# Resistance distance in directed cactus graphs

## Article Type

Research Article

## Publication Title

Electronic Journal of Linear Algebra

## Abstract

Let G = (V, E) be a strongly connected and balanced digraph with vertex set V = {1, …, n}. The classical distance dij between any two vertices i and j in G is the minimum length of all the directed paths joining i and j. The resistance distance (or, simply the resistance) between any two vertices i and j in V is defined by rij:= lii† + l†jj − 2l†ij, where l†pq is the (p, q)th entry of the Moore-Penrose inverse of L which is the Laplacian matrix of G. In practice, the resistance rij is more significant than the classical distance. One reason for this is, numerical examples show that the resistance distance between i and j is always less than or equal to the classical distance, i.e., rij ≤ dij . However, no proof for this inequality is known. In this paper, it is shown that this inequality holds for all directed cactus graphs.

## First Page

277

## Last Page

292

## DOI

10.13001/ela.2020.5093

## Publication Date

1-1-2020

## Recommended Citation

Balaji, R.; Bapat, R. B.; and Goel, Shivani, "Resistance distance in directed cactus graphs" (2020). *Journal Articles*. 481.

https://digitalcommons.isical.ac.in/journal-articles/481

## Comments

Open Access, Bronze, Green