Let G denote a directed graph with adjacency matrix Q and in- degree matrix D. We consider the Kirchhoff matrix L = D − Q, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of G. This fact has a meaningful generalization to directed graphs, as was observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace--namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights

Caughman, John S., IV

Veerman, J. J. P.

PDXScholar (Portland State University)

Portland State UniversityPDXScholarMathematics and Statistics Faculty Publications andPresentationsFariborz Maseeh Department of Mathematics andStatistics2006Kernels of Directed Graph LaplaciansJohn S. Caughman IVPortland State University, caughman@pdx.eduJ. J. P. VeermanPortland State University, veerman@pdx.eduLet us know how access to this document benefits you.Follow this and additional works at: http://pdxscholar.library.pdx.edu/mth_facPart of the Discrete Mathematics and Combinatorics CommonsThis Article is brought to you for free and open access. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications andPresentations by an authorized administrator of PDXScholar. For more information, please contact pdxscholar@pdx.edu.Citation DetailsCaughman, J.S., and Veerman, J.J.P. "Kernels of directed graph Laplacians." The Electronic Journal of Combinatorics 13.1 (2006).Kernels of Directed Graph LaplaciansJ. S. Caughman and J. J. P. VeermanDepartment of Mathematics and StatisticsPortland State UniversityPO Box 751, Portland, OR 97207.caughman@pdx.edu, veerman@pdx.eduSubmitted: Oct 28, 2005; Accepted: Mar 14, 2006; Published: Apr 11, 2006Mathematics Subject Classication: 05C50Abstract. Let G denote a directed graph with adjacency matrix Q and in-degree matrix D. We consider the Kirchho matrix L = D − Q, sometimesreferred to as the directed Laplacian. A classical result of Kirchho asserts thatwhen G is undirected, the multiplicity of the eigenvalue 0 equals the numberof connected components of G. This fact has a meaningful generalization todirected graphs, as was observed by Chebotarev and Agaev in 2005. Sincethis result has many important applications in the sciences, we oer an inde-pendent and self-contained proof of their theorem, showing in this paper thatthe algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace { namely, thenumber of reaches of the directed graph. We also extend their results by deriv-ing a natural basis for the corresponding eigenspace. The results are proved inthe general context of stochastic matrices, and apply equally well to directedgraphs with non-negative edge weights.Keywords: Kirchho matrix. Eigenvalues of Laplacians. Graphs. Stochastic matrix.1 DenitionsLet G denote a directed graph with vertex set V = f1; 2; :::; Ng and edge set E V V . To each edge uv 2 E, we allow a positive weight !uv to be assigned. The adjacencymatrix Q is the N N matrix whose rows and columns are indexed by the vertices, andwhere the ij-entry is !ji if ji 2 E and zero otherwise. The in-degree matrix D is theN N diagonal matrix whose ii-entry is the sum of the entries of the ith row of Q. Thematrix L = D − Q is sometimes referred to as the Kirchho matrix, and sometimes asthe directed graph Laplacian of G.A variation on this matrix can be dened as follows. Let D+ denote the pseudo-inverseof D. In other words, let D+ be the diagonal matrix whose ii-entry is D−1ii if Dii 6= 0the electronic journal of combinatorics 13 (2006), #R39 1and whose ii-entry is zero if Dii = 0. Then the matrix L = D+(D − Q) has nonnegativediagonal entries, nonpositive o-diagonal entries, all entries between -1 and 1 (inclusive)and all row sums equal to zero. Furthermore, the matrix S = I −L is stochastic.We shall see (in Section 4) that both L and L can be written in the form D − DSwhere D is an appropriately chosen nonnegative diagonal matrix and S is stochastic. Wetherefore turn our attention to the properties of these matrices for the statement of ourmain results.We show that for any such matrix M = D −DS, the geometric and algebraic