Random walks on edge-transitive graphs (II)

Statistics & Probability Letters 43 (1999) 25–32

Random walks on edge-transitive graphs (II)

Jos�e Luis Palaciosa; ∗, Jos�e Miguel Renomb, Pedro Berrizbeitiac

aDepartamento de C�omputo Cient ��co y Estad ��stica, Universidad Sim�on Bol ��var, Apartado 89,000 Caracas, VenezuelabSchool of Mathematics, Georgia Institute of Technology, Atlanta, Georgia

cDepartamento de Matem�aticas, Universidad Sim�on Bol ��var, Caracas, Venezuela

Received April 1998; received in revised from August 1998

Abstract

We give formulas, in terms of the number of pure k-cycles, for the expected hitting times between vertices at distancesgreater than 1 for random walks on edge-transitive graphs, extending our prior results for neighboring vertices and alsoextending results of Devroye–Sbihi and Biggs concerning distance-regular graphs. We apply these formulas to a class ofCayley graphs and give explicit values for the expected hitting times. c© 1999 Elsevier Science B.V. All rights reserved

Keywords: Edge-transitive graphs; Hitting times; Cayley graphs

1. Introduction

A simple random walk on a �nite connected undirected graph, G = (V; E) is the Markov chain Xn; n¿0,that from its current vertex v jumps to one of the d(v) neighboring vertices with uniform probability. Thehitting time (or �rst passage time) Tv of the vertex v is the minimum number of steps the random walk takesto reach that vertex: Tv = inf{n¿0: Xn = v}: The expected value of Tv when the walk starts at the vertex wis denoted by EwTv.A graph is vertex-transitive if for every pair of vertices there is an automorphism mapping one vertex onto

the other; a graph is edge-transitive if for every pair of edges, considered as undirected edges, there is anautomorphism mapping one edge onto the other; a graph is arc-transitive if the last sentence in italics isreplaced by “considered as directed edges”. The distinction between these two notions is not entirely trivial:whereas it is plain to see that if a graph G is arc-transitive then it is both vertex- and edge-transitive, thereare some elaborate examples of graphs which are both vertex- and edge-transitive but not arc-transitive (seeBouwer, 1970).It is not di�cult to prove that if all expected hitting times between neighboring vertices are equal, this

common value must be |V | − 1. In a previous paper, Palacios and Renom (1998), we gave weak conditionsunder which that occurs, namely:

∗ Corresponding author.

0167-7152/99/$ – see front matter c© 1999 Elsevier Science B.V. All rights reservedPII: S0167 -7152(98)00241 -7

26 J.L. Palacios et al. / Statistics & Probability Letters 43 (1999) 25–32

Theorem 1.1. If G is edge-transitive and regular and i∼ j then

EjTi ≡ E0T (1) = |V | − 1:

We also proved the following fact used below:

Theorem 1.2. If G is edge-transitive, then EaTb = EbTa whenever the distance between a and b, d(a; b), iseven.

In this paper we will pursue the idea of providing weak conditions under which expected hitting timesbetween vertices at distances greater than 1 are constant over all pairs of vertices in the graph at the givendistance. We will express these expected hitting times in terms of the number of pure k-cycles in the graph,i.e., cycles such that no edges other than the ones forming the cycle exist among the vertices involved inthe cycle. For the sake of simplicity we will drop the word “pure” from now on. All the graphs underconsideration will be edge-transitive and regular with common degree d.

2. Expected hitting times EaTb when d(a; b) = 2

Theorem 2.1. Let G be a regular edge-transitive graph, with common degree d. Then the number of N -cyclesper edge is a constant aN over all edges, the number of N -cycles per vertex is also a constant tN over allvertices, and if we denote by TN the total number of N -cycles in G, then we have (i) aNd = 2tN and(ii) tN |V |= NTN .

Proof. That aN is constant over all edges is an immediate consequence of the edge-transitivity. That tN isalso a constant follows from counting on any vertex the number of N -cycles in this way: on any of the dedges stemming from the given vertex there are aN N -cycles, so the total number of N -cycles on the vertex isaNd except for the fact that every N -cycle is counted twice, and we get (i) as a byproduct. Part (ii) followsfrom considering for every vertex the number of N -cycles per vertex, leading to an N -fold count of the totalnumber of N -cycles.

