Two Laplacians for the distance matrix of a graph

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1 Two Laplacians for the distance matrix of a graph Mustapha Aouchiche and Pierre Hansen GERAD and HEC Montreal, Canada CanaDAM 2013, Memorial University, June 1013 Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

2 PLAN 1 Adjacency related matrices 2 Distance Matrix 3 Distance Laplacian matrix 4 Distance signless Laplacian matrix Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

3 1. Adjacency related matrices Adjacency matrix For a graph G = (V, E) on n vertices, the adjacency matrix A = A(G) is the 01 n nmatrix indexed by the vertices of G and dened by a i,j = 1 if and only if ij E The (adjacency) spectrum (λ 1, λ 2,..., λ n) of G, with λ 1 λ 2 λ n, is the A's spectrum 2 A = Aspectrum : (3, 1, 0, 0, 2, 2) Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

4 1. Adjacency related matrices Laplacian matrix The Laplacian of G is dened by L = L(G) = Deg A, where Deg is the diagonal matrix whose diagonal entries are the degrees in G, and A the adjacency matrix of G The Laplacian spectrum (µ 1, µ 2,..., µ n) of G, with µ 1 µ 2 µ n = 0, is the L's spectrum 2 L = Lspectrum : (5, 5, 3, 3, 2, 0) Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

5 1. Adjacency related matrices Signless Laplacian matrix The signless Laplacian of G is dened by Q = Q(G) = Deg + A The Laplacian spectrum (q 1, q 2,..., q n) of G, with q 1 q 2 q n, is the Q's spectrum 2 Q = Qspectrum : (6, 4, 3, 3, 1, 1) Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

6 2. Distance matrix Denition In a connected graph G the distance d(i, j) = d G (i, j) is the length of a shortest path between i and j The distance matrix D = D(G) of a connected graph G is the n nmatrix indexed by the vertices of G and where D i,j = d(i, j) The distance spectrum or Dspectrum is denoted by ( 1, 2,..., n) with 1 2 n 2 D = Dspectrum : (7, 0, 0, 2, 2, 3) Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

7 3. Distance Laplacian Denition The transmission of a vertex i is the sum of all the distances from i to all other vertices t i = P j V d(i, j). The distance Laplacian matrix of G is dened by D L = Tr D, where Tr is the diagonal matrix whose diagonal entries are the transmissions in G The distance Laplacian spectrum or D L spectrum is denoted by ( L 1, L 2,..., L n ) with L 1 L 2 L n = 0 2 D L = D L spectrum : (10, 9, 9, 7, 7, 0) Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

8 3. Distance Laplacian Examples of distance Laplacian spectra The complete graph K n : n (n 1), 0 (also the Laplacian spectrum) The complement of an edge K n e : n + 2, n (n 2), 0 The star S n : 2n 1 (n 2), n, 0 The complete bipartite graph K a,b : 2n a (b 1), 2n b (a 1), n, 0 The complete split graph SK n,α : `n + α α 1, n n α, 0 Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

9 3. Distance Laplacian Properties For any connected graph L n = 0 (with multiplicity m(0) = 1) m( L 1 ) n 1, equality holds only for K n Among trees L 1 2n 1, equality holds only for S n For L-spectra : µ 1(G) µ 1(G e) µ 2(G) µ 2(G e) µ n(g) = µ n(g e) = 0 There is no similar result for D L spectra The D L spectra of P 6 and C 6 are ( , 15, , 11, , 0) and (13, 13, 10, 9, 9, 0), respectively If e is an edge in G such that G e is connected, then L i (G e) L i (G), for i = 1, 2,..., n L i (G) L i (K n) = n, for i = 1, 2,... n 1 Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

10 3. Distance Laplacian Transmission regular graphs A connected graph G is ktransmission regular if t i = k, for i = 1, 2,, n If G is ktransmission regular with Dspectrum ( 1, 2,..., n), then (k n,... k 1) is the D L spectrum of G Moreover, the eigenspaces are the same A 7transmission regular regular Dspectrum : (7, 0, 0, 2, 2, 3) D L spectrum : (10, 9, 9, 7, 7, 0) Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

