Strong T-Coloring of Graphs

A 𝑻-coloring of a graph 𝑮 = (𝑽,𝑬) is the generalized coloring of a graph. Given a graph 𝑮 = (𝑽, 𝑬) and a finite set T of positive integers containing 𝟎 , a 𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻. We define Strong 𝑻-coloring (S𝑻-coloring , in short), as a generalization of 𝑻-coloring as follows. Given a graph 𝑮 = (𝑽, 𝑬) and a finite set 𝑻 of positive integers containing 𝟎, a S𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻 and |𝒇(𝒖) − 𝒇(𝒘)| ≠ |𝒇(𝒙) − 𝒇(𝒚)| for any two distinct edges 𝒖𝒘, 𝒙𝒚 in 𝑬(𝑮). The S𝑻-Chromatic number of 𝑮 is the minimum number of colors needed for a S𝑻-coloring of 𝑮 and it is denoted by 𝝌𝑺𝑻(𝑮) . For a S𝑻 coloring 𝒄 of a graph 𝑮 we define the 𝒄𝑺𝑻- span 𝒔𝒑𝑺𝑻 𝒄 (𝑮) is the maximum value of |𝒄(𝒖) − 𝒄(𝒗)| over all pairs 𝒖, 𝒗 of vertices of 𝑮 and the S𝑻 -span 𝒔𝒑𝑺𝑻(𝑮) is defined by 𝒔𝒑𝑺𝑻(𝑮) = min 𝒔𝒑𝑺𝑻 𝒄 (𝑮) where the minimum is taken over all ST-coloring c of G. Similarly the 𝒄𝑺𝑻-edgespan 𝒆𝒔𝒑𝑺𝑻 𝒄 (𝑮) is the maximum value of |𝒄(𝒖) − 𝒄(𝒗)| over all edges 𝒖𝒗 of 𝑮 and the S𝑻-edge span 𝒆𝒔𝒑𝑺𝑻(𝑮) is defined by 𝒆𝒔𝒑𝑺𝑻(𝑮) = min 𝒆𝒔𝒑𝑺𝑻 𝒄 𝑮 where the minimum is taken over all ST-coloring c of G. In this paper we discuss these concepts namely, S𝑻- chromatic number, 𝒔𝒑𝑺𝑻(𝑮) , and 𝒆𝒔𝒑𝑺𝑻(𝑮) of graphs.

T-Span, T-Edge Span Critical Graphs

Given a graph 𝑮 = (𝑽, 𝑬) and a finite set 𝑻 of positive integers containing 𝟎, a 𝑻-coloring of 𝑮 is a function 𝒇 ∶ 𝑽 (𝑮) → 𝒁 + ∪ {𝟎} for all 𝒖 ≠ 𝒘 in 𝑽 (𝑮) such that if 𝒖𝒘 ∈ 𝑬(𝑮) then |𝒇(𝒖) − 𝒇(𝒘)| ∉ 𝑻. For a 𝑻-coloring 𝒇 of G, the f-span 𝒔𝒑𝑻 𝒇 (𝑮) is the maximum value of |𝒇(𝒖) − 𝒇(𝒘)| over all pairs 𝒖, 𝒘 of vertices of 𝑮. The 𝑻-span 𝒔𝒑𝑻(𝑮) is the minimum 𝒇-span over all 𝑻-colorings f of 𝑮. The 𝒇-edge span 𝒆𝒔𝒑𝑻 𝒇 (𝑮) of a 𝑻-coloring is the maximum value of 𝒇 𝒖 − 𝒇 𝒘 over all edges 𝒖𝒘 of 𝑮. The 𝑻-edge span 𝒆𝒔𝒑𝑻(𝑮) is the minimum 𝒇-edge span over all 𝑻-colorings f of 𝑮. It is known that 𝒔𝒑𝑻(𝑯) ≤ 𝒔𝒑𝑻(𝑮) and 𝒆𝒔𝒑𝑻(𝑯) ≤ e𝒔𝒑𝑻(𝑮) for every graph 𝑮. In this paper we classify which graphs containing a sub graph 𝑯 such that 𝒔𝒑𝑻 𝑯 < 𝒔𝒑𝑻(𝑮) and 𝒆𝒔𝒑𝑻(𝑯) < e𝒔𝒑𝑻(𝑮). Also we discuss the Mycielskian of 𝑻-coloring.

