Villanova Villanova Spectrum Distance-regular Walks Closed walks Structure Spectral characterizations of Twisted and odd Classical result distance-regularity of graphs Survey Polynomials Spectral Excess Preintersection numbers Odd-girth Edwin van Dam Girth Thanks Doubled Odd Doubled Grassmann Ugly DRGs Tilburg University, the Netherlands Perturbations End Modern Trends in Algebraic Graph Theory, Villanova, June 2014 Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 1 / 20
Spectrum Villanova Spectrum Distance-regular Walks Closed walks Structure A (finite simple) graph Γ on n vertices Twisted and odd Classical result Survey Polynomials Spectral Excess Preintersection ⇓ ⇑ ? numbers Odd-girth Girth Thanks Doubled Odd The spectrum (of eigenvalues) λ 1 ≥ . . . ≥ λ n Doubled Grassmann Ugly DRGs of the (a) 01-adjacency matrix A of Γ Perturbations End Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 2 / 20
Spectrum Villanova Spectrum Distance-regular Walks Closed walks Structure A (finite simple) graph Γ on n vertices Twisted and odd Classical result Survey Polynomials Spectral Excess Preintersection ⇓ ⇑ ? numbers Odd-girth Girth Thanks Doubled Odd The spectrum (of eigenvalues) λ 1 ≥ . . . ≥ λ n Doubled Grassmann Ugly DRGs of the (a) 01-adjacency matrix A of Γ Perturbations End EvD & Haemers (2003) ‘would bet’ that almost all graphs are determined by the spectrum. Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 2 / 20
Distance-regular Villanova Spectrum Distance-regular Distance-regularity: there are c i , a i , b i , i = 0 , 1 , . . . , d such that for Walks Closed walks every pair of vertices u and w at distance i : Structure Twisted and odd # neighbors z of w at distance i − 1 from u equals c i Classical result Survey # neighbors z of w at distance i from u equals a i Polynomials # neighbors z of w at distance i + 1 from u equals b i Spectral Excess Preintersection numbers Odd-girth Girth Thanks Doubled Odd Doubled Grassmann Ugly DRGs Perturbations End Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 3 / 20
Distance-regular Villanova Spectrum Distance-regular Distance-regularity: there are c i , a i , b i , i = 0 , 1 , . . . , d such that for Walks Closed walks every pair of vertices u and w at distance i : Structure Twisted and odd # neighbors z of w at distance i − 1 from u equals c i Classical result Survey # neighbors z of w at distance i from u equals a i Polynomials # neighbors z of w at distance i + 1 from u equals b i Spectral Excess Preintersection numbers Complete graphs, Strongly regular graphs, Cycles, Odd-girth Girth Thanks Hamming graphs, Johnson graphs, Grassmann graphs, Odd graphs .... Doubled Odd Doubled Grassmann Ugly DRGs Perturbations End Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 3 / 20
Distance-regular Villanova Spectrum Distance-regular Distance-regularity: there are c i , a i , b i , i = 0 , 1 , . . . , d such that for Walks Closed walks every pair of vertices u and w at distance i : Structure Twisted and odd # neighbors z of w at distance i − 1 from u equals c i Classical result Survey # neighbors z of w at distance i from u equals a i Polynomials # neighbors z of w at distance i + 1 from u equals b i Spectral Excess Preintersection numbers Complete graphs, Strongly regular graphs, Cycles, Odd-girth Girth Thanks Hamming graphs, Johnson graphs, Grassmann graphs, Odd graphs .... Doubled Odd Doubled Grassmann Ugly DRGs Fon-Der-Flaass (2002) ⇒ Almost all distance-regular graphs are not Perturbations End determined by the spectrum. Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 3 / 20
Walks A i is the distance- i adjacency matrix, A = A 1 : Villanova Spectrum Distance-regular AA i = b i − 1 A i − 1 + a i A i + c i +1 A i +1 , i = 0 , 1 , . . . , d, Walks Closed walks Structure Twisted and odd Classical result A i = p i ( A ) for a polynomial p i of degree i Survey Polynomials Spectral Excess Rowlinson (1997): A graph is a DRG iff the number of walks of length Preintersection numbers ℓ from x to y depends only on ℓ and the distance between x and y Odd-girth Girth Thanks Doubled Odd Doubled Grassmann Ugly DRGs Perturbations End Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 4 / 20
