The Wiener index is the sum of distances between all pairs of vertices in a connected graph. In this paper, explicit expressions for the expected value of the Wiener index of three types of random pentagonal chains (cf. Figure 1) are obtained.

Recently a new mechanism of proton tunneling in a prism-like water hexamer was revealed [Richardson et al., Science, 2016, 351, 1310]. The tunneling motion involves the concerted breaking of two hydrogen bonds and rotations of two nearest water molecules. Eventually, this structural transformation means flipping one of the hydrogen bonds without the creation of defects in the hydrogen bond netw...

the importance as well as the diculty of the problem of ranking fuzzy numbers is pointed out. here we consider approaches to the ranking of fuzzy numbers based upon the idea of associating with a fuzzy number a scalar value, its signal/noise ratios, where the signal and the noise are dened as the middle-point and the spread of each a-cut of a fuzzy number, respectively. we use the value of a ...

In this paper, we prove that the -labeling number of trees T, T r/2 n where n=|E(T )| and r is the radius of T. This improves the known result T eO( √ n log n) tremendously and this upper bound is very close to the upper bound T n conjectured by Snevily. Moreover, we prove that a tree with n edges and radius r decomposes Kt for some t (r + 1)n2 + 1. © 2006 Elsevier B.V. All rights reserved.

background: luteinizing hormone (lh) was secreted by the stimulating cells of the testes and ovaries in the anterior pituitary gland. the application of this hormone is in the treatment of men and women with infertility and amenorrhea respectively. m aterials and methods: in the present study the alpha and beta subunits of human lh gene were cloned into the pegfp-n1 expression vector and produc...

Abstract In this current era, the concept of nonlinearity plays an important and essential role in intuitionistic fuzzy arena. This article portrays impression different representation nonlinear pentagonal number (PIFN) its classification under scenarios. A new de-intuitification technique non-linear PIFN is addressed along with various graphical representations. Additionally, paper, we have ob...