multi-plicities of the eigenvalue zero are equal, and we nd a basis for this eigenspace (the kernelof M). Furthermore, the dimension of this kernel and the form of these eigenvectors canbe described in graph theoretic terms as follows.We associate with the matrix M a directed graph G, and write j  i if there existsa directed path from vertex j to vertex i. For any vertex j, we dene the reachable setR(j) to be the set containing j and all vertices i such that j  i. A maximal reachableset will be called a reach. We prove that the algebraic and geometric multiplicity of 0 asan eigenvalue for M equals the number of reaches of G.We also describe a basis for the kernel of M as follows. LetR1; :::Rk denote the reachesof G. For each reach Ri, we dene the exclusive part of Ri to be the set Hi = Ri n[j 6=iRj .Likewise, we dene the common part of Ri to be the set Ci = Ri n Hi. Then for eachreach Ri there exists a vector vi in the kernel of M whose entries satisfy: (i) (vi)j = 1 forall j 2 Hi; (ii) 0 < (vi)j < 1 for all j 2 Ci; (iii) (vi)j = 0 for all j 62 Ri. Taken together,these vectors v1, v2,..., vk form a basis for the kernel of M and sum to the all 1's vector 1.Due to the recent appearance of Agaev and Chebotarev's notable paper [1], we wouldlike to clarify the connections to their results. In that paper, the matrices studied havethe form M = (I − S) where  is positive and S stochastic. A simple check veriesthat this is precisely the set of matrices of the form D − DS, where D is nonnegativediagonal. The number of reaches corresponds, in that paper, with the in-forest dimension.And where that paper concentrates on the location of the Laplacian eigenvalues in thecomplex plane, we instead have derived the form of the associated eigenvectors.2 Stochastic matricesA matrix is said to be (row) stochastic if the entries are nonnegative and the row sumsall equal 1. Our rst result is a special case of Gersgorin's theorem [3, p.344].2.1 Lemma. Suppose S is stochastic. Then each eigenvalue  satises jj  1.2.2 Denition. Given any real N  N matrix M , we denote by GM the directedgraph with vertices 1; :::; N and an edge j ! i whenever Mij 6= 0: For each vertex i, setNi := fj j j ! ig. We write j  i if there exists a directed path in GM from vertex j tovertex i. Furthermore, for any vertex j, we dene R(j) to be the set containing j and allvertices i such that j  i. We refer to R(j) as the reachable set of vertex j. Finally, wesay a matrix M is rooted if there exists a vertex r in GM such that R(r) contains everyvertex of GM . We refer to such a vertex r as a root.the electronic journal of combinatorics 13 (2006), #R39 22.3 Lemma. Suppose S is stochastic and rooted. Then the eigenspace E1 associatedwith the eigenvalue 1 is spanned by the all-ones vector 1.Proof. Conjugating S by an appropriate permutation matrix if necessary, we may assumethat vertex 1 is a root. Since S is stochastic, S1 = 1 so 1 2 E1. By way of contradiction,suppose dim(E1) > 1 and choose linearly independent vectors x; y 2 E1. Suppose jxij ismaximized at i = n. Comparing the n-entry on each side of the equation x = Sx, we seethatjxnj Xj2NnSnj jxj j  jxnjXj2NnSnj = jxnj:Therefore, equality holds throughout, and jxj j = jxnj for all j 2 Nn. In fact, sincePj2Nn Snj xj = xn; it follows that xj = xn for all j 2 Nn. Since S is rooted at vertex1, a simple induction now shows that x1 = xn. So jxij is maximized at i = 1. The sameargument applies to any vector in E1 and so jyij is maximized at i = 1.Since y1 6= 0 we can dene a vector z such that zi := xi − x1y1 yi for each i. Thisvector z, as a linear combination of x and y, must belong to E1. It follows that jzij is alsomaximized at i = 1. But z1 = 0 by denition, so zi = 0 for all i. It follows that x and yare not linearly independent, a contradiction. 