No abstract

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.

In the present research, the effects of various alloying elements and microstructural constituents on the mechanical properties and corrosion behaviour have been studied for four different rebars. The microstructures of stainless steel and plain rebar primarily reveal equiaxed ferrite grains and ferrite-pearlite microstructures, respectively, with no evidence of transition zone, whereas tempered martensite at the outer rim, followed by a narrow bainitic transition zone with an internal core of ferrite-pearlite, has been observed for the thermomechanically treated (TMT) rebars. The hardness profiles obtained from this study display maximum hardness at the periphery, which decreases gradually towards the centre, thereby providing the classical U-shaped hardness profile for TMT rebars. The tensile test results confirm that stainless steel rebar exhibits the highest combination of strength (≈755 MPa) and ductility (≈27%). It has been witnessed that in Tafel plots, the corrosion rate increases for all the experimental rebars in 1% HCl solution, which is well expected because the acid solutions generally possess a higher corrosive environment than seawater (3.5% NaCl) due to their acidic nature and lower pH values. However, all the experimental results obtained from Tafel and Nyquist plots correlate well for both 1% HCl and 3.5% NaCl solutions.