Combinatorial Fiedler Theory and Graph Partition

Partition problems in graphs are extremely important in applications, as shown in the Data science and Machine learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue $a(G)$ of the Laplacian matrix $L_G$ of the graph $G$. This problem corresponds to the minimization of a quadratic form associated with $L_G$, under certain constraints involving the $\ell_2$-norm. We introduce and investigate a similar problem, but using the $\ell_1$-norm to measure distances. This leads to a new parameter $b(G)$ as the optimal value. We show that a well-known cut problem arises in this approach, namely the sparsest cut problem. We prove connectivity results and different bounds on this new parameter, relate to Fiedler theory and show explicit expressions for $b(G)$ for trees. We also comment on an $\ell_{\infty}$-norm version of the problem.


Introduction
Often real world networks contain clusters, that is, groups of points each with a large number of neighbors among them and not many connections to the outside.In Data science and Machine learning the task of clustering is very important.Given a set of data points (in some space) and their common properties measured in terms of distances, the clustering problem consists in finding subsets of these data points that are "similar".If the points to be clustered are vertices in a graph, and the edges connecting these vertices are the only information available, then the problem is called graph clustering [17,22].There are many methods for graph clustering, and one popular such method is spectral clustering.The basis is then spectral bisection where a Fiedler vector [13] is used for partitioning a graph into two connected subgraphs based on the sign of the components of the vector.This splitting may be repeated for each of the parts and thereby obtain a desired partition.
Consider an unweighted (undirected) simple graph G = (V, E) and let n = |V |.Throughout the paper we assume that G is connected.Recall that the Laplacian matrix L G is the n × n matrix L G = D G −A G where D G is the diagonal matrix with the vertex degrees on the diagonal, and A G is the adjacency matrix of G. L G is positive semidefinite and therefore it has only real, nonnegative eigenvalues.The algebraic connectivity a(G) is the second smallest eigenvalue of L G , and it is known as a connectivity measure in the graph [12,20].In particular, G is connected if and only if a(G) > 0. The matrix L G is singular, 0 is the smallest eigenvalue and a corresponding eigenvector is the all ones vector e.Therefore, by the Courant-Fischer theorem [14], a(G) = min{x T L G x : e T x = 0, ∥x∥ 2 = 1}.
Here ∥x∥ 2 = ( i x 2 i ) 1/2 is the (Euclidean) ℓ 2 -norm of x = (x 1 , x 2 , . . ., x n ) ∈ R n .By using a standard factorization L G = B T B where B is the edge-vertex incidence matrix of G, we have the alternative expression Thus, an eigenvector corresponding to the eigenvalue a(G), usually called a Fiedler vector, can be seen as assigning values to the vertices to obtain an optimal "smoothing" along edges, i.e., small difference between end points of edges, under the two normalization constraints, see [23].
These constraints assure that we avoid a constant solution x = λe for some λ and, also, the norm constraint avoids scaling (to get similar solutions) and the zero vector.We shall therefore call the minimization problem in (1) the ℓ 2 -graph smoothing problem.Here ℓ 2 refers to the fact that both the (objective) function uv∈E (x u − x v ) 2 and the norm constraint involve the Euclidean norm ℓ 2 .
The motivating question of our study is: • What happens if we modify the optimization problem (1) by changing the norm involved to the ℓ 1 -norm (the sum norm)?