2.4 Lemma. Suppose S is stochastic N N and vertex 1 is a root. Further assume N1is empty. Let P denote the principal submatrix obtained by deleting the rst row andcolumn of S. Then the spectral radius of P is strictly less than 1.Proof. Since N1 is empty, S is block lower-triangular with P as a diagonal block. Sothe spectral radius of P cannot exceed that of S. Therefore, by Lemma 2.1, the spectralradius of P is at most 1. By way of contradiction, suppose the spectral radius of P is equalto 1. Then by the Perron-Frobenius theorem (see [3, p. 508]), we would have Px = x forsome nonzero vector x.Dene a vector v with v1 = 0 and vi = xi−1 for i 2 f2; :::; Ng. We nd thatSv =0BBB@1 0    0S21...SN1P1CCCA0BB@0x1CCA =0BB@0x1CCA = v:So v 2 E1. But v1 = 0, so Lemma 2.3 implies x = 0. This contradiction completes theproof. 2.5 Corollary. Suppose S is stochastic and N  N . Assume the vertices of GS canbe partitioned into nonempty sets A, B such that for every b 2 B, there exists a 2 Awith a b in GS. Then the spectral radius of the principal submatrix SBB obtained bydeleting from S the rows and columns of A is strictly less than 1.Proof. Dene the matrix S^ byS^ =1 0u SBB;the electronic journal of combinatorics 13 (2006), #R39 3where u is chosen so that S^ is stochastic. We claim that S^ is rooted (at 1). To see this,pick any b 2 B. We must show 1 b in GS^. By hypothesis there exists a 2 A with a bin GS. Leta = x0 ! x1 !    ! xn = bbe a directed path in GS from a to b. Let i be maximal such that xi 2 A. Then thexi+1; xi entry of S is nonzero, so the xi+1 row of SBB has row sum strictly less than 1.Therefore, the xi+1 entry of the rst column of S^ is nonzero. So 1 ! xi+1 in GS^ andtherefore 1 b in GS^ as desired. So S^ is rooted, and the previous lemma gives the result.2.6 Denition. A set R of vertices in a graph will be called a reach if it is a maximalreachable set; in other words, R is a reach if R = R(i) for some i and there is no j suchthat R(i)  R(j) (properly). Since our graphs all have nite vertex sets, such maximalsets exist and are uniquely determined by the graph. For each reach Ri of a graph, wedene the exclusive part of Ri to be the set Hi = Ri n [j 6=iRj . Likewise, we dene thecommon part of Ri to be the set Ci = Ri nHi.2.7 Theorem. Suppose S is stochastic N  N and let R denote a reach of GS withexclusive part H and common part C. Then there exists an eigenvector v 2 E1 whoseentries satisfy(i) vi = 1 for all i 2 H ,(ii) 0 < vi < 1 for all i 2 C,(iii) 0 for all i 62 R.Proof. Let Y denote the set of vertices not in R. Permuting rows and columns of S ifnecessary, we may write S asS =0@ SHH SHC SHYSCH SCC SCYSY H SY C SY Y1A =0@ SHH 0 0SCH SCC SCY0 0 SY Y1ASince SHH is a rooted stochastic matrix, it has eigenvalue 1 with geometric multiplicity1. The associated eigenvector is 1H .Observe that SCC has spectral radius < 1 by Corollary 2.5. Further, notice thatS(1H ; 0C ; 0Y )T = (1H ; SCH1H ; 0Y ):T Using this, we nd that solving the equationS(1H ;x; 0C)T = (1H ;x; 0C)Tfor x amounts to solving0@ 1HSCH1H + SCCx0Y1A =0@ 1Hx0Y1A :the electronic journal of combinatorics 13 (2006), #R39 4Solving the above, however, is equivalent to solving (I − SCC)x = SCH1H : Since thespectral radius of SCC is strictly less than 1, the eigenvalues of I − SCC cannot be 0. SoI − SCC is invertible. It follows that x = (I − SCC)−1SCH1H is the desired solution.Conditions (i) and (iii) are clearly satised by (1H ;x; 0Y );T so it remains only to verify(ii). To see that the entries of x are positive, note that (I − SCC)−1 =P1i=0 SiCC ; so theentries of x are nonnegative and strictly less than 1. But every vertex in C has a pathfrom the root, where the eigenvector has value 1. So since each entry in the eigenvectorfor S must equal the average of the entries corresponding to its neighbors in GS, all entriesin C must be positive. 