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

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

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

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

Durham, July 20, 2013 Edwin van Dam – 11 / 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 Orthogonal system of predistance polynomials pi of degree i Projection normalized such that hpi, pii = pi(k) 6=0 Spectral Excess Desargues Partial linear space xpi = βi−1pi−1 + αipi + γi+1pi+1, i =0, 1,...,d, q-ary Desargues Ugly DRGs compare to Perturbations Remove vertices Remove edges AAi = bi−1Ai−1 + aiAi + ci+1Ai+1, i =0, 1,...,d, Adding edges Amalgamate Generalized Odd Proof H = Pi pi is the Hoﬀman polynomial: H(A) = J

Durham, July 20, 2013 Edwin van Dam – 11 / 24 Projection

EvDRG Dedication Spectrum Two many 1 Distance-regular hX,Y i = n tr(XY ): inner product on symmetric matrices of size n Walks Central equation Structure hp(A), q(A)i = hp, qi 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 – 12 / 24 Projection

EvDRG Dedication Spectrum Two many 1 Distance-regular hX,Y i = n tr(XY ): inner product on symmetric matrices of size n Walks Central equation Structure hp(A), q(A)i = hp, qi Twisted and odd Good conditions Project Ad onto the space A of polynomials in A: Polynomials Projection Spectral Excess Desargues d Partial linear hAd, pi(A)i hAd, pd(A)i space A = pi(A) = p (A) d X kp (A)k2 kp (A)k2 d q-ary Desargues f i=0 i d Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 12 / 24 Projection

EvDRG Dedication Spectrum Two many 1 Distance-regular hX,Y i = n tr(XY ): inner product on symmetric matrices of size n Walks Central equation Structure hp(A), q(A)i = hp, qi Twisted and odd Good conditions Project Ad onto the space A of polynomials in A: Polynomials Projection Spectral Excess Desargues d Partial linear hAd, pi(A)i hAd, pd(A)i space A = pi(A) = p (A) d X kp (A)k2 kp (A)k2 d q-ary Desargues f i=0 i d Ugly DRGs Perturbations hAd, H(A)i hAd,Ji kd Remove vertices = 2 pd(A) = pd(A) = pd(A) Remove edges kpdk pd(k) pd(k) Adding edges Amalgamate Generalized Odd where k = 1 k (u) Proof d n Pu d

Durham, July 20, 2013 Edwin van Dam – 12 / 24 Spectral Excess

EvDRG Dedication Spectrum Two many Distance-regular 2 2 Walks k k Central equation 2 2 d 2 d kd = kAdk ≥kAdk = 2 kpd(A)k = Structure f pd(k) pd(k) Twisted and odd Good conditions Polynomials Projection hence kd ≤ pd(k) with equality iﬀ Ad = pd(A) 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 – 13 / 24 Spectral Excess

EvDRG Dedication Spectrum Two many Distance-regular 2 2 Walks k k Central equation 2 2 d 2 d kd = kAdk ≥kAdk = 2 kpd(A)k = Structure f pd(k) pd(k) Twisted and odd Good conditions Polynomials Projection hence kd ≤ pd(k) with equality iﬀ Ad = pd(A) Spectral Excess Desargues Partial linear space q-ary Desargues Spectral Excess Theorem (Fiol & Garriga 1997): Ugly DRGs Perturbations Remove vertices kd ≤ pd(k) with equality iﬀ the graph is distance-regular Remove edges Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 13 / 24 Desargues

EvDRG Dedication 1 4 5 5 4 1 Spectrum Find graphs with spectrum {3 , 2 , 1 , −1 , −2 , −3 }. 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

Durham, July 20, 2013 Edwin van Dam – 14 / 24 Desargues

EvDRG Dedication 1 4 5 5 4 1 Spectrum Find graphs with spectrum {3 , 2 , 1 , −1 , −2 , −3 }. Two many Distance-regular Connected, 3-regular, bipartite on 10+10 vertices, girth 6. Walks Central equation Structure So this is the incidence graph of a partial linear space. Twisted and odd Good conditions Polynomials Diameter at most 5, with k5 ≤ 1. Projection Spectral Excess Distance distribution diagram: 20=13 + 132 + 162+? + 3+? Desargues Partial linear space k4(x)=3 so k5(x) ≤ 1. q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 14 / 24 Partial linear space

Durham, July 20, 2013 Edwin van Dam – 15 / 24 Partial linear space

EvDRG The halved graphs (the point graph and line graph of the partial linear Dedication 1 4 5 Spectrum space) have spectrum {6 , 1 , −2 }. The only graph possible is Two many J(5, 2), the complement of Petersen. Distance-regular Walks Central equation Start from point graph, and try to construct a partial linear space: Structure this can be done in more than one way: Twisted and odd Good conditions Desargues and twisted Desargues 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 – 15 / 24 Partial linear space

