Kemeny's constant and the effective graph resistance

• Published in: Linear algebra and its applications Vol. 535; pp. 231 - 244
• Main Authors: Wang, Xiangrong, Dubbeldam, Johan L.A., Van Mieghem, Piet
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15-12-2017; American Elsevier Company, Inc
• Subjects: Effective graph resistance or Kirchhoff index; Graph theory; Kemeny constant; Linear algebra; Lower bounds; Matrix; Moore–Penrose pseudo-inverse; Multiplicative degree-Kirchhoff index; Numerical analysis; Spectral graph theory; 15A63; Multiplicative degree-Kirchhoff index; Effective graph resistance or Kirchhoff index; Moore–Penrose pseudo-inverse; 15A09; Kemeny constant; Spectral graph theory; 15A18
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2017.09.003

Kemeny's constant and its relation to the effective graph resistance has been established for regular graphs by Palacios et al. [1]. Based on the Moore–Penrose pseudo-inverse of the Laplacian matrix, we derive a new closed-form formula and deduce upper and lower bounds for the Kemeny constant. Furthermore, we generalize the relation between the Kemeny constant and the effective graph resistance for a general connected, undirected graph.