Resistance distance in wheels and fans

Abstract

The wheel graph is the join of a single vertex and a cycle, while the fan graph is the join of a single vertex and a path. The resistance distance between any two vertices of a wheel and a fan are obtained. The resistances are related to Fibonacci numbers and generalized Fibonacci numbers. The derivation is based on evaluating determinants of submatrices of the Laplacian matrix.A combinatorial argument is also illustrated. A connection with the problem of squaring a rectangle is described.

Luminescence properties of MgO produced by solution combustion synthesis and doped with lanthanides and Li

“…2), the RL emission spectrum of MgO:Ce,Li is similar to the TL emission spectrum from MgO:Ce,Li reported by Bapat et al [7]. Indeed, Ce 3 + emission is associated with 5d-4f transitions of the Ce 3 + ion and is typically characterized by a double band due to splitting of the 4f 1 ground state into 2 F 5/2 and 2 F 7/2 states [31,32].…”

“…TL peaks have often been observed below 100 1C, but these are unstable at room temperature and, therefore, not suitable for dosimetry applications. Peaks above 100 1C have also been reported and a few examples from samples prepared by different methods can be found in Bapat et al [7], Dolgov et al [8], and Bos et al [9]. An extensive review of the defects in MgO and their relationship with its TL and luminescence properties can be found in Las and Stoebe [10].…”

“…The structure of the synthesized samples was confirmed by X-ray diffraction. Li co-doping was investigated as it has been observed to increase the luminescence intensity in Gd 2 O 3 :Eu 3 + [24], ZnS:Tm [25], Y 3 Al 5 O 12 :Tm [26], Y 2 O 3 :Eu [27], and MgO doped with various lanthanides [7,28,29].…”

Resistance distance in completen-partite graphs

“…Bapat and Gupta [1] gave interesting formulas in terms of Fibonacci numbers for the resistance distances in wheels and fans. In [4] the limit of the resistance distance between some pairs of vertices in fans and wheels are determined as the order of the graph increases without bound.…”

“…Thus d : V (G) → R is a function that satisfies (1) d(x, y) ≥ 0 for all x, y ∈ V (G), (2) d(x, y) = 0 if and only if x = y, (3) d(x, y) = d(y, x) for all x, y ∈ V (G), (4) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ V (G).…”