant — and on the six-vertex pair the spectrum cannot separate, it does: ℤ₁₂ against ℤ₂ ⊕ ℤ₆. Over all 1,022 Laplacian-cospectral pairs of connected graphs on eight vertices it separates 435. The adjacency polynomial, a second spectrum rather than a finer one, separates all of them.
Each spectrum hears the other's pairs
Pairs of graphs with the same Laplacian spectrum share a spanning-tree count, so the critical group could only tell them apart by how that count factors — and did on two pairs in five. Turn the census round, to the 733 pairs of eight-vertex graphs with the same adjacency spectrum, and the tree count is free to differ. It does on 717 of them, by a median of six per cent. The critical group separates exactly those 717 and not one more: on all sixteen pairs whose counts agree, including a non-cyclic one, the two graphs have the same group. And the Laplacian spectrum separates all 733, as the adjacency spectrum separated every Laplacian pair.
The group hears a switch only at two
On a regular graph the Laplacian is the degree times the identity less the adjacency matrix, so the two spectra are one piece of information and the tree count comes with them; between cospectral regular graphs the critical group is the only exact invariant left. Made by Godsil–McKay switching on four vertices, 705 distinct cospectral pairs of 3- and 4-regular graphs on 12 to 20 vertices give its answer. It separates 181 of 587 quartic pairs and none of 118 cubic ones. And on every one of the 705 the two groups agree everywhere except at the prime two: a switch is a conjugation by a matrix of halves, the group can see it only through its 2-part, and a group whose tree count has fewer than three factors of two cannot see it at all.
Long loops pay before the factor starts
The least-resistance spanning tree makes a network's loop equations well conditioned and its loops long, and the question left was what a direct solver pays for the length, and whether a tree reading the resistances only coarsely buys the conditioning back for less. Counted symbolically under minimum-degree ordering, the least tree's factor costs a median 1.2 to 2.0 times the breadth-first tree's on grids of 4 to 12 points a side, and the loop formulation goes from cheaper than the node equations at 4 × 4 to 3.8 times dearer at 12 × 12, for a conditioning two to three orders better. The price is paid in the loop matrix itself: on every network measured its factor fills in fewer entries than the breadth-first tree's. And no tree between them is a bargain. Kruskal on resistances rounded into bands pays nearly all the extra work until one band holds three quarters of the spread, and by then the conditioning has gone too.
Named alongside it
The objects these essays reach for when they reach for this one.
Spanning-treeExact ground truthCospectral graphsGraph invariantMatrix tree theoremQuadratic formGroundingNull spaceCharacteristic polynomialCondition numberEffective resistanceInvariant factors