Analogous to the classical Dirichlet problem on a (bounded) domain Ω⊆Rn, we can define the Dirichlet problem on graphs.
Problem 2.5.1 (Dirichlet problem on graphs) Let G = (V, E) and g be a function on a subset ∅ 6=∂V ⊆V (called boundary of V). Determine a function f on V, such that f|∂V =g and f is harmonic on
◦
V :=V \∂V (called interior of V), i.e., Lf = 0 on V◦.
Now it turns out that in the finite case, existence of a solution is always given, whereas uniqueness is not, when we allow graphs that are not connected1. Theorem 2.5.2 The Dirichlet problem on graphs always has a solution. It is unique iff dist(v, ∂V)<∞ for all v ∈V.
Proof: If dist(v, ∂V)<∞ for all vertices v ∈V then we can w.l.o.g. assume that the graphGis connected. (Otherwise, we would consider each connected component Gj with ∅ 6= ∂Vj separately.) Order the vertices, such that for
1Recall that in the continuous case, Ω is defined to be connected. But also if Ω consists of, say, two disjoint domains, uniqueness is still given.
some matrix M we can write
L= Le
◦
V MT
M Le∂V
!
. (2.25)
From (2.9) we know that Lev1 is positive definite. Thus, by Theorem A.1.4 (Hurwitz) detLe∂V 6= 0. So, the unique solution onV◦ is given by−(eL∂V)−1M g.
If there are vertices v with dist(v, ∂V) = ∞, then f can be set to an arbi-trary common value on these vertices. The remaining vertices have values as shown above. Hence, there also exists a solution, but it is not unique.
Note that the statement concerning the finite distances is equivalent to the statement that Le∂V is essentially diagonally dominant, defined below.
Definition 2.5.3 For a given symmetric matrix M ∈Rn×n define its graph GM = (VM, EM) by VM := {v1, . . . , vn} and EM := {{vj, vk} : Mj,k 6= 0}.
The matrix M is called essentially diagonally dominant if
|Mj,j| ≥X
k6=j
|Mj,k| for all 1≤j ≤n ,
and in each connected component of GM there exists a vertex vj, such that
|Mj,j|>X
k6=j
|Mj,k| .
The definition can also be used for a sufficient condition when the following iterative method converges to the unique solution of a Dirichlet problem on graphs. In numerical analysis, this iterative method is known as Jacobi iteration, and is used to solve systems of linear equations, see also Sect.A.3.
Definition 2.5.4 Let L ∈ Rk×k with L = D − A, where D is a regular diagonal matrix, and let x, b∈Rk. The Jacobi iteration is given by
x←D−1Ax+D−1b .
(We will, of course, apply this to L:=Le∂V, D :=De∂V and A:=Ae∂V.) With the notation from (2.25), b:=−M g and f|∂V :=g, we obtain the following formulation of the Jacobi iteration on graphs for the Dirichlet problem
∀v ∈V◦ f(v)← 1 deg(v)
X
w∈N(v)
f(w) , (2.26)
where the assignment is done for all vertices in parallel, i.e., if we write the iteration as a sequencef1, f2, . . . thenfj+1(v) := P
w∈N(v)fj(w)/deg(v).
Otherwise, if fj(w) is replaced by fj+1(w) if w has already been processed before v in some order of the vertices, the iteration is known as Gauß-Seidel iteration. Note that (2.26) sets each vertex to the current meanvalue or, equivalently, the current barycenter of its neighbors.
Theorem 2.5.5 The Jacobi iteration (Gauß-Seidel iteration) converges to the solution x of Le∂Vx=b if Le∂V is essentially diagonally dominant.
The basic recipe to prove the theorem is to show that the spectral radius ρ of D−1A is smaller than 1. Convergence (and uniqueness) then follows, e.g., from the Banach fixed point theorem with respect to some constructed norm.
For the special case that Le∂V is indeed a proper submatrix of a Laplacian matrix L, a proof is given within Theorem 5.4.5.
Note that (2.26), just as Lf|V◦ = 0 straight-forward also does, implies that for all v ∈ V◦, the solution f(v) is the meanvalue (barycenter) of its neighbors, and compare this to the meanvalue property for the continuous case, Lemma 1.2.7. Let us state this in the following lemma.
Lemma 2.5.6 (Meanvalue property on graphs) Let ∅ 6= ∂V ⊆ V and
∀v ∈V dist(v, ∂V)<∞. Let f be a function on V, harmonic on V◦. Then
∀v ∈V◦ f(v) = 1 deg(v)
X
w∈N(v)
f(w) .
Analogous to the continuous case, we can also define Perron’s method to obtain the solution of a Dirichlet problem on graphs. By means of the Gauß-Seidel iteration, and in contrast to the continuous case, this yields a con-structive method. First, we need the finite analog of the maximum principle, Lemma 1.2.9.
Lemma 2.5.7 (Maximum principle on graphs) Let ∅ 6= ∂V ⊆ V and for all v ∈ V dist(v, ∂V) <∞. Furthermore, let f be a function on V that is harmonic on V◦. Then f has no local strict maximum v ∈V◦, i.e., there is no v ∈V◦ with ∀w∈N(v) f(v)> f(w). The maximum is attained on ∂V. Proof: First, a local strict maximum v ∈ V◦ is a contradiction to the mean-value property on graphs. Now, assume that the maximum of f is not at-tained on∂V. Then there is a vertexv ∈V◦ withf(v)> f(w) for allw∈∂V. Then by the meanvalue property and induction f(w) =f(v) for all vertices
w∈V, which again is a contradiction.
Note that in the finite case, we might have local maxima, and only strict maxima in
◦
V are not possible. The lemma also holds for subharmonic func-tions, and the correspondingminimum principle also holds for superharmonic functions, defined as follows.
Definition 2.5.8 A function f on V is subharmonic (superharmonic) if
∀v ∈V◦ 1
We prove Perron’s method on graphs, the finite version of Theorem 1.5.3.
Theorem 2.5.9 (Perron’s method on graphs) Let∅ 6=∂V ⊆V and for allv ∈V dist(v, ∂V)<∞. LetF :={f : f is subharmonic and f|∂V ≤g}.
The (vertexwise) maximum f(v) := max{h(x) : h ∈ F } is harmonic on
◦
V and solves the Dirichlet problem.
Proof: Clearly, F contains the solution f and we only have to prove that there is no function h∈ F with h(v)> f(v) for some vertex v ∈V◦. Since h is subharmonic and f is (super-)harmonic, h−f is a subharmonic function with (h−f)|∂V ≤ 0 and a positive value in
◦
V. That is a contradiction to
the maximum principle on graphs, Lemma 2.5.7.
Now we can give the constructive solution method already mentioned. Set b :=−M g, f|∂V :=g (notation from (2.25)) and f|V◦ = minv∈∂V g(v). Then the Gauß-Seidel iteration remains within the class of subharmonic func-tions F (whereas the Jacobi iteration, in general, does not) and by The-orem 2.5.5 converges to the unique solution. This can be seen as the finite analog to the method discussed at the end of Sect. 1.6, with the difference that convergence is guaranteed in the finite case.
Before we take a look at an example how the Dirichlet problem on graphs can be used to approximate the classical Dirichlet problem, we consider Dirichlet’s principle on graphs. We have already used, see (2.15), the fol-lowing formulation for eigenvaluesλj and corresponding unit eigenvectors uj
λj = min
Theorem 2.5.10 (Dirichlet’s principle on graphs) Let ∅ 6= ∂V ⊆ V, for all v ∈V dist(v, ∂V)<∞, let g be a function on ∂V. The solution f of the Dirichlet problem (V, ∂V, g) minimizes the following quadratic form, i.e.,
f = arg min
The quadratic form hTLh is sometimes called energy E(h), and plays the same role as the Dirichlet integral D(h) in the continuous case.
Proof: By the discussions above, we know that the solution f is the unique function on V with f|∂V = g and Lf|V◦ = 0. Let ˆh with ˆh|∂V = g have minimum energy E(ˆh). For ε >0 and any functionh on V with h|∂V = 0
ˆhTLˆh≤(ˆh+εh)TL(ˆh+εh) = ˆhTLˆh+ 2εhTLˆh+ε2hTLh .
Thus, hTLhˆ = 0, which implies Lˆh|V◦ = 0 since h was arbitrary apart from
h|∂V = 0. Thus, ˆh=f.
The following example illustrates how the Dirichlet problem on graphs yields approximations of the solution of a classical Dirichlet problem.
Example 2.5.11 In Example1.4.2, we solved the classical Dirichlet problem (Ω, g) with Ω := B(0, ε,1) ⊆ R2 for 0 < ε < 1, g(x) = 0 for |x| = 1, and g(x) = 1 for |x| = ε. Approximate Ω by the embedding of a grid graph Gδ with a sufficiently small edge length δ >0, see Fig. 2.8. Define ∂V as the set of vertices of degree less than four, i.e., the perimeter vertices. Furthermore, set f(v) := 0 for the vertices v ∈ ∂V next to |x| = 1 and set f(v) := 1 for v ∈ ∂V next to |x| = ε. The solution of Lf = 0 on
◦
V approximates the solution of the classical Dirichlet problem.
