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Moon's classical result implies that the number of spanning trees of a complete graph Kn with n vertices containing a given spanning forest F equals nc−2∏i=1cni, where c is the number of components of F, and n1,n2,...,nc are the numbers of vertices of component of F. Dong and Ge extended this result to the complete bipartite graph, and obtain an interesting formula to count spanning trees of a complete bipartite graph Km,n containing a given spanning forest F. They also posed the problem to count spanning trees of a complete s‐partite graph containing a given spanning forest for s≥3. In this paper, we propose a technique to solve this problem. Using this technique, we obtain closed formulae to count spanning trees of complete s‐partite graphs containing a given spanning forest for s=3 and 4. Our technique also results in a new and simple proof of Dong and Ge's formula. [ABSTRACT FROM AUTHOR]

Aims: Emerging evidence suggests that cartilage intermediate layer protein 1 (CILP‐1) is associated with myocardial remodelling. However, the prognostic value of circulating CILP‐1 in patients with heart failure (HF) remains to be elucidated. This study aimed to investigate whether circulating CILP‐1 can independently predict the outcome of chronic HF. Methods and results: This prospective cohort study included 210 patients with chronic HF and left ventricular ejection fraction <50% between September 2018 and December 2019. The primary endpoint was 1 year all‐cause mortality. During the 1 year follow‐up, 28 patients died. In multivariable Cox proportional hazards regression analysis, higher CILP‐1 levels were independently associated with a higher risk of mortality after adjusting for potential confounding factors. In Kaplan–Meier analysis, patients with CILP‐1 levels above the median had a significantly higher mortality rate than those with CILP‐1 levels below the median (log‐rank P = 0.015). In addition, CILP‐1 significantly improved prognostic prediction over N‐terminal pro‐brain natriuretic peptide by an increase in net reclassification improvement (P = 0.043) and a trend towards an increase in integrated discrimination improvement (P = 0.118). Conclusions: Circulating CILP‐1 is a novel independent prognostic predictor in chronic HF. [ABSTRACT FROM AUTHOR]

Background: Research has associated nutritional status with the prognosis of cardiovascular diseases. This study aimed to investigate the prognostic value of a novel nutritional index, triglycerides × total cholesterol × body weight index (TCBI), in patients with dilated cardiomyopathy (DCM)., Methods: This retrospective cohort study enrolled 445 patients with DCM. The median follow-up period was 2.8 years, and the primary endpoint was all-cause death., Results: During follow-up, the all-cause mortality was observed in 135 out of 445 patients (30.3%). In Kaplan-Meier survival analysis, the third TCBI tertile had a lower mortality risk (T3 vs. T2 vs. T1: 16.9% vs. 35.1% vs. 38.9%; log-rank P  < 0.001). In the multivariable Cox regression analysis, patients in the third tertile were associated with a decreased mortality, whereas there was no significant difference between the T2 and T1 groups. Moreover, TCBI could significantly improve risk stratification (continuous net reclassification improvement and integrated discrimination improvement) over the Geriatric Nutritional Risk Index (GNRI) and N-terminal pro-brain natriuretic peptide (NT-proBNP)., Conclusions: TCBI is independently associated with long-term survival in patients with DCM. Combination of TCBI and other biomarkers, such as GNRI and NT-proBNP, can significantly improve prognostic prediction. Further studies with larger sample size are required to validate our results., Competing Interests: The authors declare there are no competing interests., (©2022 Wang et al.)

BACKGROUND Post-contrast acute kidney injury (PC-AKI) is a contributor to adverse outcomes after percutaneous coronary intervention (PCI). This study aimed to investigate whether fibrinogen-to-albumin ratio (FAR), a novel inflammation-based risk index, can predict the occurrence of PC-AKI in patients undergoing elective PCI. MATERIAL AND METHODS We retrospectively enrolled 291 patients who underwent elective PCI from June 2017 to June 2019. PC-AKI was defined as an increase in serum creatinine ≥0.3 mg/dL (≥26.5 μmol/L), or ≥1.5 times baseline within 48 to 72 hours after PCI. The area under the receiver-operating characteristic curve (AUC), continuous net reclassification improvement (NRI), and integrated discrimination improvement (IDI) were calculated to make comparison for PC-AKI prediction. RESULTS PC-AKI occurred in 43 patients (14.8%). FAR showed an AUC of 0.691 (95% confidence interval: 0.64-0.74; P<0.001) in predicting PC-AKI. In stepwise multivariable logistic regression, FAR was independently associated with the occurrence of PC-AKI along with hypertension, diabetes, hemoglobin, estimated glomerular filtration rate, and left ventricular ejection fraction. FAR significantly improved PC-AKI prediction over Mehran risk score in the continuous NRI and IDI, but not AUC. CONCLUSIONS FAR is independently associated with the occurrence of PC-AKI, and can significantly improve PC-AKI prediction over Mehran risk score in patients undergoing elective PCI.