Three-Dimensional Velocity and Scalar Field Measurements of an Airfoil Trailing Edge With Slot Film Cooling: The Effect of an Internal Structure in the Slot

Measurements of the 3D velocity and concentration fields were obtained using magnetic resonance imaging for a pressure-side cutback film cooling experiment. The cutback geometry consisted of rectangular slots separated by straight lands; inside each of the slots was an airfoil-shaped blockage. The results from this trailing edge configuration, the “island airfoil,” are compared to the results obtained with the “generic airfoil,” a geometry with narrower slots, wider, tapered lands, and no blockages. The objective was to determine how the narrower lands and internal blockages affected the average film cooling effectiveness and the spanwise uniformity. Velocimetry data revealed that strong horseshoe vortices formed around the blockages in the slots, which resulted in greater coolant nonuniformity on the airfoil breakout surface and in the wake. The thinner lands of the island airfoil allowed the coolant to cover a larger fraction of the trailing edge span, giving a much higher spanwise-averaged surface effectiveness, especially near the slot exit where the generic airfoil lands are widest.

An Experimental Study on Aerodynamic Performance of Turbine Nozzle Guide Vanes With Trailing-Edge Span-Wise Ejection

The present experimental study is to examine the influence of trailing-edge coolant ejection with the span-wise inclination on the aerodynamic loss of turbine nozzle guide vanes. This study uses a cascade of five vanes located in the test section of a low-speed wind tunnel. The vanes have the profile of high-pressure nozzle guide vanes, and the central vane is equipped with the internal cooling and the trailing-edge coolant ejection. The coolant is ejected through trailing-edge slots that are inclined in the span-wise direction at angles varying from 0 deg to 45 deg in 15 deg increments. The results indicate an optimum ejection rate, at which the aerodynamic loss is minimum. There is a little variation in loss as the span-wise inclination is varied when the ratio of coolant to mainstream gas mass flow rate is less than 1.5%. For higher coolant flow rates, however, the loss increases with increases in the span-wise ejection angle.

3D Velocity and Scalar Field Measurements of an Airfoil Trailing Edge With Slot Film Cooling: The Effect of an Internal Structure in the Slot

Measurements of the 3D velocity and concentration fields were obtained using magnetic resonance imaging for a pressure side cutback film cooling experiment. The cutback geometry consisted of rectangular slots separated by straight lands; inside each of the slots was an airfoil-shaped blockage. The results from this trailing edge configuration, the “island airfoil,” are compared to the results obtained with the “generic airfoil,” a geometry with narrower slots, wider, tapered lands, and no blockages. The objective was to determine how the narrower lands and internal blockages affected the average film cooling effectiveness and the spanwise uniformity. Velocimetry data revealed that strong horseshoe vortices formed around the blockages in the slots, which resulted in greater coolant non-uniformity on the airfoil breakout surface and in the wake. The thinner lands of the island airfoil allowed the coolant to cover a larger fraction of the trailing edge span, giving a much higher spanwise-averaged surface effectiveness, especially near the slot exit where the generic airfoil lands are widest.

Nonlinear Static Analysis for Progressive Collapse Potential of RC Building

In this study, the progressive collapse potential of a 10-storey concrete frame structure was investigated using nonlinear static analysis. 15 different cases were considered and their performances were compared with each other. From the nonlinear static analysis results, most of longitudinal beams in upper floors and slabs above the failure column would collapse as the results of removing an exterior column, no beams and slabs would collapse when an interior column at ground floor was removed, and only the short-span beams in superstructure would collapse when an interior column in upper floors was removed. Tie force reinforcement along floors and beams of edge span can be used to avoid the progressive failure of floors, after failure of particular column.