arXiv subjects

Randy Davila

Publications and source records attributed to Randy Davila.

At least 19 recordsLinked to original sources

arXiv2026

Geometry-Only CSL/DP Ratios and the Nonuniqueness of Decoherence Kernels

We study idealized levitated protocols that create spatial superpositions of massive test particles. For each protocol, we compare the dimensionless contrast-loss exponent of mass-proportional continuous spontaneous localization (CSL) with the Di\'osi--Penrose (DP) self-energy exponent $E_G\tau/\hbar$. We first prove that the point-particle CSL separation kernel has an exact random-unitary realization: Gaussian momentum kicks arriving at Poisson-distributed times produce the same unconditional decay of spatial coherence, although a pure state conditioned on the complete kick record remains pure. The separation kernel alone therefore specifies an operational decoherence law, not the occurrence of objective collapse. We then show that the ratio of the CSL and DP exponents is independent of particle mass and interrogation time. In the point-particle model it depends only on branch separation and an effective distance; for the standard GRW reference parameters, its resolved-superposition crossover is $x_*\approx1.91\,\mathrm{nm}$. For rigid spherical bodies with an arbitrary normalized radial mass profile, total mass, overall density scale, and interrogation time again cancel, leaving a dimensionless geometry factor. The results distinguish three requirements for a decisive experiment: detectable absolute effects, a controlled comparison of CSL and DP scales, and observables capable of discriminating physically different dynamics that share the same ensemble decoherence kernel.

quant-ph↗

arXiv2026

Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure

Let $G_n$ be the graph on $\{2,\ldots,n\}$ in which two integers are adjacent when they have a common divisor greater than one. We determine the complete asymptotic expansion of its Havel--Hakimi residue $\R(G_n)$, confirming a leading-constant prediction of Staton recorded in Fajtlowicz's \emph{Written on the Wall}. If $A=\sum_{k=2}^{\infty}(\log k)/(k^2(k-1))$, then the first two terms are $\R(G_n)=(\zeta(2)-1)n/\log n+(\zeta(2)-1-A)n/\log^2n +O(n/\log^3n)$. More precisely, the difference between $\R(G_n)$ and the prime-vertex contribution to the Caro--Wei sum is $O_\beta(n\exp\{-(\log n)^\beta\})$ for every fixed $0<\beta<1/2$; this estimate yields every coefficient in the expansion. The upper bound follows from a degree-preserving realization in which almost all relevant prime vertices are partitioned into cliques. We also prove that the unlabeled graph determines $\pi(n)$ through its simplicial true-twin classes. Stable inverses for weighted sums of the resulting degree-class counts give criteria equivalent to the Riemann hypothesis, including one involving only the Caro--Wei sum. An exact local-defect identity additionally reduces the conjectured sharp $+2$ residue bound to explicit prefix estimates.

math.CO↗

arXiv2026

The Zombie Damage Number of a Graph

In the damage variant of Cops and Robber, the \emph{damage number} $\dmg(G)$ is the number of distinct vertices damaged by the robber under optimal play, with one cop minimizing and the robber maximizing this number. We introduce the \emph{zombie damage number} $\zdmg(G)$, obtained by requiring the pursuer to move at every turn along a shortest path toward the survivor. The parameter therefore measures the cost of geodesic pursuit when the objective is containment rather than capture alone. We prove that $\dmg(G)\leq\zdmg(G)$ and characterize the graphs with $\zdmg(G)=0$. Geodesic pursuit incurs no additional damage on trees, and we determine the parameter exactly for paths, cycles, split graphs, and complete multipartite graphs. In particular, $\zdmg(C_n)=n$ for $n\geq5$, while $\zdmg(K_{n_1,\ldots,n_k})=n_1+n_2-2$ when $n_i\geq2$ for every $i$. We also develop a nonbacktracking trace argument for sparse graphs. If $G$ is connected, $\delta(G)\geq2$, and $g(G)\geq5$, then every vertex is damaged, and hence $\zdmg(G)=n(G)$. The same argument gives the sharp bound $\zdmg(G)\geq g(G)$ for every connected graph of finite girth at least five. It follows that fully subdividing each edge of a connected graph of minimum degree at least two produces a graph with maximum zombie damage. These results yield an unbounded separation from the ordinary damage number. Most of the questions leading to these results, as well as the open conjectures, arose through \textsc{Theo-Conjecture}, an advisor-supervised discovery loop combining automated conjecturing, exact computation, language-model-assisted exploration, and human mathematical judgment.

math.CO↗

arXiv2025

Graph Neural Networks vs Convolutional Neural Networks for Graph Domination Number Prediction

