A Cheeger inequality for graphs based on a reflection principle
Edward Gelernt, Diana Halikias, Charles Kenney, Nicholas F. Marshall
AA CHEEGER INEQUALITY FOR GRAPHS BASED ON AREFLECTION PRINCIPLE
EDWARD GELERNT, DIANA HALIKIAS, CHARLES KENNEY,AND NICHOLAS F. MARSHALL
Abstract.
Given a graph with a designated set of boundary vertices, we de-fine a new notion of a Neumann Laplace operator on a graph using a reflectionprinciple. We show that the first eigenvalue of this Neumann graph Laplaciansatisfies a Cheeger inequality. Introduction and Main Result
Introduction.
Suppose that G = ( V, E ) is a graph with vertices V and edges E . Let ∂V ⊆ V be a designated set of boundary vertices, and ˚ V := V \ ∂V . Wedefine the doubled graph G (cid:48) as follows. Let ˚ G = ( U, F ) be an isomorphic copy ofthe induced subgraph G [˚ V ], and let f be an isomorphism from ˚ V to U . Set F (cid:48) := (cid:8) { u, v } : u ∈ U, v ∈ ∂V, { f − ( u ) , v } ∈ E (cid:9) . Then, we define G (cid:48) := ( V (cid:48) , E (cid:48) ) where V (cid:48) := V ∪ U and E (cid:48) := E ∪ F ∪ F (cid:48) . That is tosay, G (cid:48) is defined by making an isomorphic copy of the interior of G and attachingit to the boundary vertices ∂V as in the original graph, see Figure 1. G = G = Figure 1.
A graph G , and its doubled graph G (cid:48) , where the blackand white dots denote interior and boundary vertices, respectively. Definition 1.1.
Let G (cid:48) = ( V (cid:48) , E (cid:48) ) be a doubled graph, and let f : ˚ V → U bean isomorphism as above, so that for all w ∈ ∂V and v ∈ ˚ V , { v, w } ∈ E (cid:48) ⇐⇒{ f ( v ) , w } ∈ E (cid:48) . We say that a function ϕ : V (cid:48) → R is even with respect to ∂V if ϕ ( v ) = ϕ ( f ( v )) for v ∈ ˚ V , and we say that ϕ is odd with respect to ∂V if ϕ ( v ) = − ϕ ( f ( v )) for v ∈ ˚ V , and ϕ ( v ) = 0 for v ∈ ∂V. Let L (cid:48) := D − A denote the graph Laplacian of G (cid:48) where D is the degree matrixof G (cid:48) , and A is the adjacency matrix of G (cid:48) . The following proposition characterizesthe eigenvectors of L (cid:48) as either even or odd. Mathematics Subject Classification.
Key words and phrases.
Cheeger inequality, graph Laplacian, Neumann Laplacian. a r X i v : . [ m a t h . SP ] M a y GELERNT, HALIKIAS, KENNEY, AND MARSHALL
Proposition 1.1.
The graph Laplacian L (cid:48) has | V | eigenvectors that are even withrespect to ∂V , and | ˚ V | eigenvectors that are odd with respect to ∂V ; this accountsfor all eigenvectors of L (cid:48) . Motivation.
We are motivated by the observation that the restriction of theodd and even eigenvectors of L (cid:48) to the graph G seem like natural Dirichlet andNeumann Laplacian eigenvectors for the graph G , given the respective odd andeven behavior of Dirichlet and Neumann Laplacian eigenfunctions on manifolds. Infact, the restriction of the odd eigenvectors of L (cid:48) to the graph G are eigenvectorsof the Dirichlet graph Laplacian defined by Chung in [3], and inequalities involvingthe eigenvalues of this operator have been investigated [6]. However, an operatorcorresponding to the restriction of the even eigenvectors of L (cid:48) to G has not, to ourknowledge been investigated. In [3], Chung defines the Neumann graph Laplacianby enforcing a condition that a discrete derivative vanishes on the boundary nodesof the graph, which results in different eigenvectors than those arising from the eveneigenvectors of L (cid:48) . We note that a Cheeger inequality for Chung’s definition of theNeumann graph Laplacian has recently been established by Hua and Huang [8].1.3. Odd and even eigenvectors.
The proof of Proposition 1.1 gives some initialinsight into the odd and even eigenvectors the graph Laplacian L (cid:48) on the doubledgraph G (cid:48) . Proof of Proposition 1.1.
