arXiv subjects

Sarah Allred

Publications and source records attributed to Sarah Allred.

8 recordsLinked to original sources

arXiv2026

The Deontic Gap: Large Language Models and the Modal Language of Obligation

Modal auxiliaries such as must, should, and have to mark necessity and obligation within the contexts of speaker authority and interpersonal stance. We examine whether large language models (LLMs) reproduce contemporary human patterns of deontic modal usage. Across three primary corpora, an external benchmark, two controlled replications, and a naturalistic eleven-model replication, AI-generated text consistently underuses positive deontic modals (must, should, have to, had to) relative to contemporary humans. Historical comparison with the Google Books Ngram corpus (1920-2022), used as a heuristic calibration against the published-prose record, shows that AI modal frequencies fall within the range of formal published English, whereas contemporary human modal rates in informal digital contexts often exceed twentieth-century book baselines. Phrase-level decomposition shows that the AI-human modal gap is concentrated in constructions central to interpersonal stance (should, have to, had to), while AI matches or exceeds humans on need to in instructional and question-answering contexts but not in persuasive student writing, indicating that the modal profile is genre-conditional. The findings suggest that LLM modal usage reflects the formal written resources on which these models were trained, while underusing the modal constructions through which contemporary human writers mark immediate, interpersonal obligation.

cs.CL↗

arXiv2025

Forbidding the subdivided claw as a subgraph or a minor

Let $Y$ be the subdivided claw, the $7$-vertex tree obtained from a claw $K_{1,3}$ by subdividing each edge exactly once. We characterize the graphs (finite and infinite) that do not have $Y$ as a subgraph, or, equivalently, do not have $Y$ as a minor. This work was motivated by a problem involving VCD minors. A graph $H$ is a vertex contraction-deletion minor, or VCD minor, of a graph $G$ if $H$ can be obtained from $G$ by a sequence of vertex deletions or contractions of all edges incident with a single vertex. Our result is a key step in describing $K_{1,3}$-VCD-minor-free line graphs. We also characterize graphs that forbid each subtree of $Y$. We discuss the relevance of our results for Tur\'{a}n. numbers of trees, and pathwidth and growth constants for graphs without a particular tree as a minor.

math.CO↗

arXiv2025

Enumerative Chromatic Choosability

Chromatic-choosablility is a notion of fundamental importance in list coloring. A graph is chromatic-choosable when its chromatic number is equal to its list chromatic number. In 1990, Kostochka and Sidorenko introduced the list color function of a graph $G$, denoted $P_{\ell}(G,m)$, which is the list analogue of the chromatic polynomial of $G$, $P(G,m)$. It is known that for any graph $G$ there is a positive integer $k$ such that $P_{\ell}(G,m) = P(G,m)$ whenever $m \geq k$. In this paper, we study enumerative chromatic-choosability. A graph $G$ is enumeratively chromatic-choosable when $P_{\ell}(G,m) = P(G,m)$ whenever $m \in \mathbb{N}$. We completely determine the graphs of chromatic number two that are enumeratively chromatic-choosable. We construct examples of graphs that are chromatic-choosable but fail to be enumeratively-chromatic choosable, and finally, we explore a conjecture as to whether for every graph $G$, there is a $p \in \mathbb{N}$ such that the join of $G$ and $K_p$ is enumeratively chromatic-choosable. The techniques we use to prove results are diverse and include probabilistic ideas and ideas from DP (or correspondence)-coloring.

math.CO↗

arXiv2025

Unavoidable substructures in large and infinite $2$-edge-connected graphs

In 1930, Ramsey proved that every large graph contains either a large clique or a large edgeless graph as an induced subgraph. It is well known that every large connected graph contains a long path, a large clique, or a large star as an induced subgraph. Recently Allred, Ding, and Oporowski presented the unavoidable large induced subgraphs for large and infinite $2$-connected graphs. The $2$-edge-connected (sometimes called bridgeless) graphs form an important class between connected graphs and $2$-connected graphs. In this paper we describe the unavoidable large induced subgraphs for large and infinite $2$-edge-connected graphs. Ubiquitous structures in $2$-edge-connected graphs that we call `chains of pinched super-clean ladders' play an important role in these descriptions. As consequences we obtain results on unavoidable large subgraphs, topological minors, minors, induced topological minors, induced minors, and Eulerian subgraphs in large and infinite $2$-edge-connected graphs. When appropriate we extend our results to multigraphs.

math.CO↗

arXiv2023

The Strong Spectral Property of Graphs: Graph Operations and Barbell Partitions

The utility of a matrix satisfying the Strong Spectral Property has been well established particularly in connection with the inverse eigenvalue problem for graphs. More recently the class of graphs in which all associated symmetric matrices possess the Strong Spectral Property (denoted $G^{SSP}$) were studied, and along these lines we aim to study properties of graphs that exhibit a so-called barbell partition. Such a partition is a known impediment to membership in the class $G^{SSP}$. In particular we consider the existence of barbell partitions under various standard and useful graph operations.

math.CO↗

arXiv2022

Unavoidable Induced Subgraph of Infinite 2-connected Graphs

In 1930, Ramsey proved that every infinite graph contains either an infinite clique or an infinite independent set as an induced subgraph. K\"{o}nig proved that every infinite graph contains either a ray or a vertex of infinite degree. In this paper, we establish the 2-connected analog of these results.

math.CO↗

arXiv2022

A combinatorial bound on the number of distinct eigenvalues of a graph

The smallest possible number of distinct eigenvalues of a graph $G$, denoted by $q(G)$, has a combinatorial bound in terms of unique shortest paths in the graph. In particular, $q(G)$ is bounded below by $k$, where $k$ is the number of vertices of a unique shortest path joining any pair of vertices in $G$. Thus, if $n$ is the number of vertices of $G$, then $n-q(G)$ is bounded above by the size of the complement (with respect to the vertex set of $G$) of the vertex set of the longest unique shortest path joining any pair of vertices of $G$. The purpose of this paper is to commence the study of the minor-monotone floor of $n-k$, which is the minimum of $n-k$ among all graphs of which $G$ is a minor. Accordingly, we prove some results about this minor-monotone floor.

math.CO↗

arXiv2020

Unavoidable Induced Subgraphs of Large 2-Connected Graphs

Ramsey proved that for every positive integer $n$, every sufficiently large graph contains an induced $K_n$ or $\overline{K}_n$. Among the many extensions of Ramsey's Theorem there is an analogue for connected graphs: for every positive integer $n$, every sufficiently large connected graph contains an induced $K_n$, $K_{1,n}$, or $P_n$. In this paper, we establish an analogue for 2-connected graphs. In particular, we prove that for every integer exceeding two, every sufficiently large 2-connected graph contains one of the following as an induced subgraph: $K_n$, a subdivision of $K_{2,n}$, a subdivision of $K_{2,n}$ with an edge between the two vertices of degree $n$, and a well-defined structure similar to a ladder.

math.CO↗