For simple random walk on aN-vertex graph, the mean time to cover all vertices is at leastcN log(N), wherec>0 is an absolute constant. This is deduced from a more general result about stationary finite-state reversible Markov chains. Under weak conditions, the covering time for such processes is at leastc times the covering time for the corresponding i.i.d. process.

Lower bounds for covering times for reversible Markov chains and random walks on graphs

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Reversible Markov Chains and Random Walks on Graphs

The effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights. The effective resistance comes up in many applications and fields in addition to electrical network analysis, including, for example, Markov chains and continuous-time averaging networks. In this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance, i.e., the sum of the resistances between all pairs of nodes. We show that this is a convex optimization problem and can be solved efficiently either numerically or, in some cases, analytically. We show that optimal allocation of the edge weights can reduce the total effective resistance of the graph (compared to uniform weights) by a factor that grows unboundedly with the size of the graph. We show that among all graphs with $n$ nodes, the path has the largest value of optimal total effective resistance and the complete graph has the least.

/pdf/minimizing-effective-resistance-of-a-graph-1rz525m2td.pdf

Minimizing Effective Resistance of a Graph

Functional limit theorems for multitype branching processes and generalized Pólya urns

We present a novel connectivity index for (molecular) graphs, called sum-connectivity index and give several basic properties for this index, especially lower and upper bounds in terms of graph (structural) invariants. It appears that this and the original Randic connectivity index that we call product-connectivity index are highly intercorrelated molecular descriptors, the value of the correlation coefficient being 0.991 for trees representing lower alkanes. We determine the unique tree with fixed numbers of vertices and pendant vertices with the minimum value of the sum-connectivity index, and trees with the minimum, second minimum and third minimum, and the maximum, second maximum and third maximum values of this index. Additionally, we discuss the properties of this novel connectivity index for a class of trees representing acyclic hydrocarbons.

On a novel connectivity index

We find closed-form expressions for the resistance, or Kirchhoff index, of certain connected graphs using Foster's theorems, random walks, and the superposition principle. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 135–140, 2001

Closed‐form formulas for Kirchhoff index

Most earlier mathematical studies of baseball required particular models for advancing runners based on a small set of offensive possibilities. Other efforts considered only teams with players of identical ability. We introduce a Markov chain method that considers teams made up of players with different abilities and which is not restricted to a given model for runner advancement. Our method is limited only by the available data and can use any reasonable deterministic model for runner advancement when sufficiently detailed data are not available. Furthermore, our approach may be adapted to include the effects of pitching and defensive ability in a straightforward way. We apply our method to find optimal batting orders, run distributions per half inning and per game, and the expected number of games a team should win. We also describe the application of our method to test whether a particular trade would benefit a team.

A Markov Chain Approach to Baseball

We study the resistance distance on connected undirected graphs, linking this concept to the fruitful area of random walks on graphs. We provide two short proofs of a general lower bound for the resistance, or Kirchhoff index, of graphs on N vertices, as well as an upper bound and a general formula to compute it exactly, whose complexity is that of inverting an N×N matrix. We argue that the formulas for the resistance in the case of the Platonic solids can be generalized to all distance-transitive graphs. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 29–33, 2001

Resistance distance in graphs and random walks

From an urn containing colored balls, one ball is drawn and replaced by a random number of differently colored balls, with the distribution of the added balls depending only on the color of the ball drawn. Under mild regularity conditions, the proportions of different colors will converge to deterministic limits. Two applications of this standard result are described.

Two Applications of Urn Processes The Fringe Analysis of Search Trees and The Simulation of Quasi-Stationary Distributions of Markov Chains

Building on a probabilistic proof of Foster’s first formula given by Tetali (1994), we prove an elementary identity for the expected hitting times of an ergodic N-state Markov chain which yields as a corollary Foster’s second formula for electrical networks, namely 
$$\sum {R_{ij} } \frac{{C_{iv} C_{vj} }}{{C_v }} = N - 2,$$

 where R

 ij
 is the effective resistance, as computed by means of Ohm’s law, measured across the endpoints of two adjacent edges (i,v) and (i,v), C

 iv
 and C

 iv
 are the conductances of these edges, C

 v
 is the sum of all conductances emanating from the common vertex v, and the sum on the left hand side of (1) is taken over all adjacent edges. We show how to extend Foster’s first and second formulas. As an application, we show how to use a “third formula” to compute the Kirchhoff index of a class of graphs with diameter 3.

José Luis Palacios

Papers

Closed‐form formulas for Kirchhoff index

A Markov Chain Approach to Baseball

Resistance distance in graphs and random walks

Two Applications of Urn Processes The Fringe Analysis of Search Trees and The Simulation of Quasi-Stationary Distributions of Markov Chains

Foster's Formulas via Probability and the Kirchhoff Index