The proof of this proposition is immediate from the blockstructure of the graph Laplacian L (cid:48) . Indeed, let L (cid:48) ( U, W ) denote the submatrix of L (cid:48) whose rows and columns are indexed by U ⊆ V and W ⊆ V , respectively. Wecan write L (cid:48) = X Y Y (cid:62) Z Y (cid:62) Y X , where X is the submatrix L (cid:48) (˚ V , ˚ V ), Y is the submatrix L (cid:48) (˚ V , ∂V ), and Z is thesubmatrix L (cid:48) ( ∂V, ∂V ). With this notation, the eigenvectors of L (cid:48) that are evenwith respect to ∂V are solutions to the equation X Y Y (cid:62) Z Y (cid:62) Y X uvu = µ uvu . That is to say, the vectors u and v satisfy Xu + Y v = µu and 2 Y (cid:62) u + Zv = µv .Put differently, when concatenated, u and v form an eigenvector of the matrix(1) L R := (cid:18) X Y Y (cid:62) Z (cid:19) . Observe that L R is similar to a symmetric matrix L R = (cid:18) I √ I (cid:19) (cid:18) X √ Y √ Y (cid:62) Z (cid:19) (cid:18) I √ I (cid:19) − , and thus by the Spectral Theorem, L R has | V | real eigenvectors, which give rise to | V | even eigenvectors of L (cid:48) . The eigenvectors of L (cid:48) that are odd with respect to ∂V CHEEGER INEQUALITY BASED ON A REFLECTION PRINCIPLE 3 are solutions to the equation X Y Y (cid:62) Z Y (cid:62) Y X u − u = λ u − u . Thus, each vector u such that Xu = λu gives rise to an odd eigenvector of L (cid:48) . Let L D := X. Since L D is symmetric, it follows from the Spectral Theorem that it has | ˚ V | realeigenvectors, and we conclude that L (cid:48) has | ˚ V | odd eigenvectors. (cid:3) Contribution.
In this paper, we study the operator L R defined in (1) whichwe call the reflected Neumann graph Laplacian. This operator seems to be partic-ularly natural on graphs approximating manifolds. For example, in Remark 1.1,we show that on the path graph, the eigenvectors of the Dirichlet graph Laplacian L D and reflected Neumann graph Laplacian L R are the familiar discrete sine andcosine functions. We remark that the definition of the reflected Neumann graphLaplacian L R has some similarities to the normalization used in the diffusion mapsmanifold learning method of Coifman and Lafon [7].Our main result Theorem 1.1 shows that the first eigenvalue of the normalizedreflected Neumann graph Laplacian L R defined in (2) satisfies a Cheeger inequality.The graph cuts arising from L R can differ significantly from graph cuts arisingfrom the standard normalized graph Laplacian L defined in [3]. In Figure 3, weillustrate Theorem 1.1 with an example where the first eigenvector of the Neumanngraph Laplacian L R suggests a drastically different cut than the first eigenvectorof the standard graph Laplacian, and describe how the graph cut suggested by L R is consistent with the Cheeger inequality established in Theorem 1.1. It maybe interesting to investigate the analog of other classical eigenvalue inequalitiesinvolving these definitions of L D and L R for graphs with boundary. Remark 1.1.
The operators L D and L R are particularly natural on the pathgraph. Let P n = ( V, E ) denote the path graph on n vertices, where V = { , . . . , n } and { u, v } ∈ E if and only if | u − v | = 1. If ∂V := { , n } , then the doubled graph P (cid:48) n = C n − is the cycle graph on 2 n − P = (cid:55)→ C = Figure 2.
A path graph and its doubled graph.Consider L D and L R of the path graph P n . The Dirichlet eigenvectors ϕ k andeigenvalues λ k , which satisfy L D ϕ k = λ k ϕ k for k = 1 , . . . , n −
2, are of the form λ k := 2 (cid:18) − cos (cid:18) πkn − (cid:19)(cid:19) and ϕ k ( j ) = sin (cid:18) πjkn − (cid:19) , GELERNT, HALIKIAS, KENNEY, AND MARSHALL for j = 1 , . . . , n −
2, while the Neumann eigenvectors, ψ k and µ k , which satisfy L R ψ k = µ k ψ k for k = 0 , . . . , n −
1, are of the form µ k := 2 (cid:18) − cos (cid:18) πkn − (cid:19)(cid:19) and ψ k ( j ) := cos (cid:18) πjkn − (cid:19) for j = 0 , . . . , n −
