Other Results and Open Problems
3. Open problems
Proposition 2.1. Let Γ be a graph with minimum degree δ and maximum degree
∆. Then
α2(G) ≤ min{|i : λi≥ ∆|, |i : λi≤ δ|}.
P roof. Suppose that the graph has a maximum independent set U = {1, 2, . . . , α2} with the vertices which are mutually at distance greater than 2. Then the matrix A2has a principal submatrix of the form diag(δ1, . . . , δα2). Hence, interlacing leads to
α2(G) ≤ min{|i : λi≥ ∆|, |i : λi≤ δ|}.
2
3. Open problems
Problem 3.1. Prove or disprove that, given any graph Γ = (V, E), we can find a matrix M with entries muv = 0 when uv 6∈ E such that the upper bound (23) is sharp.
Let B = S>AS be the quotient matrix of A with respect to a partition P Then we have the following known facts:
(1) The eigenvalues of B, ev B = {µ1, µ2, . . . , µm}, interlace the eigenvalues of A, ev A = {λ1, λ2, . . . , λn}.
(2) If the interlacing is tight, then P is equitable.
(3) If P is a distance partition, then P is equitable if and only if the interlacing is (2, 1)-exact in the sense of [6], that is µ1= λ1, µ2= λ2 and µm= λn. Problem 3.2. Find necessary and sufficient conditions for a partition P being equitable in terms of the bandwidth b of its quotient matrix B. Note that Fact 3 above would correspond to the case b = 3.
In Chapter 4, the result shown in Theorem 4.1 suggests the following question:
Problem 3.3. Prove or disprove: A regular bipartite graph Γ with predistance polynomial pd−1is distance-regular if and only if Ad−1= pd−1(A).
References
[1] A. Abiad, C. Dalf´o and M.A. Fiol, Algebraic characterizations of regularity properties in bipartite graphs, submitted.
[2] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin-Cummings, London, 1984.
[3] S. Bang and J.H. Koolen, Some Interlacing Results for the Eigenvalues of Distance-regular graphs, Des. Codes Cryptogr. 34 (2005), 173–186.
[4] N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, 1974.
[5] N. Biggs, Intersection matrices for linear graphs, Combinatorial Mathematics and its Appli-cations, ed. D.J.A. Welsh, 15–23, Academic Press, London, 1971.
[6] B. Bollob´as and V. Nikiforov, Graphs and Hermitian matrices: eigenvalue interlacing, Dis-crete Math. 289 (2004), 119–127.
[7] B. Bollob´as and V. Nikiforov, Graphs and Hermitian matrices: exact interlacing, Discrete Math. 308 (2008), no. 20, 4816–4821.
[8] A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin-New York, 1989.
[9] A.E. Browner and J.H. van Lint, Strongly regular graphs and partial geometries, Enumeration and designs, Proc. silver Jubilee Conf. on Combinatorics, Waterloo 1982, eds. D. M. Jackson
& S. A. Vanstone, Academic Press, Toronto. pp. 85–122.
[10] A.E. Brouwer and D.M. Mesner, The connectivity of strongly regular graphs, Europ. J. Comb.
6 (1985), 215–216.
[11] A.E. Brouwer and W.H. Haemers, The Gewirtz graph - An exercise in the theory of graph spectra, Europ. J. Comb. 14 (1993), 397–407.
[12] A.E. Brouwer and W.H. Haemers, A lower bound for the Laplacian eigenvalues of a graph -proof of a conjecture by Guo, Linear Algebra Appl. 429 (2008), 2131–2135.
[13] M. C´amara, J. F`abrega, M.A. Fiol and E. Garriga, Some families of orthogonal polynomials of a discrete variable and their applications to graphs and codes, Electron. J. Combin. 16(1) (2009), #R83.
[14] G. Chen, G. Davis, F. Hall, Z. Li, K. Patel and M. Stewart, An interlacing result on normal-ized Laplacians, SIAM J. Discrete Math. 18 (2004), 353–361.
[15] R. Courant and D. Hilbert, Methoden der Mathematischen Physic, Springer, Berlin, 1924.
[16] D.M. Cvetkovi´c, Spectrum of the graph of n-tuples, Publ. Elektrotehn. Fak. Univ. Beograd, Ser. Mat. Fiz. 274-301 (1969), 91–95.
[17] D.M. Cvetkovi´c, M. Doob and H. Sachs, Spectra of Graphs. Theory and Application, VEB Deutscher Verlag der Wissenschaften, Berlin, second edition, 1982.
[18] C. Dalf´o, M.A. Fiol and E. Garriga, On k-walk-regular graphs, Electron. J. Combin. 16(1) (2009), #R47.
[19] C. Dalf´o, E.R. van Dam, M.A. Fiol, E. Garriga and B.L. Gorissen, On almost distance-regular graphs, J. Combin. Theory Ser. A 118 (2011), 1094–1113.
[20] C.M. Da Fonseca, Interlacing properties for Hermitian matrices whose graph is a given tree, J. Matrix Anal. Appl. 27 (2005), no. 1, 130–141.
[21] E.R. van Dam and W.H. Haemers, Eigenvalues and the diameter of graphs, Linear and Multilinear Algebra 39 (1993), 33–44.