We investigate machine learning approaches to approximating the \emph{domination number} of graphs, the minimum size of a dominating set. Exact computation of this parameter is NP-hard, restricting classical methods to small instances. We compare two neural paradigms: Convolutional Neural Networks (CNNs), which operate on adjacency matrix representations, and Graph Neural Networks (GNNs), which learn directly from graph structure through message passing. Across 2,000 random graphs with up to 64 vertices, GNNs achieve markedly higher accuracy ($R^2=0.987$, MAE $=0.372$) than CNNs ($R^2=0.955$, MAE $=0.500$). Both models offer substantial speedups over exact solvers, with GNNs delivering more than $200\times$ acceleration while retaining near-perfect fidelity. Our results position GNNs as a practical surrogate for combinatorial graph invariants, with implications for scalable graph optimization and mathematical discovery.

cs.LG↗

arXiv2025

In Reverie Together: Ten Years of Mathematical Discovery with a Machine Collaborator

We present four open conjectures in graph theory generated by the automated conjecturing system \texttt{TxGraffiti}. Each conjecture is concise, grounded in natural graph invariants, and empirically validated across hundreds of graphs. Despite extensive effort, these statements remain unresolved--defying both proof and counterexample. They are not only mathematical challenges but creative expressions--born of symbolic pattern recognition and mathematician-defined heuristics, refined through years of human dialogue, and now offered back to the community as collaborative artifacts. These conjectures invite not only formal proof, but also reflection on how machines can evoke wonder, spark curiosity, and contribute to the raw material of discovery. By highlighting these problems, we aim to inspire both human mathematicians and AI systems to engage with them--not only to solve them, but to reflect on what it means when machines participate meaningfully in the creative process of mathematical thought.

cs.DM↗

arXiv2025

Connected forcing density and related problems

A connected forcing set of a graph is a zero forcing set that induces a connected subgraph. In this paper, we introduce and study CF-dense graphs -- graphs in which every vertex belongs to some minimum connected forcing set. We identify several CF-dense graph families and investigate the relationships between CF-density and analogous notions in zero forcing and total forcing. We also characterize CF-dense trees and give a formula for the number of distinct connected forcing sets in trees. Finally, we analyze when CF-density is preserved under graph operations such as Cartesian products, joins, and coronas.

math.CO↗

arXiv2025

Independence, induced subgraphs, and domination in $K_{1,r}$-free graphs

Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $\alpha_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with \emph{chromatic number} at most~$k$, we prove that if $G$ is $K_{1,r}$-free, then $\alpha_{\mathcal{F}}(G) \le (r-1)k\gamma(G)$, where $\gamma(G)$ is the \emph{domination number}. When $\mathcal{F}$ is the family of empty graphs, this bound simplifies to $\alpha(G) \le 2\gamma(G)$ for $K_{1,3}$-free (claw-free) graphs, where $\alpha(G)$ is the \emph{independence number} of $G$. For $d$-regular graphs, this is further refined to the bound $\alpha(G) \le 2\left(\frac{d+1}{d+2}\right)\gamma(G)$, which is tight for $d \in \{2, 3, 4\}$. Using Ramsey theory, we extend this framework to edge-hereditary graph families, showing that for $K_{1,r}$-free graphs, we have $\alpha_{\mathcal{F}}(G) \le r(K_r, \mathcal{F^*})\gamma(G)$, where $\mathcal{F^*}$ is the set of graphs not in $\mathcal{F}$. Specializing to $K_q$-free graphs, we show $\alpha_{\mathcal{F}}(G) \le (r(K_q, K_r) - 1)\gamma(G)$. Finally, for the \emph{$k$-independence number} $\alpha_k(G)$, we prove that if $G$ is $K_{1,r}$-free with order $n$ and minimum degree $\delta \ge k+1$, \[ \alpha_k(G) \le \left( \frac{(r-1)(k+1)}{\delta - k + (r-1)(k+1)} \right) n, \] and this bound is sharp for all parameters.

math.CO↗

arXiv2024

A Characterization of Claw-Free Graphs using Zero Forcing Invariants

We prove that the \emph{standard zero forcing number} $Z(G)$ and the \emph{positive semidefinite zero forcing number} $Z_+(G)$ are equal for all claw-free graphs $G$. This result resolves a conjecture proposed by the computer program \emph{TxGraffiti} and highlights a connection between these graph invariants in claw-free structures. As a corollary, we show that a graph $G$ is claw-free if and only if every induced subgraph $H \subseteq G$ satisfies $Z(H) = Z_+(H)$.

