In this paper, we first review local counting methods for perimeter estimation of piecewise smooth binary figures on square, hexagonal, and triangular grids. We verify that better perimeter estimates, using local counting algorithms, can be obtained using hexagonal or triangular grids. We then compare surface area estimates using local counting techniques for binary three-dimensional volumes un...

Let Γ be a triangle-free distance-regular graph with diameter d ≥ 3, valency k ≥ 3 and intersection number a2 6= 0. Assume Γ has an eigenvalue with multiplicity k. We show that Γ is 1-homogeneous in the sense of Nomura when d = 3 or when d ≥ 4 and a4 = 0. In the latter case we prove that Γ is an antipodal cover of a strongly regular graph, which means that it has diameter 4 or 5. For d = 5 the ...

Equidistributive eigenvectors of the adjacency matrix of a symmetric graph have equal weight on equivalent vertices. Real and complex coefficients are in general insufficient to achieve this property for all components of a degenerate manifold of an arbitrary graph. Quaternionic vector coefficients are shown to be necessary for equidistributivity of fourand fivefold degeneracies of the dodecahe...