The ℓ 1 -norm of x = (x 1 , x 2 , . . ., x n ) ∈ R n is defined by ∥x∥ 1 = i |x i |.In fact, often in mathematics different norms may be used in the study of some (approximation) problem.A well-known such example is the linear approximation problem min x∈R n ∥Cx − b∥ where C is an m × n (real) matrix, b ∈ R m and ∥ • ∥ is some vector norm.When the norm is ℓ 2 we obtain the least squares problem, and for the norms ℓ 1 and ℓ ∞ one may use linear programming to solve the problem.Therefore, it is important to understand the properties of solutions, and how they depend on the choice of norm.Therefore, our main goal is to consider a new graph smoothing problem which we call ℓ 1graph smoothing.It is similar to ℓ 2 -graph smoothing except that we change the norm from the ℓ 2 -norm to the ℓ 1 -norm.Let G = (V, E) be a given graph with at least one edge.The ℓ 1 -graph smoothing problem is the following optimization problem Clearly the minimum here is attained by some x as the constraints define a compact set and the function to be minimized is continuous.Note that the constraint set is not a convex set.An optimal solution x in (2) will be called an ℓ 1 -Fiedler vector.Then A main contribution of our paper is indicated in Figure 1 (where x ⊥ e means e T x = 0).The two graph smoothing problems are indicated in the two last columns of the figure.One sees how the two problems are quite similar.Based on several intermediate results we establish that optimal solutions in the new problem correspond to so-called sparsest cuts.This also means that strong connections, with bounds, between the two optimal values a(G) and b(G) may be found.Moreover, there are important consequences in terms of computational complexity.A main contribution of this paper is to show that there is a very natural optimization approach that is underlying sparsest cuts.Thus the problem (2) can be handled using a combinatorial approach.For related combinatorial approaches to Perron values of trees, see [3,4].We will discuss the computational complexity for these graph smoothing problems, although this is not done in any detail.However, some remarks are given on the complexity of approximation problems related to b(G), and we believe more theoretical work can be done here.The approaches discussed in [7] for approximate graph coloring may be of interest in this connection.
The remaining part of this Introduction is devoted to some more results from spectral partitioning and relevant spectral graph theory.Finally, an overview of the next sections is given.
In [26] one studies the maximal error in spectral bisection where the two parts in the partition have the same size.In [15] the authors investigate graphs having Fiedler vectors with unbalanced sign patterns such that a partition can result in two connected subgraphs that are distinctly different in size.They also characterize graphs with a Fiedler vector having exactly one negative component.Motivated by these results we recall some facts concerning spectral partitioning and Cheeger bounds.For a much deeper discussion of these areas, we refer to the lectures notes [24] and [25], and the references found there.Consider again a graph G = (V, E) and let n = |V |.Let S ⊆ V be a nonempty vertex set, where S ̸ = V .Let δ(S) denote the set of edges uv where u ∈ S and v ̸ ∈ S, this is called the cut induced by S. Cuts are important objects in graph theory, combinatorial optimization as well as in applications.Define the relative cut size which is the size of the cut relative to the size of the vertex set S. This is an important notion in this paper.The isoperimetric number of G, also called Cheeger's constant, is the parameter where the minimum is taken over all nonempty subsets S of V with |S| ≤ ⌊n/2⌋.
A basic treatment of the isoperimetric number and its properties may be found in [19].A related notion [18] is the edge density of a cut, defined as follows, This concept represents the density of the edges in G between the set S and its complement, compared to the number of edges in a complete bipartite graph (with vertex set S and its complement).This parameter ρ(S) is also called the sparsity of the cut.A sparsest cut is a cut which minimizes ρ(S).For more literature related to these concepts, see [10].
A small calculation shows the following relation between edge density and relative cut sizes for each subset S (with ∅ ⊂ S ⊂ V ) Below we give some inequalities relating edge density and Laplacian eigenvalues, in particular the algebraic connectivity.
Theorem 1.1 ( [18]).Let G be a graph of order n.For any nonempty subset S of vertices of G, S ̸ = V , the edge density is uniformly bounded below and above by where λ 1 is the largest eigenvalue of L(G).
In [11] one characterized the graphs for which a(G) = ρ(S), for some subset S of vertices.
There is another upper bound on the minimal density of cuts in terms of a(G).

Theorem 1.2 ([18]
).Let G = (V, E) be a graph of order n with at least two edges.Then where d max (G) is the maximal vertex degree in G.
This upper bound is a strong discrete version of the well-known Cheeger's inequality from differential geometry [8], bounding the first eigenvalue of a Riemannian manifold.It appeared in [1,2] and later, as an improved edge version in [19].
