2021/11/03 by R. Balaji, Balaji, R., R. B. Bapat +3 · 1 citation
Computer Science · Engineering · Mathematics · #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Graph theory and applications #Matrix Theory and Algorithms #Molecular Junctions and Nanostructures
paper · pdf · doi:10.48550/arxiv.2111.02051
openalex publication_date 2021/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a connected graph with V(G)=\1,\dotsc,n\. Then the resistance distance between any two vertices i and j is given by rij:=lii^† + ljj^†-2 lij^†, where lij^† is the (i,j)\rm th entry of the Moore-Penrose inverse of the Laplacian matrix of G. For the resistance matrix R:=[rij], there is an elegant formula to compute the inverse of R. This says that R-1=-(1)/(2)L + (1)/(τ' R τ) ττ', where τ:=(τ1,\dotsc,τn)'~~and~~ τi:=2- ∑_\j ∈ V(G):(i,j) ∈ E(G)\ rij~~~i=1,\dotsc,n. A far reaching generalization of this result that gives an inverse formula for a generalized resistance matrix of a strongly connected and matrix weighted balanced directed graph is obtained in this paper. When the weights are scalars, it is shown that the generalized resistance is a non-negative real number. We also obtain a perturbation result involving resistance matrices of connected graphs and Laplacians of digraphs.