arXiv subjects

Bob Lutz

Publications and source records attributed to Bob Lutz.

12 recordsLinked to original sources

arXiv2026

Quantum-Based Resilient Routing in Networks: Minimizing Latency Under Dual-Link Failures

Network optimization problems represent large combinatorial search spaces that grow exponentially with network size, making them computationally intensive to solve. This paper addresses the latency-resilient Layer 3 routing optimization problem in telecommunications networks with predefined Layer 1 optical links. We formulate this problem as a graph-based optimization problem with the objective of minimizing latency, creating vertex-disjoint paths from each site to the internet backbone, and maximizing overall resiliency by limiting the impact of dual-link failures. By framing the problem as finding two disjoint shortest paths, coupled together with a resiliency component to the objective function, we establish a single formulation to produce optimal path design. The mathematical formulation was adapted to solve the problem using quantum approximate optimization algorithm (QAOA) executed over both quantum simulator and quantum hardware. QAOA was tested on a toy graph topology with 5 vertices and 7 edges and considering two limiting scenarios respectively representing independent (uncorrelated) link failures and highly correlated failure for one pair of edges. Both explored scenarios produced the optimal network design-corresponding to the valid solution with highest frequency of occurrence and minimum energy state, hence, validating the proposed formulation for optimizing Layer 3 routing on quantum systems of the future.

cs.ET↗

arXiv2022

Matroids arising from electrical networks

This paper introduces Dirichlet matroids, a generalization of graphic matroids arising from electrical networks. We present four main theorems. First, we exhibit a matroid quotient involving geometric duals of networks embedded in surfaces with boundary. Second, we characterize the Bergman fans of Dirichlet matroids as subfans of graphic Bergman fans. Third, we prove an interlacing result on the real zeros and poles of the trace of the response matrix. And fourth, we bound the coefficients of the precoloring polynomial of a network by the coefficients of the associated chromatic polynomial.

math.CO↗

arXiv2020

Discrete homotopy of token configurations

This paper studies graphical analogs of symmetric products and unordered configuration spaces in topology. We do so from the perspective of the discrete homotopy theory introduced by Barcelo et al. Our first result is a combinatorial version of a theorem of P. A. Smith, which says that the fundamental group of any nontrivial symmetric product of $X$ is isomorphic to $H_1(X)$. Our second result gives conditions under which the n-strand braid group of a graph is isomorphic to its discrete analog.

math.CO↗

arXiv2020

Higher discrete homotopy groups of graphs

This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if $G$ is a graph containing no 3- or 4-cycles, then the $n$th discrete homotopy group $A_n(G)$ is trivial for all $n\geq 2$. Second we exhibit for each $n\geq 1$ a natural homomorphism $ψ:A_n(G)\to \mathcal{H}_n(G)$, where $\mathcal{H}_n(G)$ is the $n$th discrete cubical singular homology group, and an infinite family of graphs $G$ for which $\mathcal{H}_n(G)$ is nontrivial and $ψ$ is surjective. It follows that for each $n\geq 1$ there are graphs $G$ for which $A_n(G)$ is nontrivial.

math.CO↗

arXiv2018

Koszulness and supersolvability for Dirichlet arrangements

We prove that the cone over a Dirichlet arrangement is supersolvable if and only if its Orlik-Solomon algebra is Koszul. This was previously shown for four other classes of arrangements. We exhibit an infinite family of cones over Dirichlet arrangements that are combinatorially distinct from these other four classes.

math.CO↗

arXiv2018

Electrical networks and hyperplane arrangements

This paper studies \emph{Dirichlet arrangements}, a generalization of graphic hyperplane arrangements arising from electrical networks and order polytopes of finite posets. We generalize descriptions of combinatorial features of graphic arrangements to Dirichlet arrangements, including characteristic polynomials and supersolvability. We apply these results to visibility sets of order polytopes and fixed-energy harmonic functions on electrical networks.

math.CO↗

arXiv2017

A supercharacter approach to Heilbronn sums

Various algebraic properties of Heilbronn's exponential sum can be deduced through the use of supercharacter theory, a novel extension of classical character theory due to Diaconis-Isaacs and Andre. This perspective yields a variety of formulas and provides a method for computing the number of solutions to Fermat-type congruences.

math.NT↗

arXiv2016

Graphical cyclic supercharacters for composite moduli

Recent work has introduced the study of graphical properties of cyclic supercharacters, functions $\mathbb{Z}/n\mathbb{Z}\to \mathbb{C}$ whose values are exponential sums with close connections to Gauss sums and Gaussian periods. Plots of these functions exhibit striking features, some of which have been previously explained when the modulus $n$ is a power of an odd prime. After reviewing this material, we initiate the graphical study of images of cyclic supercharacters in the case of composite $n$.

math.NT↗

arXiv2015

Gauss' hidden menagerie: from cyclotomy to supercharacters

Gaussian periods, when viewed appropriately, exhibit a dazzling and eclectic host of visual qualities. This brief survey reviews the historical context and summarizes our current knowledge of graphical properties of Gaussian periods.

math.NT↗

arXiv2014

The graphic nature of Gaussian periods

Recent work has shown that the study of supercharacters on abelian groups provides a natural framework within which to study certain exponential sums of interest in number theory. Our aim here is to initiate the study of Gaussian periods from this novel perspective. Among other things, our approach reveals that these classical objects display dazzling visual patterns of great complexity and remarkable subtlety.

math.NT↗

arXiv2013

Four quotient set gems

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the Monthly, despite its intense coverage of quotient sets over the years.

math.NT↗

arXiv2012

Two remarks about nilpotent operators of order two

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.

math.FA↗