Laplacian for Graphs Exercises 1.

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1 Laplacian for Graphs Exercises 1. Instructor: Mednykh I. A. Sobolev Institute of Mathematics Novosibirsk State University Winter School in Harmonic Functions on Graphs and Combinatorial Designs January, 2014 Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

2 Laplacian matrix. Laplacian spectrum The graphs under consideration are supposed to be unoriented and finite. They may have loops, multiple edges and to be disconnected. Let a uv be the number of edges between two given vertices u and v of G. The matrix A = A(G) = [a uv ] u,v V (G), is called the adjacency matrix of the graph G. Let d(v) denote the degree of v V (G), d(v) = u a uv, and let D = D(G) be the diagonal matrix indexed by V (G) and with d vv = d(v). The matrix L = L(G) = D(G) A(G) is called the Laplacian matrix of G. It should be noted that loops have no influence on L(G). The matrix L(G) is sometimes called the Kirchhoff matrix of G. Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

3 Laplacian polynomial and Laplacian spectrum We denote by µ(g, x) the characteristic polynomial of L(G). We will call it the Laplacian polynomial. Its roots will be called the Laplacian eigenvalues (or sometimes just eigenvalues) of G. They will be denoted by λ 1 (G) λ 2 (G)... λ n (G), (n = V (G) ), always enumerated in increasing order and repeated according to their multiplicity. We note that λ 1 is always equal to 0. Graph G is connected if and only if λ 2 > 0. If G consists of k components then λ 1 (G) = λ 2 (G) =... = λ k (G) = 0 and λ k+1 (G) > 0. Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

4 Preliminary results The following theorems help a lot when dealing with computation of Laplacian polynomials of various graphs. Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

5 Theorem (Eigenvalues of circulant matrix) Let v = (v 0, v 1,..., v n 1 ) be a row vector in C n, and V = circ{v}. If ε is primitive n th root of unity, then v 0 v 1... v n 2 v n 1 v n 1 v 0... v n 3 v n 2 n 1 det V = det..... n 1.. = ( ε jl v j ). v 2 v 3... v 0 v 1 l=0 j=0 v 1 v 2... v n 1 v 0 Corollary Eigenvalues of circulant matrix V is given by the formulae n 1 λ l = ε jl v j, l = 0,..., n 1. j=0 Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

6 Theorem (Kel mans) Let X 1 X 2 denote the join of X 1 and X 2, i.e. the graph obtained from the disjoint union of X 1 and X 2 by adding all possible edges uv, u V (X 1 ), v V (X 2 ). Then µ(x 1 X 2, x) = x(x n 1 n 2 ) (x n 1 )(x n 2 ) µ(x 1, x n 2 )µ(x 2, x n 1 ). where n 1 and n 2 are orders of X 1 and X 2, respectively and µ(x, x) is the characteristic polynomial of the Laplacian matrix of X. Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

7 Theorem (M. Fiedler (1973)) The Laplacian eigenvalues of the Cartesian product X 1 X 2 of graphs X 1 and X 2 are equal to all the possible sums of eigenvalues of the two factors: λ i (X 1 ) + λ j (X 2 ), i = 1,..., V (X 1 ), j = 1,..., V (X 2 ). Using this theorem we can easily determine the spectrum of lattice graphs. The m n lattice graph is just the Cartesian product of paths, P m P n. Below we will show that the spectrum of path-graph P k is l (k) i So P m P n has eigenvalues = 4 sin 2 πi, i = 0, 1,..., k 1. 2k λ i,j = l (m) i +l (n) j = 4 sin 2 πi πj +4 sin2, i = 0, 1,..., m 1, j = 0, 1,..., n 1. 2m 2n Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

8 Exercises Exercise 1.1. Find Laplacian spectrum of the complete graph on n vertices K n. Answer: {0 1, n n 1 }. Exercise 1.2. Find Laplacian spectrum of the complete bipartite graph K n,m. Answer: {0 1, n m 1, m n 1, (m + n) 1 }. Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

9 Exercise 1.3. Find Laplacian spectrum of the cycle graph C n. Answer: {2 2 cos( 2πk n ), k = 0,..., n 1}. Exercise 1.4. Find Laplacian spectrum of the path graph P n. Answer: {2 2 cos( πk n ), k = 0,..., n 1}. Exercise 1.5. Show that Laplacian polynomial of the path graph P n has the following form µ(p n, x) = x U n 1 ( x 2 2 ), where U n 1 (x) = second kind. sin(n arccos x) sin(arccos x) is the Chebyshev polynomial of the Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

10 Exercise 1.6. Find Laplacian spectrum of the wheel graph W n = K 1 C n. Answer: {0, n + 1, 3 2 cos 2πk n, k = 1,..., n 1}. Exercise 1.7. Find Laplacian spectrum of the fan graph F n = K 1 P n. Answer: {0, n + 1, 3 2 cos πk n, k = 1,..., n 1}. Exercise 1.8. Show that the Laplacian polynomial of the fan graph F n = K 1 P n is given by the formula µ(f n, x) = x(x n 1)U n 1 ( x 3 2 ), where U n 1 (x) is the Chebyshev polynomial of the second kind. Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

11 Exercise 1.9. Find Laplacian spectrum of the cylinder graph P m C n. Exercise Find Laplacian spectrum of the Moebius ladder graph M n. Moebius ladder graph is a cycle graph C 2n with additional edges, connecting opposite vertices in cycle. Mednykh I. A. (Sobolev Institute of Math) Laplacian for Graphs January / 11

Laplacians of Graphs, Spectra and Laplacian polynomials

Mednykh A. D. (Sobolev Institute of Math) Laplacian for Graphs 27 June - 03 July 2015 1 / 30 Laplacians of Graphs, Spectra and Laplacian polynomials Alexander Mednykh Sobolev Institute of Mathematics Novosibirsk