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In [3,4] the authors found formulas for the number of spanning 2-forests (equivalently for the resistance distance) between any pair of vertices in a "straight linear 2-tree" and in a "linear 2-tree with a single bend." The main result of this paper generalizes these formulas to linear 2-trees with any number of bends. We also give alternative and perhaps more transparent versions of these formulas in terms of block matrices. As an application we find the resistance distance between the extremal vertices of the "corrugated linear 2-tree" and use this result to determine its asymptotic rate of growth as the number of vertices goes to infinity. Finally we note that our formulas give a different way to determine all the resistance distances in a "fan graph." [ABSTRACT FROM AUTHOR]

Abstract: In this work, we consider a transmission problem across highly conductive layers of the pre-fractal Sierpinski gasket type. We apply a homogenization approach to obtain a convergence of solutions to the transmission problem for a fixed iteration of the Sierpinski gasket. We show Mosco convergence of a sequence of energy functionals to an energy functional, supported on the highly conductive layer that incorporates a singular term on this layer. Moreover, from this convergence we can obtain results related to the convergence of approximating solutions to the limit solutions. [Copyright &y& Elsevier]