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Explicit tough Ramsey graphs
 Proceedings of the International Conference on Relations, Orders and Graphs: Interaction with Computer Science 2008, Nouha Editions
"... Dedicated to our parents on the occasion of their 30th anniversary. A graph G is ttough if any induced subgraph of it with x> 1 connected components is obtained from G by deleting at least tx vertices. Chvátal conjectured that there exists an absolute constant t0 so that every t0tough graph is pan ..."
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Dedicated to our parents on the occasion of their 30th anniversary. A graph G is ttough if any induced subgraph of it with x> 1 connected components is obtained from G by deleting at least tx vertices. Chvátal conjectured that there exists an absolute constant t0 so that every t0tough graph is pancyclic. This conjecture was disproved by Bauer, van den Heuvel and Schmeichel by constructing a t0tough trianglefree graph for every real t0. For each finite field�q with q odd, we consider graphs associated to the finite Euclidean plane and the finite upper half plane over�q. These graphs have received serious attention as they have been shown to be Ramanujan (or asymptotically Ramanujan) for large q. We will show that for infinitely many q, these graphs provide further counterexamples to Chvátal’s conjecture. They also provide a good constructive lower bound for the Ramsey number R(3, k).
Families of Ramanujan graphs and quaternion algebras
, 2007
"... 2. Quaternion algebras and superspecial points 3 2.1. Elliptic curves and the quaternion algebra Bp, ∞ 3 2.2. Abelian varieties with real multiplication and superspecial points 4 ..."
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Cited by 5 (2 self)
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2. Quaternion algebras and superspecial points 3 2.1. Elliptic curves and the quaternion algebra Bp, ∞ 3 2.2. Abelian varieties with real multiplication and superspecial points 4
Some Ramanujan hypergraphs associated to GL(n, Fq
 Proc. Amer. Math. Soc
, 2001
"... Abstract. We present examples of hypergraphs constructed from homogeneous spaces of finite general linear groups. These hypergraphs are constructed using an invariant analogue of a hypervolume and their spectra are analyzed to see if they are Ramanujan in the sense of W.C. W. Li and P. Solé. 1. ..."
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Abstract. We present examples of hypergraphs constructed from homogeneous spaces of finite general linear groups. These hypergraphs are constructed using an invariant analogue of a hypervolume and their spectra are analyzed to see if they are Ramanujan in the sense of W.C. W. Li and P. Solé. 1.
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 43 (2009), Pages 295–305 Finite Euclidean graphs over �2r are nonRamanujan
"... Graphs are attached to � d 2 r where �2 r is the ring with 2r elements using an analogue of Euclidean distance. M. R. DeDeo (2003) showed that these graphs are nonRamanujan for r � 4. In this paper, we will show that finite Euclidean graphs attached to �d 2r are non Ramanujan for r � 2 except for r ..."
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Graphs are attached to � d 2 r where �2 r is the ring with 2r elements using an analogue of Euclidean distance. M. R. DeDeo (2003) showed that these graphs are nonRamanujan for r � 4. In this paper, we will show that finite Euclidean graphs attached to �d 2r are non Ramanujan for r � 2 except for r =2andd=2, 3. Together with the results in Medrano et al. (1998), this implies that finite Euclidean graphs over �pr for p prime are nonRamanujan except for the smallest cases. 1