Garrett Kepler | Searcharxiv

arXiv2026

Some Spectral Properties of the $B_{\alpha}$ Matrix of a Graph

Recently, Samanta et al. introduced a unifying perspective to common graph-associated matrices through the matrix $B_{\alpha}(G)=\alpha A(G)+(1-\alpha)L(G)$ where $\alpha\in [0,1]$. We resolve an open problem regarding the positive semi-definiteness of $B_{\alpha}$ and similar matrices. We also explore eigenvector properties of $B_{\alpha}$ through edge and vertex removal. As a consequence, we provide a basis for results on the common graph-associated matrices.

math.CO↗

arXiv2025

Algorithmic Accountability in Small Data: Sample-Size-Induced Bias Within Classification Metrics

Evaluating machine learning models is crucial not only for determining their technical accuracy but also for assessing their potential societal implications. While the potential for low-sample-size bias in algorithms is well known, we demonstrate the significance of sample-size bias induced by combinatorics in classification metrics. This revelation challenges the efficacy of these metrics in assessing bias with high resolution, especially when comparing groups of disparate sizes, which frequently arise in social applications. We provide analyses of the bias that appears in several commonly applied metrics and propose a model-agnostic assessment and correction technique. Additionally, we analyze counts of undefined cases in metric calculations, which can lead to misleading evaluations if improperly handled. This work illuminates the previously unrecognized challenge of combinatorics and probability in standard evaluation practices and thereby advances approaches for performing fair and trustworthy classification methods.

cs.LG↗