Steiner Distance in Product Networks

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In this paper, we investigate the Steiner distance and Steiner $k$-diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner $k$-diameter of some networks.Comment: 29 pages, 4 figure

Constructing disjoint Steiner trees in Sierpi\'{n}ski graphs

Let $G$ be a graph and $S\subseteq V(G)$ with $|S|\geq 2$. Then the trees $T_1, T_2, \cdots, T_\ell$ in $G$ are \emph{internally disjoint Steiner trees} connecting $S$ (or $S$-Steiner trees) if $E(T_i) \cap E(T_j )=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for every pair of distinct integers $i,j$, $1 \leq i, j \leq \ell$. Similarly, if we only have the condition $E(T_i) \cap E(T_j )=\emptyset$ but without the condition $V(T_i)\cap V(T_j)=S$, then they are \emph{edge-disjoint Steiner trees}. The \emph{generalized $k$-connectivity}, denoted by $\kappa_k(G)$, of a graph $G$, is defined as $\kappa_k(G)=\min\{\kappa_G(S)|S \subseteq V(G) \ \textrm{and} \ |S|=k \}$, where $\kappa_G(S)$ is the maximum number of internally disjoint $S$-Steiner trees. The \emph{generalized local edge-connectivity} $\lambda_{G}(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $G$. The {\it generalized $k$-edge-connectivity} $\lambda_k(G)$ of $G$ is defined as $\lambda_k(G)=\min\{\lambda_{G}(S)\,|\,S\subseteq V(G) \ and \ |S|=k\}$. These measures are generalizations of the concepts of connectivity and edge-connectivity, and they and can be used as measures of vulnerability of networks. It is, in general, difficult to compute these generalized connectivities. However, there are precise results for some special classes of graphs. In this paper, we obtain the exact value of $\lambda_{k}(S(n,\ell))$ for $3\leq k\leq \ell^n$, and the exact value of $\kappa_{k}(S(n,\ell))$ for $3\leq k\leq \ell$, where $S(n, \ell)$ is the Sierpi\'{n}ski graphs with order $\ell^n$. As a direct consequence, these graphs provide additional interesting examples when $\lambda_{k}(S(n,\ell))=\kappa_{k}(S(n,\ell))$. We also study the some network properties of Sierpi\'{n}ski graphs

A study on capacity of signalized intersection under snow-ice conditions based on classical model modification

Abstract As the common climatic phenomena frequently occurring in northern China in winter, snow and ice weathers have made great influence to the capacity of signalized intersection. By starting from snow and ice conditions, this paper makes improved correction about the calculation of capacity of signalized intersection respectively based on HCM Method and Method of Stopping Line. On the basis of HCM Method, the author comprehensively considers factors influencing the capacity of signalized intersection under snow and ice conditions, and calibrates the integrated parameters influencing the capacity of signalized intersection under snow and ice conditions; grounded on Method of Stopping Line, the author then calibrates various parameters and makes classified study about each parameter separately according to snow weather and freezing condition, finally giving out parameter values under various states. With the help of the revised model, the paper calculates the capacity of signalized intersection and then makes the comparison between traffic capacities under various snow and ice conditions and that with capacity of signalized intersection under normal weather, hence with strong practical significance

Efficacy of parathyroid autotransplantation in endoscopic total thyroidectomy with CLND

PurposeTo evaluate the safety and efficacy of autologous parathyroid transplantation in laparoscopic total thyroidectomy combined with central lymph node dissection (CLND).MethodsRetrospective analysis of clinical data of 152 patients admitted to the General Surgery Department of Gansu Provincial People’s Hospital who underwent endoscopic total thyroidectomy combined with CLND from June 2018 to December 2021. The intraoperative parathyroid glands were divided into the orthotopic preservation group (non-transplantation group) and the immediate active autologous transplantation group (transplantation group) according to the different treatment management of parathyroid glands during operation. The levels of Ca2+ in parathyroid blood and the incidence of hypoparathyroidism were compared between the two groups before operation and 1 day, 3 day, 1 week, 1 month, 3 months and 6 months after operation.ResultsThere was no significant difference in PTH between the preoperative transplantation group compared and the non-transplantation group (P>0.05); The PTH in the transplantation group was lower than that of the non-transplantation group 1 and 3 d after surgery, and the difference was statistically significant (P<0.05); No statistically significant difference in PTH between patients in the transplantation group compared with those in the non-transplantation group at 1 week postoperatively (P>0.05); PTH was significantly higher in the transplant group than in the non-transplant group at 1, 3 and 6 months after surgery, with statistically significant differences (P<0.05); there was no statistically differences (P>0.05) in serum Ca2+ between the preoperative, 1d, 3d and 1 week postoperative transplantation group and the non-transplantation group; Blood Ca2+ was significantly higher in the transplant group than in the non-transplant group at 1, 3 and 6 months after surgery, with statistically significant differences (P<0.05); The rate of temporary hypoparathyroidism in the transplantion group was higher than that in the non-transplantion group, and the rate of permanent hypoparathyroidism was lower than that in the non-transplantion group (P=0.044); There was no significant difference in the concentration of PTH in the cephalic vein of the elbow between the transplanted side and the non-transplanted side at 1 day and 3 days postoperatively (P>0.05); the concentration of PTH in cephalic vein of the elbow was significantly higher than that in non-transplanted side at 1 week, 1 month, 3 months and 6 months postoperatively (P<0.001); the number central area dissection and metastasis dissection in the transplantation group were significantly higher than those in the non-transplantation group (P<0.05).ConclusionsMost autologous parathyroid glands, having functional parathyroid autograft, is helpful to the occurrence of hypoparathyroidism after endoscopic total thyroidectomy with CLND, and it is an effective strategy to prevent permanent hypoparathyroidism, and more thorough area dissection is beneficial to the disease prognosis

Effect of Thermal Treatments on the Transduction Behaviors of Conductometric Hydrogen Gas Sensors Integrated with HCl-Doped Polyaniline Nanofibers

We present the effect of thermal treatments on the transduction behaviors of HCl-doped polyaniline (PANI) nanofibers integrated in conductometric devices upon exposure to 1% H2 (carried by N2). After drying in N2 at 25ºC for 12 hours, devices showed a ~10% decrease in electrical resistance upon exposure to 1% H2. However, devices subject to 12-hour drying in N2 at 25ºC followed by further thermal treatments in N2 at 100ºC, 164ºC or 200ºC for 30 minutes showed different transduction behaviors. Specifically, devices subject to thermal treatments at 100ºC and 164ºC showed a decrease in electrical resistance by ~7% and \u3c0.5%, respectively. More interestingly, the device subject to thermal treatment at 200ºC showed a transduction behavior with opposite polarity, i.e. a ~5% increase in electrical resistance upon exposure to 1% H2. SEM, FTIR and TGA were employed to investigate the effect of thermal treatments on the morphology and chemical characteristics of HCl-doped polyaniline nanofibers. The results indicated that the change in the devices? interesting transduction behaviors might be related to the thermal treatment effects on the HCl-doped PANI nanofibers in (i) removal of adsorbed water, and (ii) crosslinking and/or degradation of polymer backbones