3 Matrices of the form D −DSWe now consider matrices of the form D − DS where D is a nonnegative diagonalmatrix and S is stochastic. We will determine the algebraic multiplicity of the zeroeigenvalue. We begin with the rooted case.3.1 Lemma. Suppose M = D −DS, where D is a nonnegative diagonal matrix and Sis stochastic. Suppose M is rooted. Then the eigenvalue 0 has algebraic multiplicity 1.Proof. Let M = D − DS be given as stated. First we claim that, without loss ofgenerality, Sii = 1 whenever Dii = 0. To see this, suppose Dii = 0 for some i. If Sii 6= 1,let S 0 be the stochastic matrix obtained by replacing the ith row of S by the ith row ofthe identity matrix I, and let M 0 = D−DS 0. Observe that M = M 0, and this proves ourclaim. So we henceforth assume thatSii = 1 whenever Dii = 0: (1)Next we claim that, given (1), ker(M) must be identical with ker(I−S). To see this, notethat if (I − S)v = 0 then clearly Mv = D(I − S)v = 0. Conversely, suppose Mv = 0.Then D(I − S)v = 0 so the vector w = (I − S)v is in the kernel of D. If w has a nonzeroentry wi then Dii = 0. Recall this implies Sii = 1 and the ith row of I − S is zero. Butw = (I − S)v, so wi must be zero. This contradiction implies w must have no nonzeroentries, and therefore (I − S)v = 0. So M and I − S have identical nullspaces as desired.By Lemma 2.3, S1 = 1, so M1 = 0. Therefore the geometric multiplicity, and hencethe algebraic multiplicity, of the eigenvalue 0 must be at least 1. By way of contradiction,suppose the algebraic multiplicity is greater than 1. Then there must be a nonzero vectorx and an integer d  2 such thatMd−1x 6= 0 and Mdx = 0:Now, since kerM = ker(I − S), Lemma 2.3 and the above equation imply that Md−1xmust be a multiple of the vector 1. Scaling Md−1x appropriately, we nd there exists avector v such thatMv = −1:the electronic journal of combinatorics 13 (2006), #R39 5Suppose Re(vi) is maximized at i = n. Comparing the n-entries above, we ndDnnRe(vn) + 1 = DnnXj2NnSnjRe(vj)  DnnRe(vn)Xj2NnSnj = DnnRe(vn);which is clearly impossible. 3.2 Theorem. Suppose M = D − DS, where D is a nonnegative diagonal matrix andS is stochastic. Then the number of reaches of GM equals the algebraic and geometricmultiplicity of 0 as an eigenvalue of M .Proof. Let R1; :::;Rk denote the reaches of GM and let Hi denote the exclusive part of Rifor each 1  i  k, and let C = [ki=1Ci denote the union of the common parts of all thereaches. Simultaneously permuting the rows and columns of M , D, and S if necessary,we may write M = D −DS asM =0BBBBB@DH1H1(I − SH1H1) 0    0 00. . .    0 0....... . .......0 0    DHkHk(I − SHkHk) 0−DCCSCH1       −DCCSCHk DCC(I − SCC)1CCCCCAThe characteristic polynomial det(M − I) is therefore given bydet(DH1H1(I − SH1H1)− I)   det(DHkHk(I − SHkHk)− I)  det(DCC(I − SCC)− I):By Lemma 3.1, each submatrix DH1H1(I − SH1H1) has eigenvalue 0 with algebraic andgeometric multiplicity 1. But observe that DCC has nonzero diagonal entries since C isthe union of the common parts Ci, so DCC(I − SCC) is invertible by Corollary 2.5. Thetheorem now follows. We now oer the following characterization of the nullspace.3.3 Theorem. Suppose M = D−DS, where D is a nonnegative N N diagonal matrixand S is stochastic. Suppose GM has k reaches, denoted R1; :::;Rk, where we denote theexclusive and common parts of each Ri by Hi, Ci respectively. Then the nullspace of Mhas a basis γ1; γ2; :::; γk in RN whose elements satisfy:(i) γi(v) = 0 for v 62 Ri;(ii) γi(v) = 1 for v 2 Hi;(iii) γi(v) 2 (0; 1) for v 2 Ci ;(iv)Pi γi = 1N .the electronic journal of combinatorics 13 (2006), #R39 6Proof. Let M = D −DS be given as stated. As in the proof of Theorem 3.2 above, wemay assume without loss of generality thatSii = 1 whenever Dii = 0: (2)We further observe, as in the proof of Theorem 3.2, that M and I − S have identicalnullspaces, given (2).Notice that the diagonal entries of a matrix do not aect the reachable sets in theassociated graph, so the reaches of GI−S are identical with the reaches of GS. Furthermore,scaling rows by nonzero constants also leaves the corresponding graph unchanged, soGM = GD(I−S) = GI−S. Therefore the reaches of GM are identical with the reaches ofGS.Applying Theorems 2.7 and 3.2, we nd that the nullity of the matrix M equals kand the nullspace of M has a basis satisfying (i){(iii). To see (iv), observe that the all 1'svector 1 is a null vector for M , and notice that the only linear combination of these basisvectors that