Abstract
Kemeny's constant κ(G) of a connected undirected graph G can be interpreted as the expected transit time between two randomly chosen vertices for the Markov chain associated with G. In certain cases, inserting a new edge into G has the counter-intuitive effect of increasing the value of κ(G). In the current work we identify a large class of graphs exhibiting this “paradoxical” behavior – namely, those graphs having a pair of twin pendent vertices. We also investigate this phenomenon in the context of random graphs, showing that it occurs for almost all connected planar graphs. To establish these results, we make use of a connection between Kemeny's constant and the resistance distance of graphs.
Highlights
A useful technique to get insight into the combinatorial structure of a graph is to imagine a random walker moving along its edges
The long-term and short-term behaviors of the discrete stochastic process associated with the random walk are strictly linked to both local and global properties of the graph
We will exploit an intriguing connection of Kemeny’s constant with the so-called resistance distance of graphs, which simulates the behavior of electrical resistance in electric circuits ([9,11,12,13])
Summary
A useful technique to get insight into the combinatorial structure of a graph is to imagine a random walker moving along its edges. In [10], one example of “paradoxical” graphs is presented, namely those trees with at least 4 vertices having a pair of twin pendent vertices a and b (i.e., a and b are pendent vertices and they are both adjacent to a common vertex) In this case, adding the edge ab results in an increase in Kemeny’s constant (see Theorem 2.1 below). The goal of the current work is to extend the result in [10] – which concerns trees – to a larger class of graphs, identifying new instances of the Braess’ paradox To this end, we will exploit an intriguing connection of Kemeny’s constant with the so-called resistance distance ( known as effective resistance) of graphs, which simulates the behavior of electrical resistance in electric circuits ([9,11,12,13]). Mn(R) denotes the space of real square matrices of order n, and Mn1,n2 (R) denotes the space of real n1 × n2 matrices
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