The remaining paper is organized as follows.In Section 2 we study the set of feasible solutions of the ℓ 1 -graph smoothing problem and present a rewriting of the problem.The main results are then presented in Section 3, where it is shown that optimal solutions correspond to sparsest cuts with a connectivity property.The computational complexity is also settled.In Section 4 a comparison of b(G) and other parameters is made.Section 5 is devoted to examples and specific classes of graphs where explicit expressions for b(G) are found.Moreover, a computational example is shown.In the final section we briefly consider a graph smoothing problem based on the ℓ ∞ -norm.
Notation: Vectors in R n are considered as column vectors and identified with the real ntuples.The i'th component of a vector x ∈ R n is usually denoted by x i (i ≤ n).A zero matrix, or vector, is denoted by O, and an all ones vector is denoted by e (the dimension should be clear from the context).For a real number c define c + = max{c, 0} and c − = max{−c, 0}.Then K n represents the complete graph.Moreover, P n (resp.S n ) is the path (resp.the star) with n vertices.
Throughout, we let the vertex set of the graph G be V = {v 1 , v 2 , . . ., v n }, and we identify a function x ∈ R V with the vector x = (x 1 , x 2 , . . ., x n ) where x j = x(v j ) for each j ≤ n.Define F 1 as the feasible set in (2), i.e., We also define the "smoothing function" , the path with four vertices.Let x 1 = (−1/2, 0, 0, 1/2), so the only nonzeros are in the end vertices.Then The answer is yes, as will follow from later results.We remark that the algebraic connectivity of P 4 is given by a(P 4 ) = 0.5858.
A vector x ∈ F 1 will be called a feasible solution of (2).Thus, by Lemma 2.1, a feasible solution partitions the vertices into three subsets depending on the sign of each x j , ±1 or 0, and the sum of the components of x in the positive and negative part is the same in absolute value, j: We next rewrite problem (2).Define where Note that the constraints assure that for each i at least one of the two variables x 1 i and x 2 i is zero.The next result connects the two optimization problems ( 2) and ( 6).
Lemma 2.2.The following holds: Proof.(i) Properties (i) and (ii) follows by replacing each variable x j by x j = x 1 j − x 2 j where x 1 j and x 2 j are two nonnegative variables.In this construction x 1 and x 2 are only unique up to a positive additive constant in each term, but the orthogonality constraint (x 1 ) T x 2 assures uniqueness and that x 1 = x + and x 2 = x − .Thus ( 6) is a reformulation of (2).This implies that the optimal values coincide, so (iii) holds.
Later we prove that it is NP-hard to compute b(G) and a corresponding ℓ 1 -Fiedler vector.Still, the previous lemma means that computing b(G), and the corresponding optimal x, may be done by solving (6).This is a problem of minimizing a piecewise linear convex function subject to linear constraints and a "complementarity constraint" saying that x 1 i x 2 i = 0 for each i.This problem can be written as a linear programming problem with certain linear complementarity constraints corresponding to the orthogonality x 1 ⊥ x 2 , as explained next.Consider the following optimization problem with variables x 1 j , x 2 j (j ≤ n) and y ij for ij ∈ E.
In fact, the first two constraints are equivalent to |x ) and due to the minimization equality must hold here for every ij ∈ E. As mentioned, the ℓ 1 -graph smoothing problem is NP-hard, but several general integer programming based algorithms have been developed that may be used to give approximate solutions of (7).A basic reference on the linear complementarity problem is [9].We leave it as an interesting idea for further research to use this formulation in order to find approximate solutions of the ℓ 1 -graph smoothing problem.In the final section of this paper we also use a related linear programming approach to a graph smoothing problem based on the ℓ ∞ -norm.

