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1. Spectra of products of digraphs

   https://doi.org/10.13001/ela.2020.5243  

   Catral, Minerva; Ciardo, Lorenzo; Hogben, Leslie; Reinhart, Carolyn (January 2020, The Electronic Journal of Linear Algebra)

   null (Ed.)

   A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the new distance matrix, with natural extensions to the distance Laplacian and distance signless Laplacian, in addition to the new adjacency matrix, with natural extensions to the Laplacian and signless Laplacian. Various sums of Kronecker products of nonnegative matrices are introduced to model the Cartesian and lexicographic products of digraphs. The Jordan canonical form is applied extensively to the analysis of spectra and eigenvectors. The analysis shows that Cartesian products provide a method for building infinite families of transmission regular digraphs with few distinct distance eigenvalues. 

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2. Stability of the centers of group algebras of $$GL_n(q)$$

   https://doi.org/https://doi.org/10.1016/j.aim.2019.04.026  

   Wan, Jinkui; Wang, Weiqiang (April 2019, Advances in mathematics)

   The center $$Z_n(q)$$ of the integral group algebra of the general linear group $$GL_n(q)$$ over a finite field admits a filtration with respect to the reflection length. We show that the structure constants of the associated graded algebras $$\mathscr{G}_n(q)$$ are independent of $$n$$, and this stability leads to a universal stable center with positive integer structure constants which governs the algebras $$\mathscr{G}_n(q)$$ for all $$n$$. Various structure constants of the stable center are computed and several conjectures are formulated. Analogous stability properties for symmetric groups and wreath products were established earlier by Farahat-Higman and the second author. 

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3. Management of point‐of‐use and point‐of‐entry for regulatory compliance: Survey of state administrators

   https://doi.org/10.1002/aws2.1335  

   Alfredo, Katherine; Wilson, Madelyn; Roberson, Alan (March 2023, AWWA Water Science)

   Abstract The USEPA (United States Environmental Protection Agency) Lead and Copper Rule Revisions allow the use of distributed treatment approaches such as point‐of‐use (POU) and point‐of‐entry (POE) treatment for systems with 10,000 connections or less as a compliance strategy. However, this poses an opportunity for the USEPA to reevaluate system size recommendations for distributed treatment. The current research uses online surveys and semi‐structured interviews (SSIs) to highlight the general sentiment of state regulators managing POU/POE devices and inquiries. Analysis of the 43 survey responses and 13 SSIs revealed that most state regulators described systems of approximately 30–50 connections as the most successful. Resident cooperation, operation and maintenance, monitoring, and the actual implementation of distributed treatment approaches were repeatedly listed as the greatest concerns. As the use of distributed treatment continues to expand, the water sector must devote research efforts to quantitatively determining the drivers of success as well as highlighting clear indicators of potential failure. 

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4. Separation of particles of different surface energies through control of humidity

   Sansao, B.; Kellar, J.; Cross, W.; Romkes, A. (October 2020, Minerals engineering)

   null (Ed.)

   Many of the methods to classify and concentrate minerals and the subsequent extraction of metals takes place in water-based environments (aqueous solutions). Sustainable processing through the reduction of water consumption will become a key factor to make mining operations viable in the long term. In humid environments, capillary condensation of water can occur between the particle and substrate. The objective herein is to identify separation windows in which control of relative air humidity (RH) yields different substrate adhesion for hydrophilic and hydrophobic particles of different values of interfacial energy. Plasma cleaned glass beads, and trichloro(octadecyl)silane (TCOD) treated beads were poured on a plasma cleaned glass disk and an impact caused the detachment of particles. Impact tests performed under a range of RH showed that separation of plasma cleaned and TCOD treated particles can be achieved in 80% of the tests at humidity levels between 45% and 55%. The recovery of plasma cleaned particles was five times greater than TCOD treated particles at humidity levels between 50% and 55%. 

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5. Asymptotics of Closeness Centralities of Graphs

   https://doi.org/10.3390/math13233812  

   Frias, Santiago; Galindo_Silva, Adriana; Romero, Bryan; Narayan, Darren A (December 2025, Mathematics)

   Given a connected graph G with n vertices, the distance between two vertices is the number of edges in a shortest path connecting them. The sum of the distances in a graph G from a vertex v to all other vertices is denoted by SDG(v). The closeness centrality of a vertex in a graph was defined by Bavelas to be CC(v)=n−1SDG(v) and the closeness centrality of G is CC(G)=∑v∈Gn−1SDG(v). We consider the asymptotic limit of CC(G) as the number of vertices tends to infinity and provide an elegant and insightful proof of a 2025 result by Britz, Hu, Islam, and Tang, limn→∞CC(Pn)=π, using uniform convergence and Riemann sums. We applied the same technique for the union of a cycle Cm and path Pn and the union of a path and a complete graph. We prove that of all graphs, paths have the minimum closeness centrality. Next we show for any c∈[π,∞), there exists a sequence of graphs {Gn} such that limn→∞CC(Gn)=c. In addition, we investigate the mean distance of a graph, l¯(G)=1n(n−1)∑v∈V(G)SD(v) and the normalized closeness centrality, C¯C(G)=1nCC(G). We verify a conjecture of Britz, Hu, Islam, and Tang that the set of products {l¯(G)C¯C(G):Gisfiniteandconnected} is dense in [1,2). 

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