EvDRG The halved graphs (the point graph and line graph of the partial linear Dedication 1 4 5 Spectrum space) have spectrum {6 , 1 , −2 }. The only graph possible is Two many J(5, 2), the complement of Petersen. Distance-regular Walks Central equation Start from point graph, and try to construct a partial linear space: Structure this can be done in more than one way: Twisted and odd Good conditions Desargues and twisted Desargues Polynomials Projection Neighborsof12: 13,14,15,23,24,25 Spectral Excess Desargues Partial linear Make lines of size 3: {12, 13, 23}, {12, 14, 24}, {12, 15, 25} space q-ary Desargues lines ‘123’, ‘124’, ‘125’ Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 15 / 24 Partial linear space

EvDRG The halved graphs (the point graph and line graph of the partial linear Dedication 1 4 5 Spectrum space) have spectrum {6 , 1 , −2 }. The only graph possible is Two many J(5, 2), the complement of Petersen. Distance-regular Walks Central equation Start from point graph, and try to construct a partial linear space: Structure this can be done in more than one way: Twisted and odd Good conditions Desargues and twisted Desargues Polynomials Projection Neighborsof12: 13,14,15,23,24,25 Spectral Excess Desargues Partial linear Make lines of size 3: {12, 13, 23}, {12, 14, 24}, {12, 15, 25} space q-ary Desargues lines ‘123’, ‘124’, ‘125’ Ugly DRGs Perturbations Remove vertices Or (the twisted way): {12, 13, 14}, {12, 23, 24}, {12, 15, 25} Remove edges lines ‘1’, ‘2’, ‘125’ Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 15 / 24 q-ary Desargues

Durham, July 20, 2013 Edwin van Dam – 16 / 24 q-ary Desargues

EvDRG (2d − 1)-dimensional vector space over GF (q) Dedication Spectrum Two many points: (d − 1)-dimensional subspaces Distance-regular lines: d-dimensional subspaces Walks Central equation Structure Incidence graph is doubled Grassmann Twisted and odd Point and line graph are Grassmann Jq(2d − 1,d − 1) Good conditions Polynomials Projection Twist: Fix a hyperplane H Spectral Excess Desargues lines: d-dimensional subspaces not contained in H Partial linear twisted lines: (d − 2)-dimensional subspaces contained in H space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 16 / 24 q-ary Desargues

EvDRG (2d − 1)-dimensional vector space over GF (q) Dedication Spectrum Two many points: (d − 1)-dimensional subspaces Distance-regular lines: d-dimensional subspaces Walks Central equation Structure Incidence graph is doubled Grassmann Twisted and odd Point and line graph are Grassmann Jq(2d − 1,d − 1) Good conditions Polynomials Projection Twist: Fix a hyperplane H Spectral Excess Desargues lines: d-dimensional subspaces not contained in H Partial linear twisted lines: (d − 2)-dimensional subspaces contained in H space q-ary Desargues Ugly DRGs Point graph is again Jq(2d − 1,d − 1) Perturbations Remove vertices Incidence graph is cospectral to doubled Grassmann, but not drg Remove edges Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 16 / 24 q-ary Desargues

EvDRG (2d − 1)-dimensional vector space over GF (q) Dedication Spectrum Two many points: (d − 1)-dimensional subspaces Distance-regular lines: d-dimensional subspaces Walks Central equation Structure Incidence graph is doubled Grassmann Twisted and odd Point and line graph are Grassmann Jq(2d − 1,d − 1) Good conditions Polynomials Projection Twist: Fix a hyperplane H Spectral Excess Desargues lines: d-dimensional subspaces not contained in H Partial linear twisted lines: (d − 2)-dimensional subspaces contained in H space q-ary Desargues Ugly DRGs Point graph is again Jq(2d − 1,d − 1) Perturbations Remove vertices Incidence graph is cospectral to doubled Grassmann, but not drg Remove edges Adding edges Line graph is cospectral to Jq(2d − 1,d − 1) Amalgamate Generalized Odd Spectral excess theorem: line graph is distance-regular! Proof ....but it is UGLY!!!

Durham, July 20, 2013 Edwin van Dam – 16 / 24 Ugly DRGs

EvDRG Families of ‘ugly’ distance-regular graphs with unbounded diameter: Dedication Spectrum Two many Doob, Hemmeter, Ustimenko: not distance-transitive. 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 twisted Grassmann (aka vD-Koolen 2005): not even vertex-transitive. Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 17 / 24 Perturbations

EvDRG Dalf´o& EvD & Fiol (2011): Ugly (almost) distance-regular graphs Dedication Spectrum can be used to construct cospectral graphs through perturbations: Two many Distance-regular Adding and removing vertices, edges, amalgamating vertices, etc. Walks Central equation Structure Twisted and odd Good conditions Polynomials Projection The devil’s advocate (Durham, 2013): It is easy to construct Spectral Excess cospectral graphs 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 – 18 / 24 Remove vertices

EvDRG Dedication Spectrum Two many Distance-regular Walks Central equation Structure Twisted and odd Good conditions Polynomials Projection Spectral Excess Desargues Partial linear Removing vertices from the twisted Desargues graph space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 19 / 24 Remove edges

EvDRG Dedication Spectrum Two many Distance-regular Walks Central equation Structure Twisted and odd Good conditions Polynomials Projection Spectral Excess Desargues Partial linear Removing edges from the twisted Desargues graph space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 20 / 24 Adding edges

