The Fiedler Rose: On the extreme points of the Fiedler vector

1, i = j, e ij ∈ E 0, otherwise, its degree matrix is the diagonal matrix D, where

and its graph Laplacian is the matrix L = A -D.

The graph Laplacian is the discrete analog of the usual Laplacian; for example the heat equation du dt = ∆u can be translated to the discrete setting as the system of ODE

where the vector u(t) denotes the heat at each vertex at time t. Many of the standard results about the Laplacian carry over to the graph Laplacian (and indeed are easier to prove in the discrete setting!): -L is a symmetric, positive-definite matrix, and as such has real non-negative eigenvalues 0 = λ 1 < λ 2 ≤ λ 3 ≤ . . . ≤ λ n . λ 1 = 0 corresponds to the eigenvector whose entries are all the same; that 0 is an eigenvalue of multiplicity one is a consequence of the Perron-Frobenius theorem.

Therefore the first eigenvalue of interest is λ 2 . This eigenvalue was first studied extensively by Miroslav Fiedler and he referred to it as the algebraic connectivity of the graph due to its connection to connectivity properties of the graph (see [2] and [3]). In honor of Fiedler, it’s associated eigenvector has come to be known as the Fiedler vector (In general the algebraic connectivity may correspond to an eigenspace of dimension greater than 1, in which case there are many Fiedler vectors. In the example I will consider, however, this is not an issue).

The Fiedler vector has an important role in spectral graph theory due to its successful application towards the problem of graph partitioning (see [4]). But what motivates the example I will present is its relation to the discrete heat equation.

The solution to (1) is given by

where the e i are the orthonormal basis of eigenvectors of L. As λ i > 0 for i ≥ 2 and λ 1 = 0 corresponds to e 1 = 1 √ n (1, 1, . . . , 1) T , we have that u(t) → (u 0 , e 1 )e 1 . That is, the heat will eventually even out until, in the limit, the heat is constant across all verticies.

To study the long term behavior of u(t), we note that for t large, assuming λ 2 = λ 3 and (u 0 , e 2 ) = 0, u(t) ≈ (u 0 , e 1 )e 1 + (u 0 , e 2 )e -λ2t e 2 , as the other terms die out faster. That is, in the long run, u(t) has the same structure as the Fiedler vector e 2 up to a constant multiple and translation. Therefore, the Fiedler vector captures the transient behavior of the heat flow. In particular, the extreme points of u(t) will be extreme points of the Fiedler vector. Therefore, heuristically, the extreme points of the Fiedler vector should correspond to the most “insulated verticies”, since these are the verticies in which heat/cold will get trapped and u(t) will stay the hottest/coldest. On a tree, it might seem at first that the two most insulated verticies would be the ones farthest apart. This raises the following conjecture.

where d(v, w) is the graph-distance between v and w. In other words, the extreme values of the Fiedler vector are among the pairs of verticies which are the farthest apart.

Indeed, in a recent paper by Chung, Seo, Adluru, and Vorperian, [1], a similar conjecture is made (see Section 3 for the precise statement of their conjecture).

However, I will now present a counter-example to Conjecture 1.1. Consider the graph in Figure 2.

If we consider the Fiedler vector (which will be computed in the next section) for this graph and color the verticies red, whose entry in the Fiedler vector is positive, and blue, whose entry in the Fiedler vector is negative, we get Figure 2.

The graph now looks like a rose with a curved “leaf”, long “stem”, and many “petals”, hence I will refer to it as the Fiedler rose. Note that the extreme values of the Fiedler vector are not the verticies farthest apart, the tip of the leaf and stem, but are instead the stem and any of the petal verticies (the exact values of the Fiedler vector will be computed in the next section). Thus, the Fiedler rose shows false Conjecture 1.1.

I first label the verticies of the rose as depicted, letting s be the number of verticies in the stem and p be the number of petal-verticies (I will consider Fiedler roses of varying stem lengths and petal counts). >function FiedlerRose(p,s) > >N=5+p+s; %Total verticies = 5 for leaf/rose center + s for stem + n petals >L=zeros(N,N); >r=5+s; %Center rose node > >for i=1:(4+s-1), %Building the leaf and stem > j=i+1; > L(i,i)=L(i,i)-1; > L(i,j)=L(i,j)+1; > L(j,i)=L(j,i)+1; > L(j,j)=L(j,j)-1; >end > >i=4; %Connecting the stalk to the rose center >j=r; >L(i,i)=L(i,i)-1; >L(i,j)=L(i,j)+1; >L(j,i)=L(j,i)+1; >L(j,j)=L(j,j)-1; > >for j=r+1:r+p, %Building the p rose petals > i=r; > L(i,i)=L(i,i)-1; > L(i,j)=L(i,j)+1; > L(j,i)=L(j,i)+1; > L(j,j)=L(j,j)-1; >end > >[V,D]=eigs(L,3,0.001); %Grabs the 3 eigenvalues/vectors with eigenvalue closest > %to 0.001. >v=V(:,2); %We set v to be the Fiedler vector. >if v==V(:,3), e