assumes the value 1 on each of the Hi is their sum. 4 Graph LaplaciansIn this section, we simply apply our results to the Laplacians L and L of a (weighted,directed) graph, as discussed in Section 1.4.1 Corollary. Let G denote a weighted, directed graph and let L denote the (directed)Laplacian matrix L = D+(D − Q). Suppose G has N vertices and k reaches. Thenthe algebraic and geometric multiplicity of the eigenvalue 0 equals k. Furthermore, theassociated eigenspace has a basis γ1; γ2; :::; γk in RN whose elements satisfy: (i) γi(v) = 0for v 2 G−Ri; (ii) γi(v) = 1 for v 2 Hi; (iii) γi(v) 2 (0; 1) for v 2 Ci ; (iv)Pi γi = 1N .Proof. The matrix S = I − L is stochastic and the graphs G and GS have identicalreaches. The result follows by applying Theorem 3.3. We next observe that the same results hold for the Kirchho matrix L = D −Q.4.2 Corollary. Let G denote a directed graph and let L denote the Kirchho matrixL = D −Q. Suppose G has N vertices and k reaches. Then the algebraic and geometricmultiplicity of the eigenvalue 0 equals k. Furthermore, the associated eigenspace has abasis γ1; γ2; :::; γk in RN whose elements satisfy: (i) γi(v) = 0 for v 2 G−Ri; (ii) γi(v) = 1for v 2 Hi; (iii) γi(v) 2 (0; 1) for v 2 Ci ; (iv)Pi γi = 1N .Proof. One simply checks that the matrix L has the form D − DS where S is thestochastic matrix I − L from above, and D is the in-degree matrix of G. The resultfollows by applying Theorem 3.3. In numerous applications, in particular those related to dierence - or dierentialequations (see [6]), it is a crucial fact that any nonzero eigenvalue of the Laplacian has astrictly positive real part. Using some of the stratagems already exhibited, the proof ofthis fact is easy, and we include the result for completeness.the electronic journal of combinatorics 13 (2006), #R39 74.3 Theorem. Any nonzero eigenvalue of a Laplacian matrix of the form D−DS, whereD is nonnegative diagonal and S is stochastic, has (strictly) positive real part.Proof. Let  6= 0 be an eigenvalue of D − DS and v a corresponding eigenvector, so(D −DS)v = v. Thus for all i,Diivi = vi + DiiXjSijvj : (3)Suppose Dii is zero. Then vi = 0. Since  6= 0 it follows that vi = 0. Since  6= 0, thevector v is not a multiple of 1. Let n be such that jvij is maximized at i = n. Multiply vby a nonzero complex number so that vn is real. Since vn is nonzero, the above argumentshows that Dnn 6= 0. Dividing (3) for i = n by Dnn and taking the real and imaginaryparts separately, we obtainXjSnj Re (vj) = (1− Re ()Dnn)vn;XjSnj Im (vj) = − Im ()Dnnvn:The rst of these equations implies that Re()  0. Now if Re() = 0 then for all j 2 Nnwe have vj = vn and thus Im(vj) = 0. Notice that in this case, the imaginary part of must be nonzero. So in the second equation above, the left hand side is zero but the righthand side is not. The conclusion is now immediate. Acknowledgment. The authors would like to thank Gerardo Laerriere and AncaWilliams for many helpful discussions and insightful comments on this topic.References[1] R. Agaev and P. Chebotarev. On the spectra of nonsymmetric laplacian matrices.Linear Algebra App., 399:157{168, 2005.[2] P. Chebotarev and R. Agaev. Forest matrices around the laplacian matrix. LinearAlgebra App., 356:253{274, 2002.[3] R. A. Horn and Charles R. Johnson. Matrix Analysis. Cambridge University Press,Cambridge, 1985.[4] T. Leighton and R. L. Rivest. The markov chain tree theorem. Computer ScienceTechnical Report MIT/LCS/TM-249, Laboratory of Computer Scinece, MIT, Cam-bridge, Mass., 1983.[5] U. G. Rothblum. Computation of the eigenprojection of a nonnegative matrix at itsspectral radius. Mathematical Programming Study, 6:188{201, 1976.[6] J. J. P. Veerman, G. Laerriere, J. S. Caughman, and A. Williams. Flocks andformations. J. Stat. Phys., 121:901{936, 2005.the electronic journal of combinatorics 13 (2006), #R39 8

Kernels of Directed Graph Laplacians

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Kernels of Directed Graph Laplacians

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