Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

AbstractFor a nonnegative n×n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A)=n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n×n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials

Homomorphisms, representations and characteristic polynomials of digraphs

AbstractThe existence of a homomorphism between two digraphs often implies many structural properties. We collect in this paper some characterizations of various digraph homomorphisms using matrix equations and fiber partitions. We also survey the relationship among the characteristic polynomials of a digraph and its divisors. This includes an introduction of the concept of branched coverings of digraphs, their voltage assignment representations, and a decomposition formula for the characteristic polynomial of a branched cover with branch index 1. Some open problems are included

Combined effect of celecoxib and glucosamine sulfate on inflammatory factors and oxidative stress indicators in patients with knee osteoarthritis

Purpose: To investigate the combined effect of celecoxib and glucosamine sulfate on inflammatory factors and oxidative stress indicators in patients with knee osteoarthritis (KOA).Methods: Patients were randomly assigned to two groups of 60 patients each: control group and study group. The control group received celecoxib at a dose of 200 mg/kg/day, while the study group received glucosamine sulfate (500 mg/kg) in addition to celecoxib, thrice a day. Treatment in both groups lasted 8 weeks. The serum levels of tumor necrosis factor α (TNF-α), interleukin-1 (IL-1), prostaglandin-2 (PGE2), malondialdehyde (MDA), and activity of superoxide dismutase (SOD) were assayed before and after treatment. Visual analogue scale (VAS), osteoarthritis index, Lysholm knee score scale (LKSS), and adverse reactions were also evaluated.Results: After treatment, total effectiveness was significantly higher in the study group (91.33 %) than in control group (71.67 %, p < 0.05). Serum TNF-α, IL-1 and PGE2 were significantly lesser in the glucosamine sulfate-treated patients than in control group (p < 0.05). The activity of SOD was significantly higher in glucosamine sulfate-exposed patients than control patients (p < 0.05). On the other hand, VAS and WOMAC scores were markedly lower in patients given glucosamine sulfate than in control patients (p < 0.05).Conclusion: The combination of celecoxib with glucosamine sulfate effectively reduces immune inflammatory response, oxidative stress damage, and joint pain associated with KOA.Keywords: Celecoxib, Glucosamine sulfate, Osteoarthritis, Inflammatory factors, Oxidative stres

Preventive Effects of a Chinese Herbal Formula, Shengjiang Xiexin Decoction, on Irinotecan-Induced Delayed-Onset Diarrhea in Rats

Irinotecan is a well-known chemotherapy drug for the treatment of various cancers. However, delayed-onset diarrhea is a common adverse reaction, limiting the application of the drug. The study presented was designed to evaluate the preventive effects of Shengjiang Xiexin decoction (SXD) on irinotecan-induced diarrhea and to explore the possible mechanisms of this action. We established a diarrhea rat model. The condition of the rats was observed. The proliferation and apoptosis of intestinal cells were measured using immunohistochemical assays and a caspase-3 activity assay, respectively. The expression of Lgr5 and CD44 staining were used to observe intestinal stem cells (ISCs). In addition, the activity of β-glucuronidase in the rats’ feces was measured. Our results showed that the number of proliferating intestinal cells in the SXD groups was obviously higher, while the activity of caspase-3 was lower. The expression of Lgr5 and the integrated option density (IOD) of CD44 stain were increased significantly by SXD. Additionally, SXD decreased the activity of β-glucuronidase after irinotecan administration. In conclusion, SXD exhibited preventive effects on irinotecan-induced diarrhea, and this action was associated with an inhibitory effect on intestinal apoptosis and β-glucuronidase and a promotive effect on intestinal cell proliferation due to increased maintenance of ISCs

Viral infection and transmission in a large, well-traced outbreak caused by the SARS-CoV-2 Delta variant

The SARS-CoV-2 Delta variant has spread rapidly worldwide. To provide data on its virological profile, we here report the first local transmission of Delta in mainland China. All 167 infections could be traced back to the first index case. Daily sequential PCR testing of quarantined individuals indicated that the viral loads of Delta infections, when they first become PCR-positive, were on average ~1000 times greater compared to lineage A/B infections during the first epidemic wave in China in early 2020, suggesting potentially faster viral replication and greater infectiousness of Delta during early infection. The estimated transmission bottleneck size of the Delta variant was generally narrow, with 1-3 virions in 29 donor-recipient transmission pairs. However, the transmission of minor iSNVs resulted in at least 3 of the 34 substitutions that were identified in the outbreak, highlighting the contribution of intra-host variants to population-level viral diversity during rapid spread