Durham, July 20, 2013 Edwin van Dam – 21 / 24 Amalgamate

EvDRG Dedication Spectrum Two many Distance-regular Walks Central equation Structure Twisted and odd Good conditions Polynomials Projection Spectral Excess Desargues Partial linear Amalgamate vertices in the twisted Desargues graph space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof

Durham, July 20, 2013 Edwin van Dam – 22 / 24 Generalized Odd

EvDRG Dedication Spectrum Generalized odd graph (drg with a1 = ... = ad−1 =0, ad 6=0) Two many Distance-regular No odd cycles of length less than 2d +1 (almost bipartite) 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

Durham, July 20, 2013 Edwin van Dam – 23 / 24 Generalized Odd

EvDRG Dedication Spectrum Generalized odd graph (drg with a1 = ... = ad−1 =0, ad 6=0) Two many Distance-regular No odd cycles of length less than 2d +1 (almost bipartite) Walks Central equation Structure EvD & Haemers (2011): A regular graph with d +1 distinct Twisted and odd Good conditions eigenvalues and odd-girth 2d +1 is a generalized odd graph Polynomials Projection Lee & Weng (2012) extended this for non-regular graphs 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 – 23 / 24 Generalized Odd

EvDRG Dedication Spectrum Generalized odd graph (drg with a1 = ... = ad−1 =0, ad 6=0) Two many Distance-regular No odd cycles of length less than 2d +1 (almost bipartite) Walks Central equation Structure EvD & Haemers (2011): A regular graph with d +1 distinct Twisted and odd Good conditions eigenvalues and odd-girth 2d +1 is a generalized odd graph Polynomials Projection Lee & Weng (2012) extended this for non-regular graphs Spectral Excess Desargues Partial linear Sketch of short proof (EvD & Fiol 2012): space q-ary Desargues Ugly DRGs Recall xpi = βi−1pi−1 + αipi + γi+1pi+1, i =0, 1,...,d, Perturbations Remove vertices α ,i

Durham, July 20, 2013 Edwin van Dam – 23 / 24 Generalized Odd

EvDRG Dedication Spectrum Generalized odd graph (drg with a1 = ... = ad−1 =0, ad 6=0) Two many Distance-regular No odd cycles of length less than 2d +1 (almost bipartite) Walks Central equation Structure EvD & Haemers (2011): A regular graph with d +1 distinct Twisted and odd Good conditions eigenvalues and odd-girth 2d +1 is a generalized odd graph Polynomials Projection Lee & Weng (2012) extended this for non-regular graphs Spectral Excess Desargues Partial linear Sketch of short proof (EvD & Fiol 2012): space q-ary Desargues Ugly DRGs Recall xpi = βi−1pi−1 + αipi + γi+1pi+1, i =0, 1,...,d, Perturbations Remove vertices α ,i

Durham, July 20, 2013 Edwin van Dam – 23 / 24 Proof

EvDRG ℓ ℓ A = i θi Ei (spectral decomposition) Dedication P Spectrum Two many Odd powers (ℓ =1, 3,..., 2d − 1) have zero diagonal Distance-regular Walks E A2 Central equation is and hence have constant diagonal, so the graph is regular 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 – 24 / 24 Proof

EvDRG ℓ ℓ A = i θi Ei (spectral decomposition) Dedication P Spectrum Two many Odd powers (ℓ =1, 3,..., 2d − 1) have zero diagonal Distance-regular Walks E A2 Central equation is and hence have constant diagonal, so the graph is regular Structure Twisted and odd Good conditions Hoﬀman polynomial: H(A) = pi(A) = J Polynomials Pi Projection Spectral Excess u, v at distance d: pd(A)uv =1 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 – 24 / 24 Proof

EvDRG ℓ ℓ A = i θi Ei (spectral decomposition) Dedication P Spectrum Two many Odd powers (ℓ =1, 3,..., 2d − 1) have zero diagonal Distance-regular Walks E A2 Central equation is and hence have constant diagonal, so the graph is regular Structure Twisted and odd Good conditions Hoﬀman polynomial: H(A) = pi(A) = J Polynomials Pi Projection Spectral Excess u, v at distance d: pd(A)uv =1 Desargues Partial linear space If dist(u, v) and d have diﬀerent parity: pd(A)uv =0 q-ary Desargues Ugly DRGs dist u, v

Durham, July 20, 2013 Edwin van Dam – 24 / 24 Proof

EvDRG ℓ ℓ A = i θi Ei (spectral decomposition) Dedication P Spectrum Two many Odd powers (ℓ =1, 3,..., 2d − 1) have zero diagonal Distance-regular Walks E A2 Central equation is and hence have constant diagonal, so the graph is regular Structure Twisted and odd Good conditions Hoﬀman polynomial: H(A) = pi(A) = J Polynomials Pi Projection Spectral Excess u, v at distance d: pd(A)uv =1 Desargues Partial linear space If dist(u, v) and d have diﬀerent parity: pd(A)uv =0 q-ary Desargues Ugly DRGs dist u, v

THE END Durham, July 20, 2013 Edwin van Dam – 24 / 24