We will say that a graph G satis�es condition C2 if the following requirement is met:For every four vertices a; b; x; y with d(a; b)=d(x; y)=2 there is an automorphism �2 with �2{a; b}={x; y}.It is worth noticing that this hypothesis resembles that of edge-transitivity (undirected edges) and not of

arc-transitivity; it is also weaker than 2-transitivity as de�ned in Buckley and Harary (1990).Now we can prove

Theorem 2.2. If G is a regular edge-transitive graph satisfying condition C2, then EaTb is a constant,denoted by E0T (2) over all a; b with d(a; b) = 2, and

E0T (2) = |V | − 1 + |V | − 1− dd− a3 − 1 = E0T (1) +

|V | − 1− dd− a3 − 1 ; (1)

where a3 is the number of triangles per edge, and d is the common degree of all vertices.

Proof. The existence of �2 guarantees that for every four vertices a; b; x; y with d(a; b) = d(x; y) = 2 eitherEaTb=ExTy or EaTb=EyTx. But this fact, together with Theorem 1.2, implies that EaTb is constant over all a; bwith d(a; b)=2. Also, because of Theorem 1.1, we can assume that if d(a; b)=1 then EaTb ≡ E0T (1)=|V |−1.

J.L. Palacios et al. / Statistics & Probability Letters 43 (1999) 25–32 27

Conditioning on the second jump of the walk, we get

|V |= E0T0 = dd2(2) +

2t3d2(2 + E0T (1)) +

d2 − 2t3 − dd2

(2 + E0T (2)):

The �rst summand of the right-hand side corresponds to excursions where the walk goes back and forth alongthe same edge, the second where the walk visits two edges of a triangle – there are two possible ways to dothat – and the third considers all other ways. Using Theorem 2:1(i), replacing E0T (1) = |V | − 1, and solvingfor E0T (2) gives the desired result.

Example 1. In case there are no triangles in the graph, the formula reduces to E0T (2) = d=(d− 1)(|V | − 2).This is the case of the d-cube, where E0T (2)=d=(d−1)(2d−2), and of the dodecahedron, where E0T (2)=27.

Remarks. Devroye and Sbihi (1990) and Biggs (1993) show that if G is a distance-regular graph, then theexpected hitting times between vertices depend only on the distance d between these vertices, and in fact,they give the formulas

E0T (i) = di−1∑j=0

(kj+1 + · · ·+ kD)bjkj

;

where E0T (i) is the common value for all expected hitting times between vertices at distance i, D is thediameter of the graph, k0 = 1,

bi = |{x∈V : d(x; v) = 1; d(x; w) = i + 1; d(v; w) = i}|for any pair v; w and

ki = |{x∈V : d(x; v) = i}|for any v. The ki’s satisfy the relation

n= 1 + k1 + k2 + · · ·+ kD:In particular, for i = 2 their formula reads

E0T (2) = |V | − 1 + |V | − 1− db1

;

so comparing with our formula, we should have

b1 = d− a3 − 1;which is indeed true, because if v and w are neighbors,

{z: d(z; v) = 1}= {w} ∪ {z: d(z; v) = 1; (z; w) = 1} ∪ {z: d(v; w) = 1; d(z; w) = 2};and replacing the cardinalities of these sets we get the equation d= 1 + a3 + b1.We prefer our expression because we can apply formulas which count the number of triangles in certain

Cayley graphs, as we shall see in the next section.It is easy to construct small examples of arc-transitive graphs where E0T (2) is not constant, and thus by

the results of Devroye and Sbihi (1990) and Biggs (1993), these cannot be distance-transitive. For instance,if in the 3-cube we de�ne the vertices of a new graph G as the midpoints of each edge, and on each faceof the cube we unite the four midpoints by means of a 4-cycle (i.e., we draw an inscribed square on eachface of the cube), then G is a 12-vertex arc-transitive graph (the latter easily veri�ed with rotations of theoriginal cube) where the expected hitting times between vertices at distance two are either 15 or 16. This is amore economical example of an arc-transitive non-distance-transitive graph than the one proposed in Buckley


and Harary (1990, p. 167), the generalized Petersen graph on 16 vertices. The latter, by the way, is a graphwhere expected hitting times are constant over pairs of vertices such that the distances separating them areeither 1 or 2, but not 3.