math.CO↗

arXiv2024

The \emph{Optimist}: Towards Fully Automated Graph Theory Research

This paper introduces the \emph{Optimist}, an autonomous system developed to advance automated conjecture generation in graph theory. Leveraging mixed-integer programming (MIP) and heuristic methods, the \emph{Optimist} generates conjectures that both rediscover established theorems and propose novel inequalities. Through a combination of memory-based computation and agent-like adaptability, the \emph{Optimist} iteratively refines its conjectures by integrating new data, enabling a feedback process with minimal human (\emph{or machine}) intervention. Initial experiments reveal the \emph{Optimist}'s potential to uncover foundational results in graph theory, as well as to produce conjectures of interest for future exploration. This work also outlines the \emph{Optimist}'s evolving integration with a counterpart agent, the \emph{Pessimist} (a human \emph{or machine} agent), to establish a dueling system that will drive fully automated graph theory research.

cs.AI↗

arXiv2024

Automated conjecturing with \emph{TxGraffiti}

\emph{TxGraffiti} is a data-driven, heuristic-based computer program developed to automate the process of generating conjectures across various mathematical domains. Since its creation in 2017, \emph{TxGraffiti} has contributed to numerous mathematical publications, particularly in graph theory. In this paper, we present the design and core principles of \emph{TxGraffiti}, including its roots in the original \emph{Graffiti} program, which pioneered the automation of mathematical conjecturing. We describe the data collection process, the generation of plausible conjectures, and methods such as the \emph{Dalmatian} heuristic for filtering out redundant or transitive conjectures. Additionally, we highlight its contributions to the mathematical literature and introduce a new web-based interface that allows users to explore conjectures interactively. While we focus on graph theory, the techniques demonstrated extend to other areas of mathematics.

math.CO↗

arXiv2024

Comparing the $p$-independence number of regular graphs to the $q$-independence number of their line graphs

Let $G$ be a simple graph and let $L(G)$ denote the \emph{line graph} of $G$. A \emph{$p$-independent} set in $G$ is a set of vertices $S \subseteq V(G)$ such that the subgraph induced by $S$ has maximum degree at most $p$. The \emph{$p$-independence number} of $G$, denoted by $\alpha_p(G)$, is the cardinality of a maximum $p$-independent set in $G$. In this paper, and motivated by the recent result that independence number is at most matching number for regular graphs~\cite{CaDaPe2020}, we investigate which values of the non-negative integers $p$, $q$, and $r$ have the property that $\alpha_p(G) \leq \alpha_q(L(G))$ for all r-regular graphs. Triples $(p, q, r)$ having this property are called \emph{valid $\alpha$-triples}. Among the results we prove are: \begin{itemize} \item $(p, q, r)$ is valid $\alpha$-triple for $p \geq 0$, $q \geq 3$ , and $r\geq 2$. \item $(p, q, r)$ is valid $\alpha$-triple for $p \leq q < 3$ and $r\geq 2$. \item $(p, q, r)$ is valid $\alpha$-triple for $p \geq 0$, $q = 2$, and $r$ even. \item $(p, q, r)$ is valid $\alpha$-triple for $p \geq 0$, $q = 2$, and $r$ odd with $r = \max \Big \{ 3, \frac{17(p+1)}{16}\Big \}$. \end{itemize} We also show a close relation between undetermined possible valid $\alpha$-triples, the Linear Aboricity Conjecture, and the Path-Cover Conjecture.

math.CO↗

arXiv2024

Estimating the stability number of a random graph using convolutional neural networks

Graph combinatorial optimization problems are widely applicable and notoriously difficult to compute; for example, consider the traveling salesman or facility location problems. In this paper, we explore the feasibility of using convolutional neural networks (CNNs) on graph images to predict the cardinality of combinatorial properties of random graphs and networks. Specifically, we use image representations of modified adjacency matrices of random graphs as training samples for a CNN model to predict the stability number of random graphs; where the stability number is the cardinality of a maximum set of vertices in a graph that contains no pairwise adjacency between vertices. The model and results presented in this study suggest potential for applying deep learning in combinatorial optimization problems previously not considered by simple deep learning techniques.

cs.LG↗

arXiv2024

Artificial intelligence and machine learning generated conjectures with TxGraffiti

\emph{TxGraffiti} is a machine learning and heuristic based artificial intelligence designed to automate the task of conjecturing in mathematics. Since its inception, TxGraffiti has generated many surprising conjectures leading to publication in respectable mathematical journals. In this paper we outline the machine learning and heuristic techniques implemented by TxGraffiti. We also recall its contributions to the mathematical literature and announce a new online version of the program available for anyone curious to explore conjectures in graph theory.

