On a conjecture of the Randić index

AbstractThe Randić index of a graph G is defined as R(G)=∑u∼v(d(u)d(v))−12, where d(u) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u, v. A conjecture on R(G) for connected graph G is as follows: R(G)≥r(G)−1, where r(G) denotes the radius of G. We proved that the conjecture is true for biregular graphs, connected graphs with order n≤10 and tricyclic graphs

On the Uniqueness of Convex Central Configurations in the Planar 44-Body Problem

In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture that there exists a unique convex central configuration for any four fixed positive masses in a given order belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem. Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its surrounding neighborhood.Comment: 30 pages,2 figure

The Maximal Total Irregularity of Bicyclic Graphs

In 2012, Abdo and Dimitrov defined the total irregularity of a graph G=(V,E) as irrtG=1/2∑u,v∈VdGu-dGv, where dGu denotes the vertex degree of a vertex u∈V. In this paper, we investigate the total irregularity of bicyclic graphs and characterize the graph with the maximal total irregularity among all bicyclic graphs on n vertices

On a conjecture of the Randić index

AbstractThe Randić index of a graph G is defined as R(G)=∑u∼v(d(u)d(v))−12, where d(u) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u, v. A conjecture on R(G) for connected graph G is as follows: R(G)≥r(G)−1, where r(G) denotes the radius of G. We proved that the conjecture is true for biregular graphs, connected graphs with order n≤10 and tricyclic graphs

Transcriptional up-regulation of relaxin-3 by Nur77 attenuates β-adrenergic agonist-induced apoptosis in cardiomyocytes.

The relaxin family peptides have been shown to exert several beneficial effects on the heart, including anti-apoptosis, anti-fibrosis, and anti-hypertrophy activity. Understanding their regulation might provide new opportunities for therapeutic interventions, but the molecular mechanism(s) coordinating relaxin expression in the heart remain largely obscured. Previous work demonstrated a role for the orphan nuclear receptor Nur77 in regulating cardiomyocyte apoptosis. We therefore investigated Nur77 in the hopes of identifying novel relaxin regulators. Quantitative real-time PCR (qRT-PCR) and enzyme-linked immunosorbent assay (ELISA) data indicated that ectopic expression of orphan nuclear receptor Nur77 markedly increased the expression of latexin-3 (RLN3), but not relaxin-1 (RLN1), in neonatal rat ventricular cardiomyocytes (NRVMs). Furthermore, we found that the -adrenergic agonist isoproterenol (ISO) markedly stimulated RLN3 expression, and this stimulation was significantly attenuated in Nur77 knockdown cardiomyocytes and Nur77 knockout hearts. We showed that Nur77 significantly increased RLN3 promoter activity via specific binding to the RLN3 promoter, as demonstrated by electrophoretic mobility shift assay (EMSA) and chromatin immuno-precipitation (ChIP) assays. Furthermore, we found that Nur77 overexpression potently inhibited ISO-induced cardiomyocyte apoptosis, whereas this protective effect was significantly attenuated in RLN3 knockdown cardiomyocytes, suggesting that Nur77-induced RLN3 expression is an important mediator for the suppression of cardiomyocyte apoptosis. These findings show that Nur77 regulates RLN3 expression, therefore suppressing apoptosis in the heart, and suggest that activation of Nur77 may represent a useful therapeutic strategy for inhibition of cardiac fibrosis and heart failure. © 2018 You et al

A Gene Expression Signature Predicts Survival of Patients with Stage I Non-Small Cell Lung Cancer

BACKGROUND: Lung cancer is the leading cause of cancer-related death in the United States. Nearly 50% of patients with stages I and II non-small cell lung cancer (NSCLC) will die from recurrent disease despite surgical resection. No reliable clinical or molecular predictors are currently available for identifying those at high risk for developing recurrent disease. As a consequence, it is not possible to select those high-risk patients for more aggressive therapies and assign less aggressive treatments to patients at low risk for recurrence. METHODS AND FINDINGS: In this study, we applied a meta-analysis of datasets from seven different microarray studies on NSCLC for differentially expressed genes related to survival time (under 2 y and over 5 y). A consensus set of 4,905 genes from these studies was selected, and systematic bias adjustment in the datasets was performed by distance-weighted discrimination (DWD). We identified a gene expression signature consisting of 64 genes that is highly predictive of which stage I lung cancer patients may benefit from more aggressive therapy. Kaplan-Meier analysis of the overall survival of stage I NSCLC patients with the 64-gene expression signature demonstrated that the high- and low-risk groups are significantly different in their overall survival. Of the 64 genes, 11 are related to cancer metastasis (APC, CDH8, IL8RB, LY6D, PCDHGA12, DSP, NID, ENPP2, CCR2, CASP8, and CASP10) and eight are involved in apoptosis (CASP8, CASP10, PIK3R1, BCL2, SON, INHA, PSEN1, and BIK). CONCLUSIONS: Our results indicate that gene expression signatures from several datasets can be reconciled. The resulting signature is useful in predicting survival of stage I NSCLC and might be useful in informing treatment decisions

THE SIGNLESS LAPLACIAN SEPARATOR OF GRAPHS

The signless Laplacian separator of a graph is defined as the difference between the largest eigenvalue and the second largest eigenvalue of the associated signless Laplacian matrix. In this paper, we determine the maximum signless Laplacian separators of unicyclic, bicyclic and tricyclic graphs with given order

On the Spectral Properties of Line Distance Matrices

Abstract In Similarly, we obtain bounds on the energy of line distance matrix. The spread of the spectrum of line distance matrix is considered. Introductio

The majorization theorem of connected graphs

AbstractLet π=(d1,d2,…,dn) and π′=(d1′,d2′,…,dn′) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π◁π′, if and only if π≠π′,∑i=1ndi=∑i=1ndi′, and ∑i=1jdi⩽∑i=1jdi′ for all j=1,2,…,n. If the degree of vertex v is (resp. not) equal to 1, then we call v a pendant (resp. non-pendant) vertex of G. We use Cπ to denote the class of connected graphs with degree sequence π. Suppose π and π′ are two non-increasing c-cyclic degree sequences. Let G and G′ be the graphs with greatest spectral radii in Cπ and Cπ′, respectively. In this paper, we shall prove that if π◁π′,G and G′ have the same number of pendant vertices, and the degrees of all non-pendant vertices of G′ are greater than c, then ρ(G)<ρ(G′)