1. Thus, the path graph doubling procedure defined in § Notation and definitions.
Suppose that G = ( V, E ) is a graph with vertices V and edges E . Let ∂V ⊆ V be a designated set of boundary vertices, and set˚ V = V \ ∂V . We can write the adjacency matrix A of the graph G as the blockmatrix A = (cid:18) A A A (cid:62) A (cid:19) , where A = A (˚ V , ˚ V ), A = A (˚ V , ∂V ), and A = A ( ∂V, ∂V ). Motivated byProposition 1.1 we define the reflected adjacency matrix R by R := (cid:18) A A A (cid:62) A (cid:19) . With this notation, the reflected Neumann Laplacian L R can be defined by L R = D − R, where D = diag( R(cid:126) (cid:126) L R by(2) L R := D − / L R D − / . Main result.
In this section, we present our main result Theorem 1.1. Whilethe matrix L R is not in general symmetric, it is similar to a symmetric matrix;indeed, if Q := (cid:18) I | ˚ V | I | ∂V | (cid:19) , then Q / L R Q − / is symmetric, positive-definite, and has the eigenvector D / Q / (cid:126) λ R of L R satisfies λ R := inf x (cid:62) D / Q / (cid:126) x (cid:62) Q / L R Q − / xx (cid:62) x . Let E ( U, W ) := {{ u, w } ∈ E : u ∈ U, w ∈ W } , that is, E ( U, W ) is the set of edgesbetween U and W . We define a measure m ( U, W ) on this set of edges by m ( U, W ) = | E ( U, W ) | − | E ( U ∩ ∂V, W ∩ ∂V ) | , and we define the volume vol( U ) of U ⊆ V byvol( U ) := (cid:88) u ∈ U m ( { u } , V ) . The following theorem is our main result.
CHEEGER INEQUALITY BASED ON A REFLECTION PRINCIPLE 5
Theorem 1.1.
Suppose that G = ( V, E ) is a graph with a designated set of bound-ary vertices ∂V ⊆ V , and define the Cheeger constant h R by (3) h R := min S ⊆ V m ( S, V \ S )min { vol( S ) , vol( V \ S ) } . Then, (cid:112) λ R ≥ h R ≥ λ R / , where λ R is the first nontrivial eigenvalue of L R . Recall that the standard Cheeger inequality is constructive in the sense thata cut that achieves the upper bound on the Cheeger constant can be determinedfrom the eigenfunction corresponding to the first eigenvalue of the normalized graphLaplacian L , see [1, 2]. Specifically, a partition that achieves the upper bound canbe determined by dividing the vertices into two groups based on if the value ofthe first eigenvector is more or less than some threshold; for a detailed expositionsee for example [3, 4]. Similarly, the result of Theorem 1.1 is constructive in thesense that a cut which achieves the upper bound on h R can be determined from theeigenvector ψ R of L R that corresponds to λ R . In the following remark, we presentan example where the cut arising from ψ R differs significantly from the cut arisingfrom the first eigenvector ψ of the standard normalized graph Laplacian L . Remark 1.2.
Graph cuts arising from ψ R can differ significantly from graph cutsarising from ψ . Indeed, on the left of Figure 3 we illustrate a graph whose verticesare colored by greyscale values proportional to ψ . On the right of Figure 3 weillustrate the same graph except several vertices have been designated as boundaryvertices (indicated by squares) and the color of the vertices is proportional to ψ R .Observe that ψ suggests cutting the graph by a vertical line into two equal parts,while ψ R suggests cutting the graph by a horizontal line into two equal parts. ψ = ψ R = Figure 3.
The same graph with vertices colored proportional to ψ (left) and colored proportional to ψ R (right), where the squaresin the right graph denote boundary vertices.The fact that ψ R suggests a horizontal cut of the graph is illustrative of Theorem1.1. Indeed, it is straightforward to check that the horizontal cut suggested by ψ R minimizes the cut measure m ( S, V \ S ) / (vol( S ) , vol( V \ S ) } ) from (3). In contrast,the vertical cut suggested by ψ minimizes the standard cut measure, which is equiv-alent to the measure m ( S, V \ S ) / (vol( S ) , vol( V \ S ) } ) in the case that all verticesare interior vertices. Of course, Theorem 1.1 only guarantees that the measure ofthe cut arising from the eigenvector ψ R is an upper bound for h R with value at most GELERNT, HALIKIAS, KENNEY, AND MARSHALL
Figure 4.
A barbell shaped graph whose vertices are colored pro-portional to ψ R , where squares in the graph denote boundary ver-tices. √ λ R ; however, in this simple example the cut arising from ψ R actually obtainsthis minimum. Remark 1.3.
