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We derive formulas for the expected hitting times of general random walks on graphs, in terms of voltages, with very elementary electric means. Under this new light we revise bounds and hitting times for birth-and-death Markov chains and for walks on graphs with cutpoints, and give some exact computations on the necklace graph. We also prove Tetali’s formula for hitting times without making use of the reciprocity principle. In fact this principle follows as a corollary of our argument that also yields as corollaries the triangular inequality for effective resistances and the reversibility of the sum of hitting times around a tour.

On a finite connected undirected graph

The hitting time

The beginner’s handbook when studying random walks on graphs from the viewpoint of electric networks is the work of Doyle and Snell [

The commute time

In this paper we want to depart from these expressions of expected hitting times in terms of effective resistances and give them instead in terms of voltages, using as tools the material in Doyle and Snell’s book, which relies on basic facts of electricity such as Ohm’s law and Kirchhoff’s law. We present this alternative expression for two reasons: on the one hand, this representation in terms of voltages leads to very simple proofs of many known results, giving also new insights into new results and computations which are simpler than those involving effective resistances; on the other hand, using Tetali’s formula means accepting the deep result of the reciprocity theorem for electric networks, which is outside the realm of Doyle and Snell. With voltages, we will prove Tetali’s result without making use of the reciprocity principle. In fact this principle will follow as a corollary of our argument that also yields as corollaries the triangular inequality for effective resistances and the reversibility of the sum of hitting times around a tour.

We begin with some basic facts from Doyle and Snell that we state as a lemma. Consider a general random walk on a finite graph, let

One has

One also has the relationship

We refer the reader to the book for details, though we can sketch here the main ideas: for (

Now we can prove our first main result.

Let

For (

We remark that the term

For SRW on

We get (

The bound (

The next result, found in Xu and Yau [

If

The unit current that exits at

Notice that in the previous corollary we have

For SRW, if the sizes of the sets

As in the proof of the previous corollary,

The bound (

For our next application, let us consider now the (possibly infinite) linear graph on the integers with conductances

In the linear graph one has

We use (

The corollary just proved dispatches all hitting times of birth-and-death Markov chains, as discussed in Palacios and Tetali [

Let

By symmetry, all vertices at the same distance from the origin share the same voltage and thus can be shorted. We get thus a (finite) linear graph whose

The bound

Now we look at SRW on graphs with cutpoints. We say that

With the notation of the preceding paragraph one has

In particular, if

All vertices in

Notice that (

Suppose that

Use (

Now we can give a compact expression for the expected hitting times on trees, with a compact proof perhaps simpler than the ones given by Haiyan and Fuji [

Let

In a tree every edge is a cut edge, so we apply (

We consider now the necklace graph, an example of a 3-regular graph with dimension 1 which is useful for providing extreme parameters of random walks on graphs (see [

Necklace graph.

For that purpose we place a battery between vertices 1 and 8 so that

It is worth remarking that, in this graph, as in other symmetric graphs, there is a kind of Noetherian principle, whereby symmetry implies conservation, namely, conservation of the sum of voltages at symmetric vertices:

It is not entirely clear how the computation of

In this section we prove Tetali’s result (

We will show that the reciprocity principle, used in Tetali’s original proof, in fact follows as a corollary of the superposition principle, and we will obtain also as corollaries the triangular inequality for effective resistances and the reversibility of the sum of hitting times around a tour, in a formula that generalizes (

For any

Install three batteries: one between

Since there is no net flow of current through the network, the voltage at every node due to the superposition of the three batteries is the same:

It is clear that

Since the voltage is a nonnegative quantity, (

It is plain to see that in the proof of the above theorem we have only used the superposition principle for electric networks. With the arguments in our proof we can in fact prove the reciprocity principle that was used in Tetali’s original proof of his result (

For any

Translating the words into formulas, we want to show that

The authors declare that there is no conflict of interests regarding the publication of this paper.