ℓ 1 -graph smoothing and sparsest cuts
In this section we investigate the ℓ 1 -graph smoothing problem closer and establish strong properties of the optimal solutions.This leads to a connection to sparsest cuts.Recall that we assume that the graph G is connected.The proof gives a construction based on the relative cut size ξ(S) defined in (3).Theorem 3.1.Let x be an ℓ 1 -Fiedler vector with the maximum number of zeros.Then the subgraph induced by {v ∈ V : x v > 0} is connected and the subgraph induced by {v ∈ V : Proof.We shall first prove that the subgraph induced by The proof is by contradiction, so assume the subgraph induced by V + is not connected.Then there must exist disjoint subsets S 1 and S 2 of V + such that (i) no edge joins S 1 and S 2 , We discuss different cases.Case 1: ξ(S 1 ) > ξ(S 2 ).Let ϵ be a "suitably small" positive number; in fact, ϵ < min{x v : v ∈ V + } works.Define y ∈ R V based on x as follows: Moreover, v:yv≤0 Case 2: ξ(S 1 ) < ξ(S 2 ).By symmetry of S 1 and S 2 this can be treated by similar arguments and a contradiction is derived.
Case 3: ξ(S 1 ) = ξ(S 2 ).We use the same construction of the vector y as in Case 1, but we let ϵ = min{x v : v ∈ S 1 }.Then y ∈ F 1 , and from (8) we see that f 1 (y) = f 1 (x), so y is also an ℓ 1 -Fiedler vector.However, y has at least one more zero than x, and this contradicts the choice of x (initially in the proof).
Thus, in each case we obtained a contradiction, which proves that the subgraph induced by V + := {v ∈ V : x v > 0} is connected.The proof that the subgraph induced by {v ∈ V : Next we give a main result which shows an explicit formula for b(G) which is of a combinatorial nature.This will give a strong connection to the notions presented in the Introduction.
We say that the pair (S 1 , S 2 ) is a quasi-bipartition of V if Moreover, when (S 1 , S 2 ) is optimal in (9), the corresponding vector x (S 1 ,S 2 ) is an ℓ 1 -Fiedler vector x.
Proof.Let x be an ℓ 1 -Fiedler vector.Let κ(x) be the number of distinct positive elements in x (i.e., the cardinality of the set of positive components).Choose x (optimal) with κ(x) smallest possible.Define Both S + and S − are nonempty.Also define We now prove that there is an optimal solution of (2) where all positive x v 's are equal.If M 1 = M 2 , there is nothing to prove, so assume M 1 > M 2 .Let ϵ 1 be a "small" number in absolute value and let Observe that the relationship between ϵ 1 and ϵ 2 assures that Thus, , and for all other edges ∆ uv = 0. Let N a , N b , N c , N d be the number of edges in categories a, b, c, d, respectively.Then By inserting the expression for ϵ 2 above we obtain Moreover, there exists an ϵ * > 0 such that for all ϵ 1 with |ϵ 1 | < ϵ * , the vector x ϵ lies in F 1 .We must have η = 0, otherwise we could let ϵ 1 be small enough and with opposite sign as η and then So f 1 (x ϵ ) < f 1 (x) which contradicts the optimality of x.Therefore, η = 0 and Now, let ϵ 1 > 0 and increase ϵ 1 until ∆ uv becomes 0 for some edge for which it was previously positive.This happens if either the smallest value in S + has been decreased to 0, or when largest value has been decreased to the second largest, or the smallest value has been increased to the second smallest (or both of these occur simultaneously).This x ϵ 1 is also an ℓ 1 -Fiedler vector.As κ(x ϵ 1 ) < κ(x), this contradicts our choice of x.Thus, by contradiction, it follows that M 1 = M 2 , so all positive components in x are equal.
Finally, among all ℓ 1 -Fiedler vectors whose positive components coincide with that of x, we proceed to treat the negative components in exactly the same manner as the first part of the proof.As a result, we find an ℓ 1 -Fiedler vector where all the negative components have the same value, and all the positive components have the same value.Let x denote this vector and define S 1 = {v : x v > 0}, S 2 = {v : x v < 0} and S 0 = {v : x v = 0}.Let [S i : S j ] be the set of edges uv such that u ∈ S i , v ∈ S j , where i, j ∈ {0, 1, 2}.Then From this calculation we also see that for any )), and therefore f 1 (x ′ ) ≥ b(G).This proves the theorem.