Here we visualize the first eigenfunction ψ R of the reflected Neumanngraph Laplacian L R on a classic barbell shaped graph, see Figure 4. Observe that inFigure 4 the maximum and minimum value of the eigenvector occur at an interiorvertex. This feature of the eigenvectors is interesting in the context of spectralclustering, where extreme values of the eigenvectors often correspond to the centerof clusters.1.7. Future Directions.
One future direction for this work is the problem ofselecting boundary vertices in a principled way. How the boundary is selected maydepend on the application at hand. In a social network graph, boundary verticescould correspond to individuals with many connections outside the network. Inthe context of manifold learning, where the vertices of the graph are points in R n ,boundary vertices could be selected based on the number of points within some ε -neighborhood of each vertex. On the other hand, when a graph is given bysampling from a pre-defined manifold with boundary, vertices selected from somecollar neighborhood of the boundary could be designated as boundary vertices.Another future direction arises from generalizing the setup under which our workwas done. Our graph doubling procedure inputs a graph with boundary and outputsa larger graph, containing the original graph as an induced subgraph, which has aspecial Z symmetry. Could similar Cheeger results be proven for other reflectionprocedures? For example, what if n − § P n the eigenfunctions of the reflected Neumann Laplacian L R are ofthe form ψ k ( j ) = cos( πjk/ ( n − ψ k ( j ) = cos( π ( j +1 / k/n ) is also important in numerical analysis; it could be interesting to developa graph doubling procedure whose Neumann eigenvectors on the path graph arethese vectors. CHEEGER INEQUALITY BASED ON A REFLECTION PRINCIPLE 7 Proof of Main Result
Summary.
The proof of Theorem 1.1 is divided into two lemmas: first, inLemma 2.1 we show that λ R ≤ h R , and second, in Lemma 2.2 we show that h R / ≤ λ R . The structure of our argument is similar to classical Cheeger inequalityproofs, see [3, 5].2.2. Proof of Theorem 1.1.Lemma 2.1 (Trivial direction) . We have λ R ≤ h R . Proof of Lemma 2.1.
Recall that L R := D − / Q / L R Q − / D − / . First, we observe that QL R can be written as QL R = L − L ∂ , where L = (cid:18) diag( A (cid:126) A (cid:126) − A − A − A (cid:62) diag( A (cid:62) (cid:126) A (cid:126) − A (cid:19) , and L ∂ := (cid:18) A (cid:126) − A (cid:19) . Observe that L is the standard graph Laplacian of G , while L ∂ is the graph Lapla-cian of the vertex induced subgraph G [ ∂V ]. Fix a subset S ⊆ V , and let χ S be theindicator function for S . Define x := Q / D / χ S − χ (cid:62) S DQ(cid:126) (cid:126) (cid:62) DQ(cid:126) D / Q / (cid:126) . By construction, we have x (cid:62) D / Q / (cid:126) λ N ≤ x (cid:62) D − / Q / L R Q − / D − / xx (cid:62) x = χ (cid:62) S QL R χ S χ (cid:62) S DQχ S (cid:16) (cid:126) − χ (cid:62) S DQχ S (cid:126) (cid:62) DQ (cid:17) = χ (cid:62) S ( L − L ∂ ) χ S (cid:16) (cid:126) (cid:62) DQ(cid:126) (cid:17)(cid:0) χ (cid:62) S DQχ (cid:62) S (cid:1) (cid:16) χ (cid:62) V \ S DQχ V \ S (cid:17) ≤ · χ (cid:62) S ( L − L ∂ ) χ S min (cid:110)(cid:0) χ (cid:62) S DQχ (cid:62) S (cid:1) , (cid:16) χ (cid:62) V \ S DQχ V \ S (cid:17)(cid:111) = 2 · m ( S, V \ S )min { vol( S ) , vol( V \ S ) } . Since this inequality holds for all subsets S ⊆ V , we conclude that λ R ≤ h R , aswas to be shown. (cid:3) GELERNT, HALIKIAS, KENNEY, AND MARSHALL
Lemma 2.2 (Nontrivial direction) . We have λ R ≥ h R . Proof of Lemma 2.2.
Recall that λ R = inf x (cid:62) D / Q / (cid:126) x (cid:62) L R xx (cid:62) x = inf y (cid:62) DQ(cid:126) y (cid:62) QL R yy (cid:62) QDy .