11 3. Distance Laplacian Graphs of diameter 2 Let G be a graph of diameter D = 2, (µ 1, µ 2,..., µ n = 0) its L-spectrum and ( 1, 2,..., n = 0) its D L spectrum. Then i = 2n µ n i, for i = 1, 2,..., n 1. Moreover, the Leigenspaces and Deigenspaces coincide A 7transmission regular regular Lspectrum : (5, 5, 3, 3, 2, 0) D L spectrum : (10, 9, 9, 7, 7, 0) Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

12 3. Distance Laplacian Similarities with the algebraic connectivity For the Laplacian L [Fiedler, 1973] : µ n 1 = 0 if and only if G is disconnected The multiplicity of 0 in the Lspectrum of G equals the number of connected components of G µ n 1 is called algebraic connectivity For the distance Laplacian D L : n is a D L eigenvalue of G if and only if the complement G is disconnected The multiplicity of n in the D L spectrum of G is 1 less than the number of connected components of G Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

13 3. Distance Laplacian Similarities with the algebraic connectivity Corollaries : 1 (G) n with equality if and only if G = K n If G is bipartite and n is a distance Laplacian eigenvalue of G, then G is complete bipartite The star S n is the only tree for which n is a distance Laplacian eigenvalue If the maximum degree = n 1, then n is a D L eigenvalue with multiplicity at least n (number of vertices of degree ) Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

14 4. Distance signless Laplacian Denition The transmission of a vertex i is the sum of all the distances from i to all other vertices t i = P j V d(i, j). The distance Laplacian matrix of G is dened by D Q = Tr + D, where Tr is the diagonal matrix whose diagonal entries are the transmissions in G The distance Laplacian spectrum or D Q spectrum is denoted by ( Q 1, Q 2,..., Q n ) with Q 1 Q 2 Q n 2 D Q = D Q spectrum : (14, 7, 7, 5, 5, 4) Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

15 4. Distance signless Laplacian Examples of distance signless Laplacian spectra For K n : 2n 2, n 2 (n 1) (also the signless Laplacian spectrum) «3n 2± (n 2) For K n e : 2 +16, n 2 (n 2) 2 «5n 8± 9n For S n : 2 32n+32, 2n 5 (n 2) 2 «5n 8± 9(a b) For K a,b : 2 +4ab, 2n a 4 (b 1), 2n b 4 (a 1) 2 Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

16 4. Distance signless Laplacian Properties For Q-spectra : q 1(G) q 1(G e) q 2(G) q 2(G e) q n(g) q n(g e) There is no similar result for D Q spectra The D Q spectra of P 6 and C 6 are ( , , , , , ) and (18, 9, 9, 8, 5, 5), respectively If e is an edge in G such that G e is connected, then Q i (G e) Q i (G), for i = 1, 2,..., n Q 1 (G) Q 1 (Kn) = 2n 2 with equality if and only if G = K n Q i (G) Q i (K n) = n 2, for i = 2, 3,..., n Q 2 (G) n 2 with equality if and only if G = K n Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

17 4. Distance signless Laplacian Transmission regular graphs 2Tr min 2Tr Q 1 (G) 2Tr max with equalities if and only if G is a transmission regular graph If G is ktransmission regular with Dspectrum ( 1, 2,..., n), then (k + 1, k + 2,..., k + n) is the D Q spectrum of G Moreover, the eigenspaces are the same A 7transmission regular regular Dspectrum : (7, 0, 0, 2, 2, 3) D L spectrum : (14, 7, 7, 5, 5, 4) Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

18 4. Distance signless Laplacian Bipartite components For the signless Laplacian : 0 is a Qeigenvalue of G if and only if G contains a bipartite component or an isolated vertex The multiplicity of 0 is equal to the number of bipartite components and isolated vertices For the distance signless Laplacian : If n 2 is a D Q eigenvalue of G with multiplicity m, then G contains at least m components, each of which is bipartite or an isolated vertex There exist graphs with a bipartite complement for which n 2 is not a D Q eigenvalue Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

19 4. Distance signless Laplacian Bipartite components G (left) with n = 5, Q > 3 and G (right) bipartite Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

20 The Petersen graph and its spectra Aouchiche & Hansen CanaDAM 2013, Memorial University, June / 20

A note on a conjecture for the distance Laplacian matrix

Electronic Journal of Linear Algebra Volume 31 Volume 31: (2016) Article 5 2016 A note on a conjecture for the distance Laplacian matrix Celso Marques da Silva Junior Centro Federal de Educação Tecnológica