We investigate the phase behavior of a system of hard equilateral and right-angled triangles in two dimensions using Monte Carlo simulations. Hard equilateral triangles undergo a continuous isotropic-triatic liquid crystal phase transition at packing fraction ϕ = 0.7. Similarly, hard right-angled isosceles triangles exhibit a first-order phase transition from an isotropic fluid phase to a rhomb...

The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph $G 1 K_2$ to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.

Triangles are an important building block and distinguishing feature of real-world networks, but their structure is still poorly understood. Despite numerous reports on the abundance of triangles, there is very little information on what these triangles look like. We initiate the study of degreelabeled triangles — specifically, degree homogeneity versus heterogeneity in triangles. This yields n...

The rendering process reduces the 3D geometry to a set of 2D triangles in the image plane. The projection from 3D is standard. The projected triangles are clipped and sorted so that the final collection of triangles in a pixel do not overlap each other. So at frame f , the image plane holds a collection of pixels Pk, k = 1, . . .M , each of which contains a set of 2D nonoverlapping triangles {T...

Grigore Ro su and Joseph Goguen Department of Computer Science & Engineering University of California at San Diego

A triple of vertices in a graph is a frustrated triangle if it induces an odd number of edges. We study the set Fn ⊂ [0, n 3  ] of possible number of frustrated triangles f (G) in a graph G on n vertices. We prove that about two thirds of the numbers in [0, n3/2] cannot appear in Fn, and we characterise the graphs G with f (G) ∈ [0, n3/2]. More precisely, our main result is that, for each n ≥...

In [15], Shiu proved that if a and q are arbitrary coprime integers, then there exist arbitrarily long strings of consecutive primes which are all congruent to a modulo q. We generalize Shiu’s theorem to imaginary quadratic fields, where we prove the existence of “bubbles” containing arbitrarily many primes which are all, up to units, congruent to a modulo q.