The extremal unicyclic graphs of the revised edge Szeged index with given diameter

Let $G$ be a connected graph. The revised edge Szeged index of $G$ is defined as $Sz^{\ast}_{e}(G)=\sum\limits_{e=uv\in E(G)}(m_{u}(e|G)+\frac{m_{0}(e|G)}{2})(m_{v}(e|G)+\frac{m_{0}(e|G)}{2})$, where $m_{u}(e|G)$ (resp., $m_{v}(e|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $m_{0}(e|G)$ is the number of edges equidistant from both ends of $e$, respectively. In this paper, the graphs with minimum revised edge Szeged index among all the unicyclic graphs with given diameter are characterized.Comment: arXiv admin note: text overlap with arXiv:1805.0657

No mixed graph with the nullity η(G~)=∣V(G)∣−2m(G)+2c(G)−1\eta(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1

A mixed graph $\widetilde{G}$ is obtained from a simple undirected graph $G$, the underlying graph of $\widetilde{G}$, by orienting some edges of $G$. Let $c(G)=|E(G)|-|V(G)|+\omega(G)$ be the cyclomatic number of $G$ with $\omega(G)$ the number of connected components of $G$, $m(G)$ be the matching number of $G$, and $\eta(\widetilde{G})$ be the nullity of $\widetilde{G}$. Chen et al. (2018)\cite{LSC} and Tian et al. (2018)\cite{TFL} proved independently that $|V(G)|-2m(G)-2c(G) \leq \eta(\widetilde{G}) \leq |V(G)|-2m(G)+2c(G)$, respectively, and they characterized the mixed graphs with nullity attaining the upper bound and the lower bound. In this paper, we prove that there is no mixed graph with nullity $\eta(\widetilde{G})=|V(G)|-2m(G)+2c(G)-1$. Moreover, for fixed $c(G)$, there are infinitely many connected mixed graphs with nullity $|V(G)|-2m(G)+2c(G)-s$ $( 0 \leq s \leq 3c(G), s\neq1 )$ is proved

On the extremal cacti with minimum Sombor index

Let $H$ be a graph with edge set $E_H$. The Sombor index and the reduced Sombor index of a graph $H$ are defined as $SO(H) = \sum\limits_{uv\in E_H}\sqrt{d_{H}(u)^{2}+d_{H}(v)^{2}}$ and $SO_{red}(H) = \sum\limits_{uv\in E_H}\sqrt{(d_{H}(u)-1)^{2}+(d_{H}(v)-1)^{2}}$, respectively. Where $d_{H}(u)$ and $d_{H}(v)$ are the degrees of the vertices $u$ and $v$ in $H$, respectively. A cactus is a connected graph in which any two cycles have at most one common vertex. Let $\mathcal{C}(n, k)$ be the class of cacti of order $n$ with $k$ cycles. In this paper, the lower bound for the Sombor index of the cacti in $\mathcal{C}(n, k)$ is obtained and the corresponding extremal cacti are characterized when $n\geq 4k-2$ and $k\geq 2$. Moreover, the lower bound of the reduced Sombor index of cacti is obtained by similar approach

The extremal unicyclic graphs with given diameter and minimum edge revised Szeged index

Let $H$ be a connected graph. The edge revised Szeged index of $H$ is defined as $Sz^{\ast}_{e}(H) = \sum\limits_{e = uv\in E_H}(m_{u}(e|H)+\frac{m_{0}(e|H)}{2})(m_{v}(e|H)+\frac{m_{0}(e|H)}{2})$, where $m_{u}(e|H)$ (resp., $m_{v}(e|H)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than to vertex $v$ (resp., $u$), and $m_{0}(e|H)$ is the number of edges equidistant from $u$ and $v$. In this paper, the extremal unicyclic graphs with given diameter and minimum edge revised Szeged index are characterized

N 2,N 2,N 5,N 5-Tetra­kis(2-chloro­ethyl)-3,4-dimethyl­thio­phene-2,5-dicarboxamide

In the title compound, C16H22Cl4N2O2S, the two imide groups adopt a trans arrangement relative to the central thienyl ring, so the four terminal 2-chloro­ethyl arms adopt different orientations. In the crystal, mol­ecules are linked by weak C—H⋯Cl and C—H⋯O hydrogen bonds into a three-dimensional network

N′2,N′5-Bis[(E)-2-hy­droxy­benzyl­idene]-3,4-dimethyl­thio­phene-2,5-dicarbohydrazide

In the title mol­ecule, C22H20N4O4S, both C=N bonds are in an E conformation. The benzene rings form dihedral angles of 12.10 (13) and 25.17 (12)° with the thio­phene ring. The dihedral angle between the two benzene rings is 17.59 (14)°. There are two intra­molecular O—H⋯N hydrogen bonds. In the crystal, N—H⋯O hydrogen bonds connect mol­ecules into chains along [010]

