Throughout this paper, we will let denote a graph with vertex set V and (undirected) edge set E, and we will let, and vol(G) denote the sum of the degrees of all the vertices of G. A random walk on a graph is a sequence of vertices where is chosen uniformly at random among the neighbors of. Random walks on graphs are well-studied, and considerable literature exists about them. See in particular [2] and [3] for good surveys, especially in the use of spectral techniques in studying random walks on graphs.

The adjacency matrix A of G is the matrix with rows and columns indexed by V given by

It is a well-known fact that the entry of is the number of walks of length k starting at vertex u and ending at vertex v. Define D to be the diagonal matrix with rows and columns indexed by V with, where denotes the degree of vertex v. A random walk on a graph G is a Markov process with transition probability matrix, so

Given any starting probability distribution on the vertex set V, the resulting expected distribution after applying k random walk steps is given by. Here we are considering and as row vectors in.

Note that, in general, P is not symmetric for an irregular graph, but is similar to the symmetric matrix. Thus, the eigenvalues of P are real, and if we order them as, then it is easy to see that with eigenvector, and. By Perron-Frobenius theory, if the matrix P is irreducible, then we have that, and if P is aperiodic, then. The matrix P being irreducible and aperiodic corresponds to the graph G being connected and non-bipartite.

The stationary distribution for a random walk on G is given by

The stationary distribution has the important property the, so that a random walk with initial distribution will stay at at each step. An important fact about the stationary distribution is that if G is a connected graph that is not bipartite, then for any initial distribution on, we have

for all v (see [2] ).

Knowing that a random walk will converge to some stationary distribution, a fundamental question to consider is to determine how quickly the random walk approaches the stationary distribution, or in other words, to determine the mixing rate. In order to make this question precise, we need to consider how to measure the distance between two distribution vectors.

Several measures for defining the mixing rate of a random walk have been given (see [3] ). Classically, the mixing rate is defined in terms of the pointwise distance (see [2] ). That is, the mixing rate is

Note that a small mixing rate corresponds to fast mixing. Alternatively, the mixing rate can be considered in terms of the standard (Euclidean) norm, the relative pointwise distance, the total variation distance, or the -squared distance. In general, these measures can yield different distances, but spectral bounds on the mixing rate are essentially the same for each. See [3] for a detailed comparison of each. For our purposes, we will primarily be concerned with the -squared distance, which will be defined below.

The mixing rate of a random walk is directly related to the eigenvalues of P.

Theorem 1 (Corollary 5.2 of) Let G be a connected non-bipartite graph with transition probability matrix P, and let the eigenvalues of P be. Then the mixing rate is.

Thus, the smaller the eigenvalues of P, the faster the random walk converges to its stationary distribution.

A non-backtracking random walk on G is a sequence of vertices where is chosen randomly among the neighbors of such that for. In other words, a non-backtracking random walk is a random walk in which a step is not allowed to go back to the immediately previous state. A non-back- tracking random walk on a graph is not a Markov chain since, in any given state, we need to remember the previous step in order to take the next step. In order for this to be well-defined, we assume throughout the remainder of the paper that the minimim degree of G is at least 2.

Define to be the transition probability matrix for a k-step non-back- tracking random walk on the vertices. That is is the probability that a non-backtracking random walk starting at vertex u ends up at vertex v after k steps. Note that, where is the transition matrix for an ordinary random walk on G. However, is not simply since a non-backtracking random walk is not a Markov chain.

This process can be turned into a Markov chain, however, by changing the state space from the vertices of the graph to the directed edges of the graph. That is, replace each edge in E with two directed edges (one in each direction). Then the non-back- tracking random walk is a sequence of directed edges where if, and then and. That is, the non-back- tracking condition restricts the walk from moving from an edge to the edge going in the opposite direction. Denote the set of directed edges by. The transition probability matrix for this process we will call. Observe that

Note that is a matrix. Note also that is the transition matrix for a walk with k steps on the directed edges.

Lemma 1. Given any graph G, the matrix as defined above is doubly stochastic.

Proof. Observe first that the rows of the matrix sum to 1, as it is a transition probability matrix. In addition, the columns of sum to 1. To see this, consider the column indexed by the directed edge.

The entry of this column corresponding to the row indexed by is if and if. Since this is equal to. Otherwise, the entry is 0.

Thus the column sum is

as claimed. □

Define the distribution by

where is the vector of length 2m with each entry equal to 1.

Lemma 2. Let be any distribution on the directed edges of G. If the matrix is irreducible and aperiodic, then

as.