The next corollary says that if x is an ℓ 1 -Fiedler vector then x has no component equal to zero.
Proof.Choose an ℓ 1 -Fieldler vector vector x.Partition the set of vertices into S + = {v : x v > 0}, S − = {v : x v < 0} and S 0 = {v : x v = 0}.We recall that S + and S − are nonempty and we want to prove that S 0 = ∅.Consider the following edge sets Assume that S 0 ̸ = ∅.
• Let S ′ = S + ∪ S 0 and consider the quasi-bipartition (S ′ , S − ).Then • Let S ′′ = S − ∪ S 0 and consider the quasi-bipartition (S + , S ′′ ).Then The next theorem sums up the results above.It connects b(G) to the minimum edge density and also shows that we may restrict to connected subgraphs when computing b(G). where is the edge density of the cut δ(S) and the minimum is taken for nonempty subsets S of V such that S ̸ = V and both S and its complement induce connected subgraphs of G.
Proof.This follows by combining Theorem 3.2 and Corollary 3.3: we have that the quasibipartition (S 1 , S 2 ) must be a partition, i.e., S 1 ∪ S 2 = V as from Corollary 3.3 any ℓ 1 -Fiedler vector has no components equal to zero.Therefore δ(S 1 ) = δ(S 2 ).The argument in the proof of Theorem 3.1 assures connectedness of the two subgraphs.The definition of edge density of a cut and the relation in (4) give the desired formula for b(G).
Thus we arrive at the important insight: • up to a multiplicative constant, namely n/2, the optimal value b(G) in the ℓ 1 -graph smoothing problem coincides with the smallest edge density of a cut in G.
This gives a very intuitive interpretation of the partitioning of a graph G according to the positive and negative values in the ℓ 1 -Fiedler vector: it corresponds to a cut of smallest edge density, also called a sparsest cut.In fact, for such a sparsest cut δ(S) a corresponding ℓ 1 -Fiedler vector is x = x S = (x v : v ∈ V ) given by where S = V \ S. Thus, this correspondence is underlying when we later refer to a solution of the ℓ 1 -graph smoothing problem or the sparsest cut problem, i.e., one solution can be converted into the other.Due to this close connection to the sparsest cut, we can now conclude that computing an ℓ 1 -Fiedler vector is a computationally hard problem.
Corollary 3.5.The computation of b(G) and a corresponding ℓ 1 -Fiedler vector is NP-hard.
Proof.It was shown in [6] that computing min{ρ(S) : ∅ ⊂ S ⊂ V } is NP-hard.Therefore, due to Theorem 3.4, the desired conclusion follows.
As mentioned, in [6] it is shown that the sparsest cut is NP-hard.The paper further contains a number of results showing that the problem is polynomially solvable for certain classes of graphs.For instance, this applies to graphs with bounded treewidth.Furthermore, for cactus graphs, i.e., connected graphs where each edge is in at most one cycle, there is an algorithm for finding a sparsest cut which is linear in n, the number of vertices.This class includes trees, which we return to below.In [5] an efficient approximation algorithm for the sparsest cut was established, and it gives an O( √ n) approximation (to the minimum value).In [27] there is whole chapter devoted to the sparsest cut problem.Here the connection to a certain linear programming problem is shown.This is a multicommodity network flow problem where the goal is to maximize throughput for a given set of demands given by origin/destination (OD-) pairs and the corresponding flow demand.The special case where all demands are 1 and every pair is an OD-pair leads to an upper bound of the throughput which is the edge density.Therefore the minimum edge density, and a sparsest cut, corresponds to a bottleneck of the multicommodity flow problem.In this approach linear programming duality is combined with some basic results on embeddability of ℓ 1 -metric spaces.This leads to a very important approximation algorithm with approximation error O(log n); for the details we refer to [27] or [25].
Sparsest cuts are used in applications.For instance, in [28] sparsest cuts (which is called ratio cuts there) are used in image segmentation.They also show that the sparsest cut problem is polynomially solvable in planar graphs, which is of interest in image analysis.In [21] one considered a segmention (or decomposition) problem in image analysis, using a model based on partial differential equations and total variation norm (L 1 -norm).