(E)-3-(9-Anthr­yl)-1-(4-fluoro­phen­yl)-2-(4-nitro-1H-imidazol-1-yl)prop-2-en-1-one

In the title compound, C26H16FN3O3, the dihedral angle between the anthryl and fluoro­phenyl groups is 37.8 (1)°. With respect to the imidazolyl group, the twist angles between the imidazolyl group and the anthryl unit and between the imidazoly group and the fluoro­phenyl group are 64.4 (1) and 74.5 (1)°, respectively

Dual Energy Spectral CT Imaging for Colorectal Cancer Grading: A Preliminary Study

ObjectivesTo assess the diagnostic value of dual energy spectral CT imaging for colorectal cancer grading using the quantitative iodine density measurements in both arterial phase (AP) and venous phase (VP).Methods81 colorectal cancer patients were divided into two groups based on their pathological findings: a low grade group including well (n = 13) and moderately differentiated cancer (n = 24), and a high grade group including poorly differentiated (n = 42) and signet ring cell cancer (n = 2). Iodine density (ID) in the lesions was derived from the iodine-based material decomposition (MD) image and normalized to that in the psoas muscle to obtain normalized iodine density (NID). The difference in ID and NID between AP and VP was calculated.ResultsThe ID and NID values of the low grade cancer group were, 14.65±3.38mg/mL and 1.70±0.33 in AP, and 21.90±3.11mg/mL and 2.05± 0.32 in VP, respectively. The ID and NID values for the high grade cancer group were 20.63±3.72mg/mL and 2.95±0.72 in AP, and 26.27±3.10mg/mL and 3.51±1.12 in VP, respectively. There was significant difference for ID and NID between the low grade and high grade cancer groups in both AP and VP (all p<0.001). ROC analysis indicated that NID of 1.92 in AP provided 70.3% sensitivity and 97.7% specificity in differentiating low grade cancer from high grade cancer.ConclusionsThe quantitative measurement of iodine density in AP and VP can provide useful information to differentiate low grade colorectal cancer from high grade colorectal cancer with NID in AP providing the greatest diagnostic value

1H-1,2,4-Triazol-4-ium (3,4-dichloro­phen­yl)methane­sulfonate

In the title molecular salt, C2H4N3 +·C7H5Cl2O3S−, C—C—S angle [112.25 (18)°] deviates slightly from that expected for ideal sp 3-hybridization geometry. In the crystal, the components are linked by N—H⋯O and bifurcated N—H⋯(O,O) hydrogen bonds into chains parallel to [110]

HPV prevalence and genotype distribution in 2,306 patients with cervical squamous cell carcinoma in central and eastern China

BackgroundTo explore the positivity rate and genotype distribution of human papillomavirus (HPV) in cervical squamous cell carcinoma (CSCC) tissues in central and eastern China and to provide theoretical basis for cervical cancer screening and prophylactic HPV vaccine development in China.MethodsDNA was extracted from paraffin-embedded tissues of CSCC samples and exfoliated cervical cells of cervical cancer screening populations. 23 HPV genotypes were detected by combining polymerase chain reaction (PCR) and reverse dot hybridized gene chip detection technology in 2,306 CSCC tissues and 10,245 cervical cancer screening populations. The genotype distribution of HPV infection was analyzed.ResultsThe overall infection rate of HPVs in 2,306 CSCC patients was 92.71%. The frequency of single-type HPV infection and multiple-type HPV infection were 86.48% and 13.51%, respectively. The most common HPV genotypes detected in Chinese CSCC tissues were HPV-16, HPV-18, HPV-31, HPV-33, HPV-45, HPV-52, HPV-58, and HPV-59. The overall positivity rate of these eight high-risk HPV (HR-HPV) genotypes in HPV-positive CSCC was as high as 96.91%. Of which the positivity rate of seven HR-HPV genotypes related to nine-valent HPV vaccines in HPV-positive CSCC was 95.09%. Meanwhile, the overall infection rates of HR-HPV and low-risk HPV (LR-HPV) in female aged 35–64 years who underwent cervical cancer screening were 13.16% and 1.32%, respectively. The high-frequency HR-HPV genotypes in cervical cancer screening women were HPV-52, HPV-58, HPV-16, HPV-53, HPV-68, HPV-39, HPV-51, and HPV-56, with positivity rates of 2.25%, 1.60%, 1.31%, 1.22%, 0.93%, 0.92%, 0.78%, and 0.74%, respectively.ConclusionAmong women screened for cervical cancer in China, detecting the 8 high-frequency HR-HPV genotypes can reduce technical difficulty and reagent costs, while also improving the efficiency and effectiveness of cervical cancer screening. HPV genotyping assists gynecologists in assessing the risk of HR-HPV-positive cervical intraepithelial neoplasia and guiding them in implementing appropriate interventions. Furthermore, HPV genotyping is helpful for doctors to follow up HR-HPV-positive women and to evaluate the protective effect of HPV vaccine