Walks A i is the distance- i adjacency matrix, A = A 1 : Villanova Spectrum Distance-regular AA i = b i − 1 A i − 1 + a i A i + c i +1 A i +1 , i = 0 , 1 , . . . , d, Walks Closed walks Structure Twisted and odd Classical result A i = p i ( A ) for a polynomial p i of degree i Survey Polynomials Spectral Excess Rowlinson (1997): A graph is a DRG iff the number of walks of length Preintersection numbers ℓ from x to y depends only on ℓ and the distance between x and y Odd-girth Girth Thanks Doubled Odd Doubled Intersection numbers do not determine the graph (in general) Grassmann Ugly DRGs Perturbations End Do the eigenvalues determine distance-regularity ? Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 4 / 20
Closed walks Villanova Spectrum Distance-regular Walks Closed walks ( A ℓ ) uu = tr A ℓ = � � λ ℓ Structure i Twisted and odd u i Classical result Survey Polynomials Spectral Excess Preintersection � � p ( A ) uu = tr p ( A ) = p ( λ i ) numbers Odd-girth u i Girth Thanks Doubled Odd for every polynomial p Doubled Grassmann Ugly DRGs Perturbations All spectral information is in these equations End Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 5 / 20
Structure Villanova Spectrum Distance-regular Walks The following can be derived from the spectrum: Closed walks Structure ■ number of vertices Twisted and odd Classical result ■ number of edges Survey ■ number of triangles Polynomials number of closed walks of length ℓ ■ Spectral Excess Preintersection bipartiteness ■ numbers ■ regularity Odd-girth Girth ■ regularity + connectedness Thanks ■ regularity + girth Doubled Odd ■ odd-girth Doubled Grassmann Ugly DRGs Perturbations End Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 6 / 20
Twisted and odd Villanova Spectrum Distance-regular Distance-regularity is not determined by the spectrum Walks Closed walks Structure Twisted and odd Classical result Survey Polynomials Spectral Excess Preintersection numbers Odd-girth Girth Thanks Doubled Odd Doubled Grassmann Ugly DRGs Perturbations The (‘almost’ dr) twisted Desargues graph End (Bussemaker & Cvetkovi´ c 1976, Schwenk 1978) Note: Desargues is Doubled Petersen Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 7 / 20
Classical result If Γ is distance-regular, diameter d , valency k , girth g , distinct Villanova Spectrum eigenvalues k = θ 0 , θ 1 , . . . , θ d , satisfying one of the following Distance-regular properties, then every graph cospectral with Γ is also distance-regular: Walks Closed walks 1. g ≥ 2 d − 1 (Brouwer & Haemers 1993), Structure Twisted and odd 2. g ≥ 2 d − 2 and Γ is bipartite (EvD & Haemers 2002) , Classical result 3. g ≥ 2 d − 2 and c d − 1 c d < − ( c d − 1 + 1)( θ 1 + . . . + θ d ) (EvD&Haemers 2002) , Survey Polynomials 4. c 1 = . . . = c d − 1 = 1 (EvD & Haemers 2002) , Spectral Excess 5. a 1 = . . . = a d − 1 = 0 , a d � = 0 (Huang & Liu 1999). Preintersection numbers Odd-girth Girth Thanks Doubled Odd Doubled Grassmann Ugly DRGs Perturbations End Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 8 / 20
Classical result If Γ is distance-regular, diameter d , valency k , girth g , distinct Villanova Spectrum eigenvalues k = θ 0 , θ 1 , . . . , θ d , satisfying one of the following Distance-regular properties, then every graph cospectral with Γ is also distance-regular: Walks Closed walks 1. g ≥ 2 d − 1 (Brouwer & Haemers 1993), Structure Twisted and odd 2. g ≥ 2 d − 2 and Γ is bipartite (EvD & Haemers 2002) , Classical result 3. g ≥ 2 d − 2 and c d − 1 c d < − ( c d − 1 + 1)( θ 1 + . . . + θ d ) (EvD&Haemers 2002) , Survey Polynomials 4. c 1 = . . . = c d − 1 = 1 (EvD & Haemers 2002) , Spectral Excess 5. a 1 = . . . = a d − 1 = 0 , a d � = 0 (Huang & Liu 1999). Preintersection numbers Odd-girth Moreover, the following graphs are determined by their spectrum: Girth Thanks 1. dodecahedron and icosahedron (Haemers & Spence 1995), Doubled Odd Doubled 2. coset graph extended ternary Golay code (EvD & Haemers 2002) , Grassmann Ugly DRGs 3. Ivanov-Ivanov-Faradjev graph (EvD & Haemers & Koolen & Spence 2006) , Perturbations 4. Hamming graph H (3 , q ) , q ≥ 36 (Bang &EvD & Koolen 2008) . End Villanova, June 2, 2014 Edwin van Dam - Tilburg University – 8 / 20
Recommend
More recommend
