arXiv subjects

Lorenzo Ciardo

Publications and source records attributed to Lorenzo Ciardo.

20 records · Page 2Linked to original sources

arXiv2020

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

Kemeny's constant $κ(G)$ of a connected graph $G$ is a measure of the expected transit time for the random walk associated with $G$. In the current work, we consider the case when $G$ is a tree, and, in this setting, we provide lower and upper bounds for $κ(G)$ in terms of the order $n$ and diameter $δ$ of $G$ by using two different techniques. The lower bound is given as Kemeny's constant of a particular caterpillar tree and, as a consequence, it is sharp. The upper bound is found via induction, by repeatedly removing pendent vertices from $G$. By considering a specific family of trees - the broom-stars - we show that the upper bound is asymptotically sharp.

math.CO↗

arXiv2019

The Braess' Paradox for Pendant Twins

The 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 pendant vertices. We also investigate the occurrence of this phenomenon in random graphs, showing that almost all connected planar graphs are paradoxical. To establish these results, we make use of a connection between the Kemeny's constant and the resistance distance of graphs.

math.CO↗