Laplacian spectrum of small graphs

This is a collection of entirely unoriginal remarks about Laplacian spectrum of graphs. For an accessible overview of the subject I recommend the M.S. thesis The Laplacian Spectrum of Graphs by Michael William Newman. It also includes a large table of graphs with their spectra. Here I will avoid introducing matrices and enumerating vertices.

Let {V} be the vertex set of a graph. Write {u\sim v} if {u, v} are adjacent vertices. Given a function {f\colon V\to \mathbb R}, define {L f(v) = \sum_{u\colon u\sim v}(f(v)-f(u))}.
This is a linear operator (the graph Laplacian) on the Euclidean space {\ell^2(V)} of all functions {f\colon V\to \mathbb R} with the norm {\|f\|^2 = \sum_{v\in V} f(v)^2}. It is symmetric: {\langle L f, g\rangle = \langle f, L g\rangle } and positive semidefinite: {\langle L f, f\rangle = \frac12 \sum_{u\sim v}(f(u)-f(v))^2\ge 0}. Since equality is attained for constant {f}, 0 is always an eigenvalue of {L}.

This is the standard setup, but I prefer to change things a little and replace {\ell^2(V)} by the smaller space {\ell^2_0(V)} of functions with zero mean: {\sum_{v\in V}f(v)=0}. Indeed, {L} maps {\ell^2(V)} to {\ell^2_0(V)} anyway, and since it kills the constants, it makes sense to focus on {\ell^2_0(V)}. It is a vector space of dimension {n-1} where {n=|V|}.

One advantage is that the smallest eigenvalue is 0 if and only if the graph is disconnected: indeed, {\langle L f, f\rangle=0} is equivalent to {f} being constant on each connected component. We also gain better symmetry between {L} and the Laplacian of the graph complement, denoted {L'}. Indeed, since {L' f(v) = \sum_{u\colon u\not \sim v}(f(v)-f(u))}, it follows that {(L+L')f(v) = \sum_{u\colon u\ne v} (f(v)-f(u)) = n f(v)} for every {f\in \ell^2_0(V)}. So, the identity {L+L' = nI} holds on {\ell^2_0(V)} (it does not hold on {\ell^2(V)}). Hence the eigenvalues of {L'} are obtained by subtracting the eigenvalues of {L} from {n}. As a corollary, the largest eigenvalue of {L} is at most {n}, with equality if and only if the graph complement is disconnected. More precisely, the multiplicity of eigenvalue {n} is one less than the number of connected components of the graph complement.

Let {D} denote the diameter of the graph. Then the number of distinct Laplacian eigenvalues is at least {D}. Indeed, let {u, v} be two vertices at distance {D} from each other. Define {f_0(u) = 1} and {f_0=0} elsewhere. Also let {f_k=L^k f_0} for {k=1, 2, \dots}. Note that {f_k\in \ell_0^2(V)} for all {k\ge 1}. One can prove by induction that {f_k(w)=0} when the distance from {w} to {u} is greater than {k}, and {(-1)^k f_k(w) > 0} when the distance from {w} to {u} is equal to {k}. In particular, {f_k(v) = 0} when {k<D} and {f_D(v)\ne 0}. This shows that {f_D} is not a linear combination of {f_1, \dots, f_{D-1}}. Since {f_k = L^{k-1}f_1}, it follows that {L^{D-1}} is not a linear combination of {L^0, L^1, \dots, L^{D-2}}. Hence the minimal polynomial of {L} has degree at least {D}, which implies the claim.

Let’s consider a few examples of connected graphs.

3 vertices

There are two connected graphs: the 3-path (D=2) and the 3-cycle (D=1). In both cases we get D distinct eigenvalues. The spectra are [1, 3] and [3, 3], respectively.

