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

Abstract

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.

Perron value and moment of rooted trees

“…Example 2. The rooted star may be expressed as the rooted sum of trivial trees: S n = + n−1 i=1 E. Then, Proposition 4.2 yields the bound ρ(S n ) ≤ 1 + n, which -by virtue of the expression (1)is asymptotically sharp as n approaches infinity, while Proposition 4.3 provides the exact value µ(S n ) = n − 1 as found in [6]. 3 allow to obtain results on the Perron value and the moment of a class of rooted trees that will prove useful in Section 5.…”

“…For the case of the random walk on an undirected graph, the corresponding Kemeny's constant provides a measure of the long-run ability of the graph structure to transmit information along its edges; hence, it can be seen as a connectivity notion for the graph. In [6], the authors study this parameter in the context of the random walk on a tree and show that it can be expressed in terms of Kemeny's constant for the random walks on certain subtrees by means of a quantity called moment. The moment µ((T, r)) (or simply µ(T )) of a rooted tree (T, r) is defined by…”

A 1-Separation Formula for the Graph Kemeny Constant and Braess Edges

“…In this section we use Theorem 2.1 to show that the path graph has the largest Kemeny's constant among trees of order n. We first show that the path graph also has the largest moment among trees of order n. While this was shown by Proposition 5.2 of [8], for the sake of completeness we give a different proof using results from this paper. We note that in trees the effective resistance between vertices i, j is the graph distance between i, j (see [1]).…”

“…The notion of the moment is proposed for rooted trees in [8]. We extend this concept to the more general class of simple connected graphs.…”

“…For example, if l = 1 it is known that the edge will be Braess for all nontrivial connected graphs. However, as k increases (8) only guarantees this edge to be Braess for increasingly large graphs. For instance if k = 10 and l = 1 (8) would suggest we need m 1 > 26 which is unnecessarily large.…”