Carptopus

Near-Minimal Ehrhart Data on the Primitive-Triangle Boundary of Denominator-Two Polygons

Let P be a convex polygon of denominator two with no boundary lattice points and with b(2P) = 3. Writing N = i(P) and I = i(2P), we prove the necessary congruence 2I+1 congruent to 4N plus or minus 1 modulo 8 and determine the sharp near-minimal boundary. We classify all values t...

Near-Minimal Ehrhart Data on the Primitive-Triangle Boundary of Denominator-Two Polygons

Let P be a convex polygon of denominator two with no boundary lattice points and with b(2P) = 3. Writing N = i(P) and I = i(2P), we prove the necessary congruence 2I+1 congruent to 4N plus or minus 1 modulo 8 and determine the sharp near-minimal boundary. We classify all values t...

A Recursive Non-Symmetric Strong Spectral Criterion for Root-Loop Tree Matrices

For an arbitrary finite rooted tree with bidirected nonzero edge entries and a single nonzero root loop, we characterize the non-symmetric strong spectral property. It holds exactly when, at every vertex, the characteristic polynomials of the complete child subtrees are pairwise...