4 vertices

  • One graph of diameter 3, the path. Its spectrum is {[2-\sqrt{2}, 2, 2+\sqrt{2}]}.
  • One graph of diameter 1, the complete graph. Its spectrum is {[4, 4, 4]}. This pattern continues for other complete graphs: since the complement is the empty graph ({n} components), all {n-1} eigenvalues are equal to {n}.
  • Four graphs of diameter 2, which are shown below, with each caption being the spectrum.
4-0
1, 1, 4
4-2
1, 3, 4
4-3
2, 2, 4
4-4
2, 4, 4

Remarks:

  • The graph [1, 3, 4] has more distinct eigenvalues than its diameter.
  • The graph [2, 2, 4] is regular (all vertices have the same degree).
  • The smallest eigenvalue of graphs [1, 1, 4] and [2, 2, 4] is multiple, due to the graphs having a large group of automorphisms (here rotations); applying some of these automorphisms to an eigenfunctions for the smallest eigenvalue yields another eigenfunction.
  • [1, 3, 4] and [2, 4, 4] also have automorphisms, but their automorphisms preserve the eigenfunction for the lowest eigenvalue, up to a constant factor.

5 vertices

  • One graph of diameter 4, the path. Its spectrum is related to the golden ratio: it consists of {(3\pm \sqrt{5})/2, (5\pm \sqrt{5})/2}.
  • One graph of diameter 1, the complete one: [5, 5, 5, 5]
  • Five graphs of diameter 3. All have connected complement, with the highest eigenvalue strictly between 4 and 5. None are regular. Each has 4 distinct eigenvalues.
  • 14 graphs of diameter 2. Some of these are noted below.

Two have connected complement, so their eigenvalues are less than 5 (spectrum shown on hover):

One has both integers and non-integers in its spectrum, the smallest such graph. Its eigenvalues are {3\pm \sqrt{2}, 3, 5}.

5-15
1.585786, 3, 4.414214, 5

Two have eigenvalues of multiplicity 3, indicating a high degree of symmetry (spectrum shown on hover).

Two have all eigenvalues integer and distinct:

 

The 5-cycle and the complete graph are the only regular graphs on 5 vertices.

6 vertices

This is where we first encounter isospectral graphs: the Laplacian spectrum cannot tell them apart.

Both of these have spectrum {3\pm \sqrt{5}, 2, 3, 3} but they are obviously non-isomorphic (consider the vertex degrees):

Both have these have spectrum {3\pm \sqrt{5}, 3, 3, 4} and are non-isomorphic.

Indeed, the second pair is obtained from the first by taking graph complement.

Also notable are regular graphs on 6 vertices, all of which have integer spectrum.

6-46
1, 1, 3, 3, 4
6-71
3, 3, 3, 3, 6
6-98
2, 3, 3, 5, 5
6-108
4, 4, 4, 6, 6
6-111
6, 6, 6, 6, 6

Here [3, 3, 3, 3, 6] (complete bipartite) and [2, 3, 3, 5, 5] (prism) are both regular of degree 3, but the spectrum allows us to tell them apart.

The prism is the smallest regular graph for which the first eigenvalue is a simple one. It has plenty of automorphisms, but the relevant eigenfunction (1 on one face of the prism, -1 on the other face) is compatible with all of them.

7 vertices

There are four regular graphs on 7 vertices. Two of them are by now familiar: 7-cycle and complete graph. Here are the other two, both regular of degree 4 but with different spectra.

7-720
3, 4, 4, 5, 5, 7
7-832
3.198, 3.198, 4.555, 4.555, 6.247, 6.247

There are lots of isospectral pairs of graphs on 7 vertices, so I will list only the isospectral triples, of which there are five.

Spectrum 0.676596, 2, 3, 3.642074, 5, 5.681331:

Spectrum 0.726927, 2, 3.140435, 4, 4, 6.132637:

Spectrum 0.867363, 3, 3, 3.859565, 5, 6.273073:

Spectrum 1.318669, 2, 3.357926, 4, 5, 6.323404:

All of the triples mentioned so far have connected complement: for example, taking the complement of the triple with the spectrum [0.676596, 2, 3, 3.642074, 5, 5.681331] turns it into the triple with the spectrum [1.318669, 2, 3.357926, 4, 5, 6.323404].

Last but not least, an isospectral triple with an integer spectrum: 3, 4, 4, 6, 6, 7. This one has no counterpart since the complement of each of these graphs is disconnected.

8 vertices

Regular graphs, excluding the cycle (spectrum 0.585786, 0.585786, 2, 2, 3.414214, 3.414214, 4) and the complete one.