3. Cayley graphs

Given a group (G; ·) and a subset S of G such that S=S−1, the Cayley graph, Cay(G; S), is de�ned as thegraph whose vertex set V is the set of elements of G, and such that v and w are neighbors (denoted v∼w) i�vw−1∈ S. It is easy to see that every Cayley graph is vertex-transitive, though in general it may not possessany other symmetry properties. We will focus our attention into a subclass of these, the unitary Cayley graphs,de�ned as Cay(Zn; Un), where (Zn;+; ·) is the commutative ring of the integers modulo n, and Un is the set ofunits, i.e., those elements of Zn which possess a multiplicative inverse; these units turn out to be the non-zerointegers less than n which are relatively prime with n. In this context we can prove the following.

Theorem 3.1. In Cay(Zn; Un), the function �(x) = wx + z, where w∈Un and z ∈Zn are �xed, is a graphautomorphism.

Proof. The fact that w is a unit is crucial to prove that � is a bijection. All that remains to prove isthat � preserves adjacencies; if x∼y, then x − y∈Un and thus �(x) − �(y) = w(x − y)∈Un; i.e.,�(x)∼�(y).

Theorem 3.2. All Cay(Zn; Un) are arc-transitive.

Proof. Let x; y; a; b be vertices such that x∼y and a∼ b. Consider the function�(v) = (b− a)(x − y)−1(x − v) + a:

Then according to Theorem 3.1, � is a graph automorphism, because (b − a)(x − y)−1∈Un; also �(x) = a,�(y) = b.

Theorem 3.3. Let x; y∈Zn, x 6∼y. Let p be the smallest prime divisor of n such that x ≡ y(p). Then1. If p= 2 then d(x; y) = 2.2. If p¿ 2 then: (i) if n is odd then d(x; y) = 2; and (ii) if n is even then d(x; y) = 3.

Proof. (1) We use the Chinese Remainder Theorem (CRT) to �nd an integer z such that z 6≡ x(2) and suchthat for each odd prime divisor p of n, z satis�es z 6≡ x and z 6≡y. This is possible since p¿3. It followsthat such z satis�es z∼ x and z∼y so that d(x; y) = 2.(2) (i) Another application of the CRT gives us a z such that z 6≡ x(p), z 6≡y(p) for each prime divisor

p of n. As before z∼ x, z∼y, implying d(x; y) = 2. (ii) Again, use the CRT to �nd z such that z∼ x. Thehypothesis implies that z ≡ y(2). In particular, z 6∼y. By case 1 applied to y and z we get d(y; z) = 2, sothat d(x; y) = 3.

As a corollary we obtain the following.

Theorem 3.4. (i) If n is odd and composite, or a power of 2, then the diameter of Cay(Zn; Un) is 2. (ii) ifn= 2lm; l¿1; m odd, m¿ 1, then the diameter of Cay(Zn; Un) is 3.

The proof of the following theorem can be found in Berrizbeitia and Giudici (1996).


Theorem 3.5. For any Cay(Zn; Un), the number of triangles in the graph, T3(n), is given by the formula

T3(n) =16n3

∏p|n(1− (1=p))(1− (2=p)); (2)

where the product extends over all prime divisors p of n.

Now the idea is to insert Eq. (2) into our formula (1). In order to do that, we need to check whether aCay(Zn; Un) satis�es condition C2. This is not always the case: for instance, in Cay(Z15; U15), whose diameteris 2, there are two distinct values for EaTb when d(a; b) = 2. However, we can prove

Theorem 3.6. Cay(Zn; Un) satis�es condition C2 if (i) n= 2p, p¿ 2 prime, or (ii) n= pk , p prime.