cs.AI↗

arXiv2024

Lower bounds for the total (distance) $k$-domination number of a graph

For $k \geq 1$ and a graph $G$ without isolated vertices, a \emph{total (distance) $k$-dominating set} of $G$ is a set of vertices $S \subseteq V(G)$ such that every vertex in $G$ is within distance $k$ to some vertex of $S$ other than itself. The \emph{total (distance) $k$-domination number} of $G$ is the minimum cardinality of a total $k$-dominating set in $G$, and is denoted by $\gamma_{k}^t(G)$. When $k=1$, the total $k$-domination number reduces to the \emph{total domination number}, written $\gamma_t(G)$; that is, $\gamma_t(G) = \gamma_{1}^t(G)$. This paper shows that several known lower bounds on the total domination number generalize nicely to lower bounds on total (distance) $k$-domination.

math.CO↗

arXiv2023

Advancements in Research Mathematics through AI: A Framework for Conjecturing

In the words of the esteemed mathematician Paul Erd\"os, the mathematician's task is to \emph{prove and conjecture}. These two processes form the bedrock of all mathematical endeavours, and in the recent years, the mathematical community has increasingly sought the assistance of computers to bolster these tasks. This paper is a testament to that pursuit; it presents a robust framework enabling a computer to automatically generate conjectures - particularly those conjectures that mathematicians might deem substantial and elegant. More specifically, we outline our framework and provide evidence in the mathematical literature demonstrating its use in generating publishable research and surprising mathematics. We suspect our simple description of computer-assisted mathematical conjecturing will catalyze further research into this area and encourage the development of more advanced techniques than the ones presented herein.

math.CO↗

arXiv2022

Computer assisted discovery: Zero forcing vs vertex cover

In this paper, we showcase the process of using an automated conjecturing program called \emph{TxGraffiti} written and maintained by the second author. We begin by proving a conjecture formulated by \emph{TxGraffiti} that for a claw-free graph $G$, the vertex cover number $β(G)$ is greater than or equal to the zero forcing number $Z(G)$. Our proof of this result is constructive, and yields a polynomial time algorithm to find a zero forcing set with cardinality $β(G)$. We also use the output of \emph{TxGraffiti} to construct several infinite families of claw-free graphs for which $Z(G)=β(G)$. Additionally, inspired by the aforementioned conjecture of \emph{TxGraffiti}, we also prove a more general relation between the zero forcing number and the vertex cover number for any connected graph with maximum degree $Δ\ge 3$, namely that $Z(G)\leq (Δ-2)β(G)$+1.

math.CO↗

arXiv2021

Conjecture of TxGraffiti: Independence, domination, and matchings

TxGraffiti is an automated conjecturing program that produces graph theoretic conjectures in the form of conjectured inequalities. This program written and maintained by the second author since 2017 was inspired by the successes of previous automated conjecturing programs including Fajtlowicz's GRAFFITI and DeLaViña's GRAFFITI.pc. In this paper we prove and generalize several conjectures generated by TxGraffiti when it was prompted to conjecture on the \emph{independence number}, the \emph{domination number}, and the \emph{matching number} (and generalizations of each of these graph invariants). Moreover, in several instances we also show the proposed inequalities relating these graph invariants are sharp.

math.CO↗

arXiv2020

On a Vizing-type integer domination conjecture

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $γ(G)$. For any integer $k\ge 1$, a function $f : V (G) \rightarrow \{0, 1, . . ., k\}$ is called a \emph{$\{k\}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a $\{k\}$-dominating function is the sum of its values over all the vertices. The $\{k\}$-domination number of $G$, $γ_{\{k\}}(G)$, is defined to be the minimum weight taken over all $\{k\}$-domination functions. Brešar, Henning, and Klavžar (On integer domination in graphs and Vizing-like problems. \emph{Taiwanese J. Math.} {10(5)} (2006) pp. 1317--1328) asked whether there exists an integer $k\ge 2$ so that $γ_{\{k\}}(G\square H)\ge γ(G)γ(H)$. In this note we use the Roman $\{2\}$-domination number, $γ_{R2}$ of Chellali, Haynes, Hedetniemi, and McRae, (Roman $\{2\}$-domination. \emph{Discrete Applied Mathematics} {204} (2016) pp. 22-28.) to prove that if $G$ is a claw-free graph and $H$ is an arbitrary graph, then $γ_{\{2\}}(G\square H)\ge γ_{R2}(G\square H)\ge γ(G)γ(H)$, which also implies the conjecture for all $k\ge 2$.

math.CO↗