160 citations · 199 across the 25 of their papers we have counts for
15 papers · 1 filter
Strongly regular graph, strongly polynomial sequences, and two-level polynomials
Tristram Bogart, Santiago Jiménez-Salazar
We study the behavior of the characteristic and chromatic polynomials of certain families of graphs and their relation to homomorphism counts, focusing on polynomial dependence of…
Optimal Small Set Expanders and Their Codes
Tristram Bogart, Marcelo Fiori, Pedro Raigorodsky +1
A left-regular bipartite graph of degree is called a -small-set-expander if every subset of left vertices of size at most has at least neighbors. Such…
A differential characterization of volume polynomials of permutohedra
Tristram Bogart, Federico Castillo, Damián de la Fuente +1
We study a graded vector space of polynomials associated to a square matrix, defined by a finite difference condition along the rows. We show this space coincides with one defined…
Counting with two-level polynomials
Tristram Bogart, Kevin Woods
We examine combinatorial counting functions with two parameters, and . For fixed , these functions are (quasi-)polynomial in . As varies, the degree of this polyno…
An Ehrhart theoretic approach to generalized Golomb rulers
Tristram Bogart, Daniel Felipe Cuéllar
A Golomb ruler is a sequence of integers whose pairwise differences, or equivalently pairwise sums, are all distinct. This definition has been generalized in various ways to allow…
Bounds on Determinantal Complexity of Two Types of Generalized Permanents
Tristram Bogart, Juan Andrés Valero
We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permane…
Complex psd-minimal polytopes in dimensions two and three
Tristram Bogart, João Gouveia, Juan Camilo Torres
The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real se…
A Plethora of Polynomials: A Toolbox for Counting Problems
Tristram Bogart, Kevin Woods
A wide variety of problems in combinatorics and discrete optimization depend on counting the set of integer points in a polytope, or in some more general object constructed via…
An Algebraic Approach to Projective Uniqueness with an Application to Order Polytopes
Tristram Bogart, João Gouveia, Juan Camilo Torres
A combinatorial polytope is said to be projectively unique if it has a single realization up to projective transformations. Projective uniqueness is a geometrically compelling…
Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic
Tristram Bogart, John Goodrick, Kevin Woods
Let be polynomial functions of . For fixed , let be the numerical semigroup generated by $f_1(n),\ldots,f_k(n)…
A parametric version of LLL and some consequences: parametric shortest and closest vector problems
Tristram Bogart, John Goodrick, Kevin Woods
Given a parametric lattice with a basis given by polynomials in Z[t], we give an algorithm to construct an LLL-reduced basis whose elements are eventually quasi-polynomial in t: th…
Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior
Tristram Bogart, John Goodrick, Kevin Woods
Parametric Presburger arithmetic concerns families of sets S_t in Z^d, for t in N, that are defined using addition, inequalities, constants in Z, Boolean operations, multiplication…
Superlinear subset partition graphs with dimension reduction, strong adjacency, and endpoint count
Tristram C. Bogart, Edward D. Kim
We construct a sequence of subset partition graphs satisfying the dimension reduction, adjacency, strong adjacency, and endpoint count properties whose diameter has a superlinear a…
Hom-polytopes
Tristram Bogart, Mark Contois, Joseph Gubeladze
We study the polytopes of affine maps between two polytopes -- the hom-polytopes. The hom-polytope functor has a left adjoint -- tensor product polytopes. The analogy with the cate…
Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs
Matthias Beck, Tristram Bogart, Tu Pham
A \emph{Golomb ruler} is a sequence of distinct integers (the \emph{markings} of the ruler) whose pairwise differences are distinct. Golomb rulers can be traced back to additive nu…