Proof of (i). (1) We �rst prove that if y ≡ x+p(n) then d(x; y)¿ 2. Let z be such that z∼ x. Then z− x isodd, so z− (x+p) is even, which implies z 6∼y. Hence d(x; y)¿ 2. (Actually, we can show that d(x; y) = 3but this is irrelevant to our purpose.)(2) Next we show that if d(x; y) = 2, then x ≡ y(2) and x 6≡y(p). Indeed, if d(x; y) = 2 then x and y are

distinct, thus we cannot have both x ≡ y(2) and x ≡ y(p). On the other hand x 6∼y so we cannot have bothx 6≡y(2) and x 6≡y(p). Finally, we cannot have x 6≡y(2) and x ≡ y(p) since this would imply y ≡ x+p(n)which would contradict (1). It follows that x ≡ y(2) and x 6≡y(p) as desired.(3) Suppose d(a; b) = 2 and d(c; d) = 2. We will prove that there is w, a unit mod n, and x∈Zn such that

�w;x(a) = c and �w;x(b) = d. By (2) we know that a− b 6≡ 0(p) and c − d 6≡ 0(p). We use the CRT to �ndw satisfying

w ≡ 1(2) (3)

andw(b− a) ≡ (d− c)(p): (4)

Such w is a unit mod n= 2p.Let x= c−wa. Then �w;x(a)=wa+ x=wa+ c−wa= c and �w;x(b)=wb+ x=wb+ c−wa=w(b− a)+ c.Using (4) we get �w;x(b) ≡ (d− c) + c(p) ≡ d(p).By (2) we know b − a ≡ 0(2) and c ≡ d(2), so �w;x(b) = w(b − a) + c ≡ c(2) ≡ d(2). It follows that

�w;x(b) ≡ d(n) and the proof of (i) is complete.Proof of (ii). Note that d(x; y) = 2 if and only if x ≡ y(p). This follows from the fact that x − y is a unitmodpk , i.e., x − y 6≡ 0(p). It follows that if d(x; y) = 2 then x and y have the same set of neighbors andthus the transposition (xy) is a graph automorphism.Now suppose d(a; b) = d(c; d) = 2, let x = c − a and let �x be the translation by x (�x(y) = y + x). Then

�x is a graph automorphism and �x(a) = c. On the other hand �x(b) = b+ x ≡ a+ x ≡ d(p). It follows thatd(�x(b); d) = 2.Let � be the composition of �x with the transposition (�x(b); d). Then � is a graph automorphism with

�(a) = c and �(b) = d, �nishing the proof of (ii).

Example 2. In the case Cay(Z2p; U2p), the graphs are bipartite, thus they have no triangles, and example 1applies: E0T2 = 2(p− 1)2=(p− 2).In the case Cay(Zpk ; Upk ), we compute:

T3 ≡ T3(pk) = 16p3k(1− 1

p

)(1− 2

p

)=16p3k−2(p− 1)(p− 2);

a3 =6T3d|V | = p

k−1(p− 2)


and using formula (1) we get

E0T2 = pk − 1 + pk−1 − 1− (pk − pk−1)pk − pk−1 − pk−1(p− 2) = p

k = |V |:

This should not be surprising, since Cay(Zpk ; Upk ) has diameter 2 (see below for another argument), and thuswe know all its expected hitting times.

4. Expected hitting times EaTb when d(a; b) = 3

Along the lines of Theorem 2.2, we can give closed form formulas for expected hitting times betweenvertices whose distance is 3, in case the graph is edge-transitive, regular, and in addition to satisfying conditionC2, it satis�es:Condition C3: for every four vertices x; y; a; b such that d(x; y) = d(a; b) = 3 there is an automorphism �3

such that �3(x) = a and �3(y) = b.Now we can prove:

Theorem 4.1. If G is a graph satisfying the hypotheses of Theorem 2.2 and condition C3, then the value ofEaTb whenever d(a; b) = 3 is a constant, denoted E0T (3), and the value of this constant is

E0T (3) =d2(|V | − 3) + d− E0T (1)(d− 1 + a4)− a5E0T (2)

(d− 1)2 − a5 − a4 − a3 ; (5)

where a3; a4; a5 denote the number of, respectively, triangles, squares and pentagons per edge, and d is thecommon degree of all vertices.

