Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4m for every positive integer m. If m > 1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (4m, 2m +m, m +m, m +m). Strongly regular graphs with these parameters have been called max energy graphs, because they have maximal energy (as defined by ...

Let Γ denote a near polygon distance-regular graph with diameter d ≥ 3, valency k and intersection numbers a1 > 0, c2 > 1. Let θ1 denote the second largest eigenvalue of Γ. We show θ1 ≤ k − a1 − c2 c2 − 1 . We show the following (i)–(iii) are equivalent. (i) Equality is attained above; (ii) Γ is Q-polynomial with respect to θ1; (iii) Γ is a dual polar graph or a Hamming graph.

Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Čech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen...

We prove that any completely regular code with minimum eigenvalue in geometric graph Γ corresponds to a the clique of Γ. Studying interrelation these codes, complete characterization codes Johnson graphs J(n,w) covering radius w−1 and strength 1 is obtained. In particular this result finishes J(n,3). also classify J(n,4) only one case for eigenvalues left open.

We introduce the notion of L2-rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that L2rigidity passes to normalizers and is satisfied by nonamenable II1 factors which are non-prime, have property Γ, or are weakly rigid. As a consequence we obtain that ...

We introduce the notion of L2-rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that L2-rigidity passes to normalizers and is satisfied by nonamenable II1 factors which are non-prime, have property Γ , or are weakly rigid. As a consequence we obtain tha...

The Coxeter simplex with symbol is a compact hyperbolic 4-simplex and the related Coxeter group Γ is a discrete subgroup of Isom(H 4). The Coxeter simplex with symbol is a spherical 3-simplex, and the related Coxeter group G is the group of symmetries of the regular 120-cell. Using the geometry of the regular 120-cell, Davis [3] constructed an epimorphism Γ → G whose kernel K was torsion-free, ...

Let Γ be a distance-regular graph with diameter d. For vertices x and y of Γ at distance i, 1 ≤ i ≤ d, we define the sets Ci(x, y) = Γi−1(x) ∩ Γ(y), Ai(x, y) = Γi(x) ∩ Γ(y) and Bi(x, y) = Γi+1(x) ∩ Γ(y). Then we say Γ has the CABj property, if the partition CABi(x, y) = {Ci(x, y), Ai(x, y), Bi(x, y)} of the local graph of y is equitable for each pair of vertices x and y of Γ at distance i ≤ j. ...

The weak regular coherence (WRC) property of a finitely generated group Γ was introduced by G. Carlsson and this author to play the role of a weakening of Waldhausen’s regular coherence as part of computation of the integral K-theoretic assembly map. This property is a coarse invariant of the word metric. It was shown that groups with finite asymptotic dimension (FAD) have this property. A new ...

We characterize the macroscopic effective mechanical behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using Γ-convergence.