Comparison with other parameters
This section establishes bounds on b(G) in terms of other parameters.
By combining Theorem 3.4 and Theorem 1.1 we now obtain the following bounds on b(G).
Corollary 4.1.Let G be a graph of order n.Then where a(G) is the algebraic connectivity of G and λ 1 is the largest eigenvalue of L(G).
Let d min (G) denote the smallest degree of a vertex in G. Proof.Let S 1 and S 2 be such that the minimum in ( 9) is attained (so Next, let v be a vertex with smallest degree in G, so d v = d min (G), and let S = {v}.Then ρ(S) = d v /(n − 1) so, by Theorem 3.2, an upper bound on b(G) is n 2(n−1) d min (G).
Let mc(G) denote the cardinality of a minimum cut in G, i.e., The next result is proved similar to the upper bound in the previous corollary, by letting S be such that δ(S) is a minimum cut.
where s = |S| and δ(S) is a minimum cut in G.
This bound is of interest because a minimum cut may be found efficiently (polynomial time) by simple greedy algorithms.Moreover, such a minimum cut may be the starting point of algorithms for computing approximations to b(G).
In order to compare b(G) to the algebraic connectivity a(G) we need a well-known property of norms.
Let m = |E| be the number of edges in G.
Proof.Let x ′ be a Fiedler vector of unit length, so uv∈E (x where the last inequality is obtained from the Cauchy-Schwarz inequality.

Examples and special graphs
This section contains examples and results for special graphs.
The next example considers the complete graph, which is an extreme case in terms of the parameter b(G).• v is the central vertex.Then, ρ(S) = n−1 n−1 = 1.• v is not the central vertex.Then, ρ(S) = 3 n−1 .2. Consider all remaining subsets S, such that |S| = i ≥ 2. Again, two situations can occur: With some calculus we see that min S ρ(S) is equal to 2/(n − 2), so b(G) = n/(n − 2).
For trees we can find a sparsest cut analytically, as described next.Let T = (V, E) be a tree, and let e = uv ∈ E. Then T \ {e} consists of two disjoint trees, T e u and T e v , where T e u = (V e u , E e u ) contains u, T e v = (V e v , E e v ) contains v and V e u ∪ V e v = V .We call e = uv a center edge if ||V e u | − |V e v || is smallest possible and we let ∆(T ) denote this minimum value.For instance, let T be the path P n .if n = |V | is even, then ∆(P n ) is 0 , and there exist only one center edge.If n is odd, then ∆(P n ) is 1 and there are two center edges.If T is the star S n , then ∆(S n ) = n − 2 and all the edges of S n are center edges.
A substar S in a tree T is a vertex-induced subgraph which is a star, i.e., it consists of edges sharing a common vertex.
(a) n is even, say n = 2k for some integer k.
Then, e = e k,k+1 is the unique center edge, so C = {e}.In this case we have (b) n is odd, say n = 2k + 1 for some integer k.
A pendant edge in a graph is an edge where one of its vertices has degree 1.
Example 8. Let n 1 , n 2 , . . ., n k be positive integers.The starlike tree S(n 1 , n 2 , . . ., n k ) is the tree that results from the stars S n 1 , S n 2 , . . ., S n k by connecting their centers to an extra vertex v, see [16].Let n = k i=1 n i + 1 and define n M = max i=1,2,...,k n i .Then, , then there is a unique center edge, which is vv ℓ .Otherwise the maximum of |S i | is attained for more than one i, and the center edges constitute a substar connecting v to the center of those stars S i .
Finally, in this section, we give a computational example for a graph G.It illustrates the partitions obtained based on the Fiedler vector and the ℓ 1 -Fiedler vector, respectively.
Example 9.In this example we generated some "random" points in the plane and constructed edges between points that were closer than some given distance.This gave a graph G.The graph contained 15 vertices and 69 edges.We wanted to compare the partitions obtained from the ℓ 2 (Fiedler) versus ℓ 1 (sparsest cut) graph smoothing approach.