Proof. Conditioning on a return to the starting point in either exactly two steps or in three or more stepswe get

E0T0 =|V |

=2d2

d3+d(d− 1)d3

(3 + E0T (1)) +2t3d3(3) +

2t4d3(3 + E0T (1))

+2t5d3(3 + E0T (2)) +

d3 − 2t5 − 2t4 − 2t3 − d(d− 1)− d2d3

(3 + E0T (3)):

The �rst summand corresponds to returning to the starting point in two steps; the others correspond to walkingthree steps through, respectively, an L (i.e., walking two consecutive edges and then the third step being areversal of the second), a triangle, a square, a pentagon and anything else. Solving for E0T (3) we get

E0T (3) =d3(|V | − 3) + d2 − E0T (1)(d2 − d+ 2t4)− 2t5E0T (2)

d3 − 2t5 − 2t4 − 2t3 − 2d2 + d :

An application of Theorem 2.1(i) �nishes the proof.

Example 3. If G contains neither triangles nor pentagons then

E0T (3) = |V | − 1 + d(|V | − 2d)(d− 1)2 − a4 : (6)


This is the case of the d-cube (where t4 =(d2

)and a4 = d − 1), and in general, of any bipartite

graph.Berrizbeitia and Giudici (1996) give a closed form formula for the total number of squares T4(n) in

Cay(Zn; Un) which is considerably more involved than the one for triangles: it contains six summands similarto Eq. (2); however, when n is even the formula reduces to

T4(n) =18n�(n)

n2∏

p|n(1− (3=p) + (3=p2))− 2�(n) + 1

; (7)

where � is Euler’s function.The graphs Cay(Z2p; U2p) are bipartite, and therefore, formulas (6) and (7) apply. In fact, after some algebra

we get for these graphs

T4(2p) =p(p− 1)(p− 2)(p− 3)

4;

and inserting into (6) we get

E0T (3) =p(2p− 3)p− 2

and since we know that the diameter of Cay(Z2p; U2p) = 3, we have all the expected hitting times for thesegraphs, namely: E0T (1) = 2p− 1, E0T (2) = 2(p− 1)2=(p− 2), E0T (3) = 1 + E0T2 = p(2p− 3)=(p− 2).To justify the use of (5) in computing E0T (3) we should have veri�ed that condition C3 holds. We can

get around that with this argument: conditioning on the last jump of the walk we have

E0T (3) = E0Tp = 1 +1d

∑x∼p

E0Tx:

But since all x∼p must be at distance two from 0, all d summands in the summation are equal to E0T (2)and we get

E0T (3) = 1 + E0T (2):

A similar argument would have given us the conclusion of Example 2.

5. Final remarks

As mentioned before, Devroye and Sbihi (1990) and Biggs (1993) showed that if G is a distance-regulargraph, then the expected hitting times between any pair of vertices in the graph depend only on the distancebetween these vertices. We deal with a related but di�erent problem: we seek the weakest conditions underwhich we can guarantee that all expected hitting times between vertices at distance less than or equal to kdepend only on the distance – though the expected hitting times may not be constant over vertices at distancesgreater than k – and �nd that it is su�cient to impose conditions C1; C2; : : : ; Ck , where C1 asks that the graphbe regular and edge-transitive, and C2j and C2j+1 imitate conditions C2 and C3, mutatis mutandis, given above.Notice that because of Theorem 1.2, conditions C2j are weaker (unordered) than conditions C2j+1.We gave formulas for those common expected hitting times in terms of pure k-cycles for cases where

k64. The generalization to larger values of k is obvious but more cumbersome, and its usefulness dependson having closed form formulas for the number of pure pentagons, hexagons, etc., which are somewhat scarce(see Berrizbeitia and Giudici, 1996).


References

Berrizbeitia, P., Giudici, R.E., 1996. Counting pure k-cycles in sequences of Cayley graphs. Discrete Math. 149, 11–18.Biggs, N.L., 1993. Potential theory on distance-regular graphs. Cambridge conference in honour of Paul Erd�os.Buckley, F., Harary, F., 1990. Distance in graphs. Addison-Wesley, New York.Bouwer, I.Z., 1970. Vertex and edge transitive, but not 1-transitive, graphs. Can. Math. Bull. 13, 231–237.Devroye, L., Sbihi, A., 1990. Random walks on highly symmetric graphs. J. Theoret. Probab. 4, 497–514.Palacios, J.L., Renom, J.M., 1998. Random walks on edge transitive graphs. Statist. Probab. Lett. 37, 29–34.

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Random walks on edge-transitive graphs (II)