Balaji, R.; Bapat, R. B.; Goel, Shivani Resistance distance in directed cactus graphs. (English) Zbl 1444.05060 Electron. J. Linear Algebra 36, 277-292 (2020). Summary: Let \(G=(V,E)\) be a strongly connected and balanced digraph with vertex set \(V=\{1,\dotsc,n\}\). The classical distance \(d_{ij}\) 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 \(r_{ij}:=l_{ii}^{\dagger}+l_{jj}^{\dagger}-2l_{ij}^{\dagger}\), where \(l_{pq}^{\dagger}\) is the \((p,q)^{\text{th}}\) entry of the Moore-Penrose inverse of \(L\) which is the Laplacian matrix of \(G\). In practice, the resistance \(r_{ij}\) 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., \(r_{ij} \leq d_{ij}\). However, no proof for this inequality is known. In this paper, it is shown that this inequality holds for all directed cactus graphs. Cited in 1 ReviewCited in 5 Documents MSC: 05C20 Directed graphs (digraphs), tournaments 05C12 Distance in graphs 05C40 Connectivity 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) Keywords:strongly connected balanced digraph; directed cactus graph; Laplacian matrix; Moore-Penrose inverse; cofactor sums PDFBibTeX XMLCite \textit{R. Balaji} et al., Electron. J. Linear Algebra 36, 277--292 (2020; Zbl 1444.05060) Full Text: arXiv Link References: [1] R. Bapat. Resistance distance in graphs.Math. Student, 68:87—98, 1999. · Zbl 1194.05031 [2] R.B. Bapat and T. Raghavan.Nonnegative Matrices and Applications. Cambridge University Press, Cambridge, 1997. · Zbl 0879.15015 [3] S. Chaiken. A combinatorial proof of the all minors matrix tree theorem.SIAM J. Algebraic Discrete Methods, 3:319-329, 1982. Available at https://doi.org/10.1137/0603033. · Zbl 0495.05018 [4] J. Ding and A. Zhou. Eigenvalues of rank-one updated matrices with some applications.Appl. Math. Lett., 20:1223-1226, 2007. Available at https://doi.org/10.1016/j.aml.2006.11.016. · Zbl 1139.15003 [5] S. Goel, B. Ramamurthy, and R. Bapat. Resistance matrices of balanced directed graphs.Linear Multilinear Algebra, 2020. Available at https://doi.org/10.1080/03081087.2020.1748850. [6] R.A. Horn and C.R. Johnson.Topics in Matrix AnalysisCambridge University Press, Cambridge, 1994. [7] Y. Hou and J. Chen. Inverse of the distance matrix of a cactoid digraph.Linear Algebra Appl., 475:1—10, 2015. Available at https://doi.org/10.1016/j.laa.2015.02.002. · Zbl 1312.05083 [8] D.J.KleinandM.Randic.Resistancedistance.J.Math.Chem.,12:81—95,1993.Availableat https://doi.org/10.1007/BF01164627 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.