In the ℓ 2 approach we obtained a cut with 22 edges and the partition contained 7 and 8 vertices, respectively.In the ℓ 1 approach we obtained a cut with 12 edges and the partition contained 3 and 13 vertices, respectively.The corresponding edge densities were 0.39 and 0.33.So the sparsest edge density is 0.33.Note that the sparsest cut is quite unbalanced, but has very few edges compared to the cut in the Fiedler partition.For certain graphs this may happen, but graphs are so different that we will not make any general claims on properties of these solutions.Concerning the previous example note that each of the two solutions are optimal in the respective optimization problems, so comparing partitions is a bit artificial.However, both methods may be used for partitioning, and which one is the better is hard to say in general.It depends on the underlying application and the type of graph considered.Computationally, the Fiedler vector can be computed fast, while for larger graphs we need heuristics for finding approximate sparsest cuts.On the other hand, a sparse or sparsest cut is a reasonable notion and can be analysed for specific graph classes, while the Fiedler vector relies on an eigenvector which is not easy to give a direct combinatorial interpretation.Finally, for clustering it might be possible to combine these two methods, possibly also with some of the many other approaches known for graph clustering.

Graph smoothing in ℓ ∞ -norm and concluding remarks
We conclude the paper with some remarks.First, it also makes sense to consider the graph smoothing problem in other norms, such as the ℓ ∞ -norm (given by ∥x∥ ∞ = max i |x i |).This leads to the ℓ ∞ -graph smoothing problem In contrast to the ℓ 1 -graph smoothing problem we can solve (16) efficiently, for general graph G.
Proof.We rewrite problem (16) using a construction rather similar to what we did for the ℓ 1 -graph smoothing problem in (7).For each k ≤ n consider the linear programming problem LP(k): minimize y x i − x j ≤ y (ij ∈ E), −(x i − x j ) ≤ y (ij ∈ E), n j=1 x j = 0, −1 ≤ x j ≤ 1 (j ≤ n), x k = 1. ( Here the variables are x = (x 1 , x 2 , . . ., x n ) and y.Note that in every optimal solution of ( 17 x i = 0, which contradicts that i z i = 0. Thus, the optimal value of ( 16) is ≥ γ, and therefore equal to γ, as desired.
Thus consecutive components of x * differ in absolute value by y = 2/5.
One can also solve (16) explicitly for stars, and some other graphs.It is ongoing work to investigate the ℓ ∞ -graph smoothing problem further and to relate to the ℓ 1 -and ℓ 2 -graph smoothing problems.Moreover, it is interesting to see if the corresponding cuts obtained from the signs in an optimal x * are useful in partitioning problems.
Finally, we remark that the main contribution of the present paper was to investigate a variant of the optimization (variational) characterization of algebraic connectivity, by changing into the ℓ 1 -norm.We showed strong optimality properties that are similar to the Fiedler theory for algebraic connectivity.Also, we showed that optimal solutions correspond to sparsest cuts which gives a new way to view these combinatorial objects.We believe that further work on similar optimization problems, in different norms, would be interesting.This includes to combine the different approaches into useful algorithms for important applications in clustering problems in graphs.

Example 4 .
Consider K n the complete graph of order n.Let S ⊂ V where S ̸ = ∅, V .Define s = |S|.Thenρ(S) = s(n − s) s(n − s) = 1.which is independent of S! So, by Theorem 3.4, b(

Theorem 5 . 1 .
Let T = (V, E) be a tree.Then there exists a center edge e = uv ∈ E such that the cut δ(V e u ) is a sparsest cut, and b(T ) = (1/2) 1/|V e u | + 1/|V e v | .Proof.We apply Theorem 3.4, so a sparsest cut must be of the form C = δ(V e u ) for some edge e = uv.The corresponding edge density of the cut C is ρeasy to compute that this is minimized precisely when ||V e u | − |V e v || is smallest possible, i.e., when e is a center edge and therefore b(G) = (n/2)ρ(V e u ) = (1/2) 1/|V e u | + 1/|V e v | .