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Abstract In this paper, we investigate the degree ofh-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of theh-polynomial for various fundamental classes of graphs such as paths, cycles, and bipartite graphs. To the best of our knowledge, this study represents the first investigation into the combinatorial interpretation of this algebraic invariant. Additionally, we characterize all connected graphs in which the sum of the Castelnuovo–Mumford regularity and the degree of theh-polynomial of an edge ideal achieve its maximum value, equal to the number of vertices in the graph. 

In response to a burgeoning pediatric mental health epidemic, recent guidelines have instructed pediatricians to regularly screen their patients for mental health disorders with consistency and standardization. Yet, gold-standard screening surveys to evaluate mental health problems in children typically rely solely on reports given by caregivers, who tend to unintentionally under-report, and in some cases over-report, child symptomology. Digital phenotype screening tools (DPSTs), currently being developed in research settings, may help overcome reporting bias by providing objective measures of physiology and behavior to supplement child mental health screening. Prior to their implementation in pediatric practice, however, the ethical dimensions of DPSTs should be explored. Herein, we consider some promises and challenges of DPSTs under three broad categories: accuracy and bias, privacy, and accessibility and implementation. We find that DPSTs have demonstrated accuracy, may eliminate concerns regarding under- and over-reporting, and may be more accessible than gold-standard surveys. However, we also find that if DPSTs are not responsibly developed and deployed, they may be biased, raise privacy concerns, and be cost-prohibitive. To counteract these potential shortcomings, we identify ways to support the responsible and ethical development of DPSTs for clinical practice to improve mental health screening in children.