Let us briefly sketch why this approximates the solution of the classical Dirichlet problem. Details will be given in Sect. 2.7. Let x = (x1, x2) ∈ Ω coincide with the position of v ∈ V◦. In the following we use the symbol f both for the function on Ω and on V. We may approximate the Laplacian operator as usual by the difference quotient with respect to δ given by
−∆f(x)≈ 2f(x1, x2)−f(x1+δ, x2)−f(x1−δ, x2) δ2
+ 2f(x1, x2)−f(x1, x2+δ)−f(x1, x2−δ) δ2
= X
w∈N(v)
f(v)−f(w)
δ2 =Lf(v)/δ2 = 0 =Lf(v) . (2.27) From this we see that the action of−∆ can locally be approximated2byL/δ2, such that we can expect the function f on V should also approximate the function f on Ω. Section 2.7 shows that in this case (2 dimensions) we can globally even remove the factor δ−2 from the relation between −∆ andL.
2Note that the minus sign stems from the fact that inLf(v), the values of the neigh-borsf(w) are subtracted fromf(v) instead of vice versa.
Figure 2.8: Classical Dirichlet problem and Dirichlet problem on graphs
Table 2.3: Classical Dirichlet problem and Dirichlet problem on graphs Continuous case inRn Finite case on G
Laplacian operator ∆ Laplacian matrix L
f: Ω−→R, f:V −→R,
Ω bounded domain V vertex set of a graph G (Classical) Dirichlet problem
f ∈ C2(Ω)∩ C(Ω), f:V −→R,
f =g on ∂Ω, f =g on∂V,
∆f = 0 on Ω Lf = 0 on V◦
Existence of solution
iff allx∈∂Ω regular always given Uniqueness of solution
always given iff ∀v ∈V dist(v, ∂V)<∞ Meanvalue property
for all x∈Ω, B(x, r)⊆Ω for all v ∈V◦ f(x) = 1
rn−1ωn Z
|ξ−x|=r
f(ξ)dξ f(v) = 1 deg(v)
X
w∈N(v)
f(w) Maximum principle
if f is non-constant, if ∀v ∈V dist(v, ∂V)<∞, no local maximum; no local strict maximum, maximum attained on∂Ω maximum attained on ∂V
Perron’s method
F :={f ∈ C(Ω) : f sub., f|∂Ω≤g} F :={f : f sub., f|∂V ≤g}
if allx∈∂Ω regular, the solution is a solution is always f(x) = max{h(x) : h∈ F } f(v) = max{h(x) : h∈ F }
Dirichlet’s principle
if f ∈ C2(Ω)∩ C(Ω), f|∂Ω =g mins. if f|∂V =g minimizes D(h) =
Z
x∈Ω
|∇h(x)|2 dx , E(h) = X
(v,w)∈Eσ
(h(v)−h(w))2 , then f is the solution then f is a solution other implication is false other implication is true
In Table 2.3, we point out some more relations between continuous and finite case. We may also consider to compare the Beer-Neumann method and the iterative method mentioned at the end of Sect. 1.6 with iterative methods to solve Lf = 0, i.e., by Jacobi and Gauß-Seidel iteration.
Finally, we will consider another application of the Dirichlet problem on 3-connected graphs. A graphG has(vertex) connectivity k iff the removal of any subset C ⊆V with|C|< kleaves Gconnected. We briefly mention that the maximum k, such that G is k-connected, is bounded from below by λ2. This follows immediately from (2.13) and using a vertex set C with |C|=k that disconnects G. Anyhow, we want to go over to the following theorem.
Theorem 2.5.12 (Tutte [113]) Fixing the vertices of a face of a planar, 3-connected graph onto the corners of a strictly convex polygon C, and setting the remaining nodes to the barycenter of their neighbors, yields a planar (straight-line) embedding.
The connection to the Dirichlet problem is obvious. Define ∅ 6=∂V to be the vertices of the mentioned face. SinceG is 3-connected the condition that for allv ∈V dist(v, ∂V)<∞is clearly fulfilled, and the embedding described in the theorem is the unique embedding defined by the two functionsx1, x2onV that coincide with the positions of C on ∂V and the remaining values are determined by the two corresponding Dirichlet problems. Chapter 5 shows lots of embeddings of this type. The theorem also holds when replacing the strictly convex polygon (all inner angles are smaller than π) by a convex polygon (all inner angles are not greater than π). But strict convexity is necessary if we allow also subdivisions of 3-connected graphs (i.e., replacing an edge by edge-vertex-edge), for which the theorem still holds. Finally, it even holds when we allow cut pairs (two vertices that disconnect G), on∂V.