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Eigenvalues and Distance-Regularity of Graphs Edwin Van EvDRG EvDRG Dedication Spectrum Two many Distance-regular Walks Central equation Eigenvalues and distance-regularity of graphs Structure Twisted and odd Good conditions Polynomials Projection Spectral Excess Edwin van Dam Desargues Partial linear space q-ary Desargues Ugly DRGs Perturbations Dept. Econometrics and Operations Research Remove vertices Remove edges Tilburg University Adding edges Amalgamate Generalized Odd Proof Graph Theory and Interactions, Durham, July 20, 2013 Durham, July 20, 2013 Edwin van Dam – 1 / 24 Dedication EvDRG Dedication Spectrum Two many Distance-regular Walks Central equation Structure Twisted and odd Good conditions Polynomials Projection Spectral Excess Desargues Partial linear space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof David Gregory Durham, July 20, 2013 Edwin van Dam – 2 / 24 Spectrum EvDRG Dedication Spectrum Two many Distance-regular Walks Central equation A (finite simple) graph Γ on n vertices Structure Twisted and odd Good conditions Polynomials Projection ⇑ ? Spectral Excess ⇓ Desargues Partial linear space q-ary Desargues Ugly DRGs The spectrum (of eigenvalues) λ ≥ ... ≥ λ Perturbations 1 n Remove vertices of the (a) 01-adjacency matrix A of Γ Remove edges Adding edges Amalgamate Generalized Odd Proof Durham, July 20, 2013 Edwin van Dam – 3 / 24 Two many EvDRG Dedication Spectrum Two many Distance-regular There are 2 graphs on 30 vertices with spectrum Walks Central equation Structure 12, 2 (9×), 0 (15×), −6 (5×). Twisted and odd Good conditions Polynomials Projection Spectral Excess Desargues There are more than 60,000 graphs on 30 vertices with spectrum Partial linear space q-ary Desargues 12, 3 (10×), 0 (5×), −3 (14×). Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof Durham, July 20, 2013 Edwin van Dam – 4 / 24 Distance-regular EvDRG Dedication Spectrum Two many Distance-regularity: there are ci, ai, bi, i =0, 1,...,d such that for Distance-regular every pair of vertices u and w at distance i: Walks Central equation Structure # neighbors z of w at distance i − 1 from u equals ci Twisted and odd Good conditions # neighbors z of w at distance i from u equals ai Polynomials # neighbors z of w at distance i +1 from u equals bi Projection Spectral Excess Desargues Partial linear space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof Durham, July 20, 2013 Edwin van Dam – 5 / 24 Distance-regular EvDRG Dedication Spectrum Two many Distance-regularity: there are ci, ai, bi, i =0, 1,...,d such that for Distance-regular every pair of vertices u and w at distance i: Walks Central equation Structure # neighbors z of w at distance i − 1 from u equals ci Twisted and odd Good conditions # neighbors z of w at distance i from u equals ai Polynomials # neighbors z of w at distance i +1 from u equals bi Projection Spectral Excess Desargues Complete graphs, Strongly regular graphs (among which are regular Partial linear space complete multipartite graphs), Cycles, q-ary Desargues Ugly DRGs Hamming graphs, Johnson graphs, Grassmann graphs, Odd graphs .... Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof Durham, July 20, 2013 Edwin van Dam – 5 / 24 Distance-regular EvDRG Dedication Spectrum Two many Distance-regularity: there are ci, ai, bi, i =0, 1,...,d such that for Distance-regular every pair of vertices u and w at distance i: Walks Central equation Structure # neighbors z of w at distance i − 1 from u equals ci Twisted and odd Good conditions # neighbors z of w at distance i from u equals ai Polynomials # neighbors z of w at distance i +1 from u equals bi Projection Spectral Excess Desargues Complete graphs, Strongly regular graphs (among which are regular Partial linear space complete multipartite graphs), Cycles, q-ary Desargues Ugly DRGs Hamming graphs, Johnson graphs, Grassmann graphs, Odd graphs .... Perturbations Remove vertices Remove edges Fon-Der-Flaass (2002) ⇒ Almost all distance-regular graphs are not Adding edges determined by the spectrum. Amalgamate Generalized Odd Proof cf. EvD & Haemers (2003) ‘would bet’ that almost all graphs are determined by the spectrum. Durham, July 20, 2013 Edwin van Dam – 5 / 24 Walks EvDRG Ai is the distance-i adjacency matrix, A = A1: Dedication Spectrum Two many AAi = bi−1Ai−1 + aiAi + ci+1Ai+1, i =0, 1,...,d, Distance-regular Walks Central equation Structure Ai = pi(A) for a polynomial pi of degree i Twisted and odd Good conditions Polynomials Projection Rowlinson (1997): A graph is a DRG iff the number of walks of length Spectral Excess Desargues ℓ from x to y depends only on ℓ and the distance between x and y Partial linear space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof Durham, July 20, 2013 Edwin van Dam – 6 / 24 Walks EvDRG Ai is the distance-i adjacency matrix, A = A1: Dedication Spectrum Two many AAi = bi−1Ai−1 + aiAi + ci+1Ai+1, i =0, 1,...,d, Distance-regular Walks Central equation Structure Ai = pi(A) for a polynomial pi of degree i Twisted and odd Good conditions Polynomials Projection Rowlinson (1997): A graph is a DRG iff the number of walks of length Spectral Excess Desargues ℓ from x to y depends only on ℓ and the distance between x and y Partial linear space q-ary Desargues Ugly DRGs Perturbations Distance-regular graphs: intersection numbers ↔ eigenvalues Remove vertices Remove edges Adding edges Intersection numbers do not determine the graph (in general) Amalgamate Generalized Odd Proof Do the eigenvalues determine distance-regularity ? Durham, July 20, 2013 Edwin van Dam – 6 / 24 Central equation EvDRG Dedication Spectrum Two many Distance-regular Walks ℓ ℓ ℓ (A )uu = tr A = λi Central equation X X Structure u i Twisted and odd Good conditions Polynomials Projection Spectral Excess X p(A)uu = tr p(A)= X p(λi) Desargues Partial linear u i space q-ary Desargues Ugly DRGs for every polynomial p Perturbations Remove vertices Remove edges Adding edges All spectral information is in these equations Amalgamate Generalized Odd Proof Durham, July 20, 2013 Edwin van Dam – 7 / 24 Structure EvDRG Dedication Spectrum Two many Distance-regular The following can be derived from the spectrum: Walks Central equation ■ number of vertices Structure ■ number of edges Twisted and odd ■ Good conditions number of triangles Polynomials ■ number of closed walks of length ℓ Projection ■ bipartiteness Spectral Excess ■ Desargues regularity Partial linear ■ regularity + connectedness space ■ q-ary Desargues regularity + girth Ugly DRGs ■ odd-girth Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof Durham, July 20, 2013 Edwin van Dam – 8 / 24 Twisted and odd EvDRG Dedication Spectrum Two many Distance-regularity is not determined by the spectrum Distance-regular Walks Central equation Structure Twisted and odd Good conditions Polynomials Projection Spectral Excess Desargues Partial linear space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges The (‘almost’ dr) twisted Desargues graph Adding edges Amalgamate (Bussemaker & Cvetkovi´c 1976, Schwenk 1978) Generalized Odd Proof Note: Desargues is Doubled Petersen Durham, July 20, 2013 Edwin van Dam – 9 / 24 Good conditions EvDRG Dedication Spectrum Theorem. If Γ is distance-regular, diameter d, valency k, girth g, Two many distinct eigenvalues k = θ0, θ1,...,θd, satisfying one of the following Distance-regular Walks properties, then every graph cospectral with Γ is also distance-regular: Central equation Structure Twisted and odd Good conditions Polynomials Projection Spectral Excess Desargues Partial linear space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof Durham, July 20, 2013 Edwin van Dam – 10 / 24 Good conditions EvDRG Dedication Spectrum Theorem. If Γ is distance-regular, diameter d, valency k, girth g, Two many distinct eigenvalues k = θ0, θ1,...,θd, satisfying one of the following Distance-regular Walks properties, then every graph cospectral with Γ is also distance-regular: Central equation Structure 1. g ≥ 2d − 1 (Brouwer&Haemers), Twisted and odd 2. g ≥ 2d − 2 and Γ is bipartite (EvD&Haemers), Good conditions Polynomials 3. g ≥ 2d − 2 and cd−1cd < −(cd−1 + 1)(θ1 + ... + θd) (EvD&Haemers), Projection c ... c − (EvD&Haemers) Spectral Excess 4. 1 = = d 1 =1 , Desargues 5. Γ = dodecahedron or icosahedron (Haemers&Spence), Partial linear space 6. Γ = coset graph extended ternary Golay code (EvD&Haemers), q-ary Desargues 7. Γ = Ivanov-Ivanov-Faradjev graph (EvD&Haemers&Koolen&Spence), Ugly DRGs Perturbations 8. Γ = line graph Petersen graph or line graph Hoffman-Singleton Remove vertices graph (EvD&Haemers), Remove edges Adding edges 9. Γ = Hamming graph H(3, q), q ≥ 36 (Bang&EvD&Koolen), Amalgamate a1 ... a −1 , a Generalized Odd 10. Γ = generalized odd graph ( = = d =0 d 6=0) Proof (Huang&Liu). Durham, July 20, 2013 Edwin van Dam – 10 / 24 Polynomials EvDRG Dedication Spectrum Consider the spectrum of a k-regular graph Two many Distance-regular 1 1 Inner product hp, qi = tr(p(A)q(A)) = p(λi)q(λi) Walks n n Pi Central equation Structure on the space of polynomials mod minimal polynomial Twisted and odd Good conditions Polynomials Projection Spectral Excess Desargues Partial linear space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized
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