Degree 3 regular:

8-4326
2, 2, 2, 4, 4, 4, 6
8-6409
0.763932, 2, 4, 4, 4, 4, 5.236068
8-6579
1.438447, 2.381966, 2.381966, 3, 4.618034, 4.618034
8-6716
1.267949, 2, 2.585786, 4, 4, 4.732051,
5.414214
8-8725
2, 2, 2.585786, 2.585786, 4, 5.414214,
5.414214

Degree 4 regular

8-4575
4, 4, 4, 4, 4, 4, 8
8-9570
2.763932, 4, 4, 4, 4, 6, 7.236068
8-10188
2.438447, 3.381966, 3.381966, 5, 5.618034, 5.618034, 6.561553
8-10202
2.585786, 3.267949, 4, 4, 5.414214, 6, 6.732051
8-10819
2.585786, 2.585786, 4, 5.414214, 5.414214, 6, 6
8-10888
2, 4, 4, 4, 6, 6, 6

Degree 5 regular

8-10481
4.381966, 4.381966, 5, 5, 6.618034, 6.618034, 8
8-10975
4, 4, 6, 6, 6, 6, 8
8-11082
4, 4.585786, 4.585786, 6, 6, 7.414214, 7.414214

Degree 6 regular

8-11112
6, 6, 6, 6, 8, 8, 8

Credits: the list of graphs by Brendan McKay and NetworkX library specifically the methods read_graph6 (to read the files provided by Prof. McKay), laplacian_spectrum, diameter, degree, and draw.

Nodal lines

Wikipedia article on nodes offers this 1D illustration: a node is an interior point at which a standing wave does not move.

Standing wave and its nodes
Standing wave and its nodes

(At the endpoints the wave is forced to stay put, so I would not count them as nodes despite being marked on the plot.)

A standing wave in one dimension is described by the equation {f''+\omega^2 f=0}, where {\omega} is its (angular) frequency. The function {u(x,t) = f(x)\cos \omega t} solves the wave equation {u_{tt}=u_{xx}}: the wave vibrates without moving, hence the name. In mathematics, these are the (Dirichlet) eigenfunctions of the Laplacian.

Subject to boundary conditions {f(0)=0 = f(\pi)} (fixed ends), all standing waves on the interval {(0,\pi)} are of the form {\sin nx} for {n=1,2,3,\dots}. Their eigenvalues are exactly the perfect squares, and the nodes are equally spaced on the interval.

Things get more interesting in two dimensions. For simplicity consider the square {Q=(0,\pi)\times (0,\pi)}. Eigenfunctions with zero value on the boundary are of the form {f(x,y) = \sin mx \sin ny} for positive integers {m,n}. The set of eigenvalues has richer structure, it consists of the integers that can be expressed as the sum of two positive squares: 2, 5, 8, 10, 13, 17,…

The zero sets of eigenfunctions in two dimensions are called nodal lines. At a first glance it may appear that we have nothing interesting: the zero set of {\sin mx \sin ny} is a union of {n-1} equally spaced horizontal lines, and {m-1} equally spaced vertical lines:

Boring nodal lines
This is a square, not a tall rectangle

But there is much more, because a sum of two eigenfunctions with the same eigenvalue is also an eigenfunction. To begin with, we can form linear combinations of {\sin mx \sin ny} and {\sin nx \sin my}. Here are two examples from Partial Differential Equations by Walter Strauss:

When {f(x,y) = \sin 12x \sin y+\sin x \sin 12y }, the square is divided by nodal lines into 12 nodal domains:

Frequency 145, twelve nodal domains
Eigenvalue 145, twelve nodal domains

After slight perturbation {f(x,y) = \sin 12x \sin y+0.9\sin x \sin 12y } there is a single nodal line dividing the square into two regions of intricate geometry:

Also frequency 145, but two  nodal domains
Also eigenvalue 145, but two nodal domains

And then there are numbers that can be written as sums of squares in two different ways. The smallest is {50=1^2+7^2 = 5^2+5^2}, with eigenfunctions such as

\displaystyle    f(x,y) = \sin x\sin 7y +2\sin 5x \sin 5y+\sin 7x\sin y

pictured below.

Frequency 50
Frequency 50

This is too good not to replicate: the eigenfunctions naturally extend as doubly periodic functions with anti-period {\pi}.

Periodic extension
Periodic extension