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.