Kemeny's constant and the effective graph resistance

Abstract

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.

On Kemeny's constant for trees with fixed order and diameter

“…The problem of understanding how the structure of the network influences the value of Kemeny's constant then becomes entirely graph-theoretical. Moreover, this quantity is strongly linked to a particular metric on graphs known as resistance distance and the related Kirchhoff index and multiplicative degree-Kirchhoff index [12]. In Remark 2.1 and Remark 4.1, we comment on our results in the light of this connection.…”

“…The Kirchhoff index of a graph, defined as half the sum of the resistance distances between vertices in the graph, is related to Kemeny's constant (see [12]). In [13,Proposition 2.5.…”

Markov fundamental tensor and its applications to network analysis

“…Perhaps the most known topology index is the Kirchhoff index [13] which has found a variety of applications [34,35,36,7,37]. Kirchhoff index is also closely connected to Kemeney's constant [38,39]. The Kirchhoff index is often defined in terms of effective resistances [13], K(G) = 1 2 s,t Ω st , which is closely related to commute times, as Ω st = 1 |E| C st [40].…”

Topological Approach to Measure Network Recoverability

“…2) Effective graph resistance r G . The effective graph resistance [22], [23], [16], [24] origins from the field of electric circuit analysis, which is defined as the accumulated effective resistance between all pairs of nodes. The effective graph resistance refers to the average power dissipated in a resistor network with random infected currents, which can indicate the overall diffusivity of information spreading in a communication network.…”