Resistance distance in connected balanced digraphs

Abstract

Let D be a connected balanced digraph. The classical distance dijD from vertex i to vertex j is the length of a shortest directed path from i to j in D. Let L be the Laplacian matrix of D and L†=(lij†) be the Moore–Penrose inverse of L. The resistance distance rijD from i to j is then defined by rijD≔lii†+ljj†−2lij†. Let C be a collection of connected and balanced digraphs, each member of which is a finite union of the form D1∪D2∪....∪Dk where each Dt is a connected and balanced digraph with Dt∩(D1∪D2∪⋯∪Dt−1) being a single vertex, for all 1<t≤k. In this paper, we show that for any digraph D in C, rijD≤dijD(∗) for all i,j. This is established by partitioning the Laplacian matrix of D suitably. This generalizes the main result in Balaji et al. (2020), namely, the inequality (*) holds for any directed cactus D. Related studies have been made by many authors. For instance, in Gurvich (2022) and Gurvich and Vyalyi (2012), Vladimir Gurvich applied the results on connected balanced digraphs to semi-conductors in electrical networks. In Young et al. (2016) [12,13] , the authors study ‘effective resistance distance’ in graphs and digraphs while in Chebotarev (2011), graph geodetic distance is studied.