[22] A.B. Dieker, Interlacings for random walks on weighted graphs and the interchange process, J. Discrete Math. 24 (2010), no. 1, 191–206.
57
[23] M.A. Fiol, E. Garriga and J.L.A. Yebra, Locally pseudo-distance regular graphs, J. Combin.
Theory Ser. B 68 (1996), 179–205.
[24] M.A. Fiol and E. Garriga, From local adjacency polynomials to locally pseudo-distance-regular graphs, J. Combin. Theory Ser. B 71 (1997), 162–183.
[25] M.A. Fiol and E. Garriga, On the algebraic theory of pseudo-distance-regularity around a set, Linear Algebra Appl. 298 (1999), 115–141.
[26] M.A. Fiol, Eigenvalue interlacing and weight parameters of graphs, Linear Algebra Appl. 290 (1999), 275–301.
[27] M.A. Fiol, E. Garriga and J.L.A. Yebra, Boundary graphs: The limit case of a spectral property, Discrete Math. 226 (2001), 155–173.
[28] M.A. Fiol, On pseudo-distance-regularity. Linear Algebra Appl. 323 (2001), 145–165.
[29] M.A. Fiol, Algebraic characterizations of distance-regular graphs, Discrete Math. 246 (2002), 111–129.
[30] C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
[31] B.L. Gorrissen, Almost distance-regular graphs, Master Thesis, Tilburg University, 2009.
[32] W.H. Haemers, Eigenvalue techniques in design and graph theory (thesis Technical University Eindhoven 1979), Math. Centre Tract 121, Mathematical Centre, Amsterdam, 1980.
[33] W.H. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226-228 (1995), 593–616.
[34] W.H. Haemers, Distance-Regularity and the Spectrum of Graphs, Linear Algebra Appl. 236 (1996), 265–278.
[35] W.H. Haemers, A. Mohammadian and B. Tayfeh-Rezaie, On the sum of Laplacian eigenvalues of graphs, Linear Algebra Appl. 432 (2010), 2214–2221.
[36] C. Helmberg, B. Mohar, S. Poljak and F. Rendl, A spectral approach to bandwidth and separator problems in graphs, Linear and Multilin. Algebra 39 (1995), 73–90.
[37] A.J. Hoffman, On the polynomial of a graph, Amer. Math. Monthly 70 (1963), 30–36.
[38] A.J. Hoffman, On eigenvalues and colorings of graphs, Graph Theory and its Applications (B. Harris ed.), Academic Press, New York, 1970, pp. 79–91.
[39] Y. Hong, A bound on the spectral radius of graphs, Linear Algebra Appl., 108 (1988), 135–
139.
[40] L. Lov´asz, On the Shannon capacity of a graph, IEEE Trans. Inf. Theory 25 (1979), 1–7.
[41] D.E. Manolopoulos, D.R. Woodall and P.W. Fowler, Electronic stability of fullerenes: eigen-value theorems for leapfrog carbon clusters, J. Chem. Soc. Faraday Trans. 88 (1992), 2427–
2435.
[42] B. Mohar, Eigenvalues, Diameter, and Mean Distance in Graphs, Graphs Combin. 7 (1991), 53–64.
[43] B. Mohar, A domain monotonicity theorem for graphs and Hamiltonicity, Discrete Appl.
Math. 36 (1992), 169–177.
[44] P. Rowlinson, Linear algebra, in Graph Connections (L.W. Beineke and R.J. Wilson, eds.), Oxford Lecture Ser. Math. Appl., Vol. 5, 86–99, Oxford Univ. Press, New York, 1997.
[45] P.M. Weichsel, On distance-regularity in graphs, J. Combin. Theory Ser. B 32 (1982), 156–
161.
[46] B.F. Wu, J.Y. Shao and Y.Liu, Interlacing eigenvalues on some operations of graphs, Linear Algebra Appl. 430 (2009), no. 4, 1140–1150.
Notation
A Adjacency matrix of graph Γ auv (u, v)-entry of matrix A L Laplacian matrix of graph Γ luv (u, v)-entry of matrix L
d + 1 Number of different eigenvalues of adjacency matrix A D = D(Γ) Diameter of a graph Γ
∂(u, v) Distance between vertices u and v δ Degree of (regular) graph Γ δu Degree of vertex u
δ Average degree of graph Γ E = E(Γ) Edge set of a graph Γ Ei Eigenspace of eigenvalue θi
ecc(u) Eccentricity of vertex u
ev Γ = ev A Set of different eigenvalues of graph Γ
Γ Graph
Γk Distance-k graph of Γ
Γk(u) Set of vertices at distance k from vertex u u, v Vertices of Γ
H Hoffman polynomial
I Identity matrix
j All-1 vector
J All-1 matrix
θimi Eigenvalue of adjacency matrix A with multiplicity mi= m(θi) mu(θi) u-local multiplicity of θi
n Number of vertices in Γ
Nk(u) Set of vertices at distance at most k from u
O 0-matrix
0 0-vector
φΓ Characteristic polynomial of Γ
sp Γ = sp A Spectrum of the adjacency matrix of graph Γ tr A Trace of matrix A
V = V (Γ) Vertex set of a graph Γ
∼ Adjacency between vertices