Let g be a vector satisfying λ R = g (cid:62) QL R gg (cid:62) DQg , and g (cid:62) QD(cid:126) . Let { v , . . . , v n } be an enumeration of the vertices V so that g v ≤ ... ≤ g v n ,and set S j := { v , ..., v j } , for j = 1 , . . . , n. Let p be the largest integer such thatvol( S p ) ≤ vol( V ) /
2, that is, p := max { j ∈ { , . . . , n } : vol( S j ) ≤ vol( V ) / } . Let g + and g − denote the positive and negative parts of g − g v p , respectively. Thatis, g + v := max { g v − g v p , } and g − v := max { g v p − g v , } . Let u ∼ v denote { u, v } ∈ E and q = diag( Q ) . Then λ R = g (cid:62) ( L − L ∂ ) gg (cid:62) DQg = (cid:80) u ∼ v ( g u − g v ) − (cid:80) u ∼ vu,v ∈ ∂V ( g u − g v ) (cid:80) v g v d v q v ≥ (cid:80) u ∼ v ( g u − g v ) − (cid:80) u ∼ vu,v ∈ ∂V ( g u − g v ) (cid:80) v ( g ( v ) − g ( v p )) d v q v , where the last inequality holds because we have increased the denominator. Fromhere,(4) λ R ≥ (cid:80) u ∼ v (( g + u − g + v ) + ( g − u − g − v ) ) − (cid:80) u ∼ vu,v ∈ ∂V (( g + u − g + v ) + ( g − u − g − v ) ) (cid:80) v (( g + v ) + ( g − v ) ) d v q v , Recall that(5) a + bc + d ≥ min (cid:26) ac , bd (cid:27) , for any a, b ≥ c, d >
0. From (4), we can set a = (cid:80) u ∼ v ( g + u − g + v ) − (cid:80) u ∼ vu,v ∈ ∂V ( g + u − g + v ) , b = (cid:80) u ∼ v ( g − u − g − v ) − (cid:80) u ∼ vu,v ∈ ∂V ( g − u − g − v ) , c = (cid:80) v ( g + v ) d v q v , and d = (cid:80) v ( g − v ) d v q v . Observe that a and b are nonnegative. Indeed, a = (cid:88) u ∼ vu/ ∈ ∂V or v / ∈ ∂V ( g + u − g + v ) , which has nonnegative summands, and a similar statement holds for b. Without loss of generality, (5) implies that λ R ≥ (cid:80) u ∼ v ( g + u − g + v ) − (cid:80) u ∼ vu,v ∈ ∂V ( g + u − g + v ) (cid:80) v ( g + v ) d v q v . CHEEGER INEQUALITY BASED ON A REFLECTION PRINCIPLE 9
To simplify notation in the following, let f = g + . We begin by setting λ := (cid:80) u ∼ v ( f u − f v ) − (cid:80) u ∼ vu,v ∈ ∂V ( f u − f v ) (cid:80) v f v d v q v . Multiplying the numerator and denominator by the same term gives λ = (cid:16)(cid:80) u ∼ v ( f u − f v ) − (cid:80) u ∼ vu,v ∈ ∂V ( f u − f v ) (cid:17) (cid:16)(cid:80) u ∼ v ( f u + f v ) − (cid:80) u ∼ vu,v ∈ ∂V ( f u + f v ) (cid:17) ( (cid:80) v f v d v q v ) (cid:16)(cid:80) u ∼ v ( f u + f v ) − (cid:80) u ∼ vu,v ∈ ∂V ( f u + f v ) (cid:17) . Applying the Cauchy-Schwarz inequality in the numerator gives λ ≥ (cid:16)(cid:80) u ∼ v | f u − f v | − (cid:80) u ∼ vu,v ∈ ∂V | f u − f v | (cid:17) ( (cid:80) v f v d v q v ) (cid:16)(cid:80) u ∼ v ( f u + f v ) − (cid:80) u ∼ vu,v ∈ ∂V ( f u + f v ) (cid:17) . Next, we observe that (cid:88) u ∼ v ( f u + f v ) − (cid:88) u ∼ vu,v ∈ ∂V ( f u + f v ) = (cid:88) v f v d v q v − (cid:88) u ∼ v ( f u − f v ) − (cid:88) u ∼ vu,v ∈ ∂V ( f u − f v ) , and thus it follows that λ ≥ (cid:16)(cid:80) u ∼ v | f u − f v | − (cid:80) u ∼ vu,v ∈ ∂V | f u − f v | (cid:17) ( (cid:80) v f v d v q v ) (2 − λ ) . We want to show that (cid:88) u ∼ v | f u − f v | − (cid:88) u ∼ vu,v ∈ ∂V | f u − f v | ≥ n (cid:88) i =1 | f v i − f v i +1 | m ( S i , V \ S i ) . We can write (cid:88) u ∼ v | f u − f v | − (cid:88) u ∼ vu,v ∈ ∂V | f u − f v | = n (cid:88) i =2 i − (cid:88) j =1 (cid:18) χ E i,j − χ ∂ i χ ∂ j (cid:19) ( f v i − f v j ) , where χ E i,j = (cid:40) { v i , v j } ∈ E { v i , v j } ∈ E , and χ ∂ i = (cid:40) i ∈ ∂V v i ∈ ∂V . Note that we are justified in dropping theabsolute value signs because f v i is an increasing function of i. Next we write f v i − f v j as a telescoping series f v i − f v j = ( f v i − f v i − ) + ( f v i − − f v i − ) + ... + ( f v j +1 − f v j ) , and rearrange terms in the summation to conclude that n (cid:88) i =2 i − (cid:88) j =1 (cid:18) χ E i,j − χ ∂ i χ ∂ j (cid:19) ( f v i − f v j ) = n (cid:88) l =1 n (cid:88) k =1 n (cid:88) j =1 (cid:18)(cid:18) χ E j,k + l − χ ∂ j χ ∂ k + l (cid:19) χ j ≤ l (cid:19) ( f v l +1 − f v l ) , where χ j ≤ l = (cid:40) j ≤ l . Then, to complete this step, we note that n (cid:88) k =1 n (cid:88) j =1 (cid:18)(cid:18) χ E j,k + l − χ ∂ j χ ∂ k + l (cid:19) χ j ≤ l (cid:19) = m ( S l , V \ S l ) . Returning to our main sequence of inequalities for λ , we have λ ≥ ( (cid:80) i | f v i − f v i +1 | m ( S i , V \ S i )) (cid:80) v f v d v q v ) ≥ ( α (cid:80) ni =1 | f v i − f v i +1 | min { vol( S i ) , vol( V \ S i ) } ) (cid:80) u f ( u ) d u q v ) , where α := min ≤ i ≤ n m ( S i , V \ S i )min { vol( S i ) , vol( V \ S i ) } . Since f v i is nondecreasing, a rearrangement of the numerator of the previous ex-pression gives λ ≥ α (cid:80) i ( f v i | min { vol( S i ) , vol( V \ S i ) } − min { vol( S i +1 ) , vol( V \ S i +1 ) }| )) ( (cid:80) u f ( u ) d u q u ) . It follows that λ R ≥ λ ≥ α (cid:80) i f v i d v i q v i ) ( (cid:80) u f u d u q u ) = α ≥ h R , which completes the proof. (cid:3) Acknowledgements.
We thank the referees for their helpful comments. This re-search was supported by Summer Undergraduate Math Research at Yale (SUMRY)2018. NFM was supported in part by NSF DMS-1903015.
References [1] N. Alon. Eigenvalues and expanders.
Combinatorica , 6 (1986): 86–96.[2] J. Cheeger. A lower bound for the smallest eigenvalue of the Laplacian.
Problems in Analysis ,R. C. Gunning, editor, Princeton Univ. Press, (1970): 195-199.[3] F. Chung.
Spectral Graph Theory . CBMS Regional Conference Series in Mathematics, No. 92,American Mathematical Society, 1997.[4] F. Chung. Four Cheeger-type inequalities for graph partitioning algorithms.
Proc. ICCM , 2(2007): 751–772.[5] F. Chung. Laplacians of graphs and Cheeger’s inequalities.
Combinatorics, Paul Erdos iseighty , 2 (1996): 157–172.[6] F. Chung and K. Oden. Weighted Graph Laplacians and Isoperimetric Inequalities.
Pac. J.Appl. Math.
CHEEGER INEQUALITY BASED ON A REFLECTION PRINCIPLE 11 [7] R. R. Coifman and S. S. Lafon. Diffusion maps.
Appl. Comput. Harmon. Anal. , 21, no. 1(2006): 5–30.[8] B. Hua and Y. Huang. Neumann Cheeger Constants on Graphs.
J. Geom. Anal. , 28 (2018):2166–2184.[9] G. Strang. The Discrete Cosine Transform.
SIAM Rev. , 41 (1999): 135–147., 41 (1999): 135–147.