The girth of a thin distance-regular graph

Let Γ be a distance-regular graph of diameterd≥3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE _i ^* (x)W≤1 for 0≤i≤d, whereE _i ^* (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d≥3 and valency k≥3. Then the following are equivalent:

1. Γis a regular generalized quadrangle.
2. Γis thin and c ₃=1.

Corollary. Let Γ=(X,R) be a thin distance-regular graph with diameter d≥3 and valency k≥3. Then Γ has girth 3, 4, 6, or 8. Then girth of Γ is 8 exactly when Γ is a regular generalized quadrangle.     

Erratum: Renormalization group study of the A+B→Φ diffusion-limited reaction

A recent argument of Oerding shows that our calculation of the quantity , which determines the amplitude of the asymptotic decay of the particle density in 2<d<4, was in error. Instead it is simply given by =n ₀, the initial density, for uncorrelated initial conditions.     

Three-Dimensional Hückel Theory for closo-Carboranes

We have recently developed a 3-dimensional Hückel method for cluster compounds. The method uses a set of approximations for Coulomb, resonance, and overlap integrals very similar to those employed in the familiar 2-dimensional Hückel theory for the pi electrons of planar conjugated hydrocarbons. The method can be adapted to heteroatomic clusters by introducing heteroatomic Coulomb integrals, alpha(Y) = alpha(X) + hbeta, whereh is a parameter for heteroatom Y. In this paper, we use the 3-dimensional Hückel method to study the properties of the closo-carboranes, C(2)B(n)()(-)(2)H(n)(). We calibrate the method by choosing a value of the heteroatomic parameter h that distinguishes positional isomers by energy and gives them relative energies in rough agreement with those established by observation and ab initio calculations. We obtain modest improvement in matching ab initio relative energies of isomers by means of a three-parameter, first-order perturbation treatment. We use the calibrated method to evaluate various mechanisms proposed for the isomerizations of C(2)B(4)H(6), C(2)B(5)H(7), and C(2)B(6)H(8), all of which have been observed to undergo intramolecular isomerizations. Rearrangements of C(2)B(6)H(8) have been satisfactorily explained by a single-DSD (diamond-square-diamond) process. Those for C(2)B(5)H(7) require at least two DSD processes, concerted, consecutive, or overlapping. Several different mechanisms have been proposed for the rearrangement of C(2)B(4)H(6). In evaluating intermediate and transition state structures, the 3-dimensional Hückel method gives higher energies to those structures with a larger number of nontriangular faces, a plausible conclusion except that occasionally it is wrong. In comparison with ab initio results, the 3-dimensional Hückel method fails to give low energies for classical structures.