Proof. It follows from Lemma 1 that is a stationary distribution for. This follows because, since the columns of sum to 1, we have

Therefore, if the sequence converges, it must converge to. Now, being irreducible and aperiodic are precisely the conditions for this to converge. □

Let f be a probability distribution on the vertices of G. Then f can be turned into a distribution on as follows. Define

Conversely, given a distribution on, define a distribution g on the vertices by

Thus, given any starting distribution on the vertex set of G, we can compute the distribution after k non-backtracking random walk steps as follows. First compute the distribution on the directed edges as above, then compute, then is given by. The following proposition tells us that this converges to the same stationary distribution as an ordinary random walk on a graph.

Theorem 2. Given a graph G and a starting distribution on the vertices of G, define to be the distribution on the vertices after k non-backtracking

random walk steps. Define the distribution by (note that

this is the stationary distribution for an ordinary random walk on G). Then if the matrix is irreducible and aperiodic, then for any starting distribution on V, we have

Proof. As described above, take the distribution on vertices to the corresponding distribution on directed edges. Then define. Then by Lemma 2,

converges to. Now, and observe that

So pulling the distribution on directed edges back to a distribution on the vertices yields. Thus the result follows. □

Definition 1. The -squared distance for measuring convergence of a random walk is defined by

Notice that since,

Theorem 3. Let be the eigenvalues of. Then the convergence rate for the non-backtracking random walk with respect to the -squared distance is bounded above by

Proof. We have

Observe that is orthogonal, which is the eigenvector for, so we see that

Therefore,

□

The transition probability matrix for the walk on directed edges can be thought of as a transition matrix for a random walk on a directed line graph of the graph G. In this way, theory for random walks on directed graphs can be applied to analyze non-back- tracking random walks. Random walks on directed graphs have been studied by Chung in [12] by way of a directed version of the normalized graph Laplacian matrix. In [12] , the Laplacian for a directed graph is defined as follows. Let P be the transition probability matrix for a random walk on the directed graph, and let be its Perron vector, that is,. Then let be the diagonal matrix with the entries of along the diagonal. Then the Laplacian for the directed graph is defined as

This produces a symmetric matrix that thus has real eigenvalues. Those eigenvalues are then related to the convergence rate of a random walk on the directed graph. In particular, the convergence rate is bounded above by, where is the second smallest eigenvalue of (see Theorem 7 of [12] ).

Applying this now to non-backtracking random walks, define as before. Then as seen above, is the constant vector with for all v. Then the directed Laplacian for a non-backtracking walk becomes

Then Theorem 1 of [12] , applied to the matrix as defined, gives the Rayleigh quotient for a function by

From this it is clear that is positive semidefinite with smallest eigenvalue. If are the eigenvalues of, then Theorem 7 from [12] implies that the convergence rate for the corresponding random walk is bounded above by

We remark that for an ordinary random walk on an undirected graph G, the convergence rate is also on the order of, where now denotes the normalized Laplacian of the undirected graph G. Note that

where denotes the Rayleigh quotient with respect to

, and

with given above.

The following result shows that the Laplacian bound does not give an improvement for non-backtracking random walks over ordinary random walks.

Proposition 1. Let G be any graph, and let be the normalized graph Laplacian and the non-backtracking Laplacian defined above. Then we have

Proof. Let be the function orthogonal to D that achieves the minimum in the Rayleigh quotient for. So

Define by. Observe that

So is orthogonal to . Therefore

□

The transition probability matrix defined above is a weighted version of an important matrix that comes up in the study of zeta functions on finite graphs. We define B to be the matrix with rows and columns indexed by the set of directed edges of G as follows.

The matrix B can be thought of as a non-backtracking edge adjacency matrix, and the entries of describe the number of non-backtracking walks of length k from one directed edge to another, in the same way that the entries of powers of the adjacency matrix, , count the number of walks of length k from one vertex to another. The expression is closely related to zeta functions on finite graphs. A result known as Ihara’s Theorem further relates such zeta functions to a determinant expression involving the adjacency matrix. While we will not go into zeta functions on finite graphs in this paper, the following result equivalent to Ihara’s theorem will be of interest to us.

Ihara’s Theorem. For a graph G on n vertices and m edges, let B be the matrix defined above, let A denote the adjacency matrix, D the diagonal degree matrix, and I the identity. Then

We remark that the expression is the characteristic polynomial of B evaluated at, multiplied by the appropriate power of. In this way the complete spectrum of the matrix B is given by the reciprocals of the roots of the polynomial. Numerous proofs of this result exist in the literature [7] - [11] . For completeness, we will include here an elementary proof that uses only basic linear algebra. To the knowledge of the author, this proof is original. To begin, we will need a lemma giving a well-known property of determinants.

Lemma 3. Let M be a matrix, N a matrix, and A an invertible matrix. Then

Proof. Note that

Taking determinants of both sides gives the result. □

Proof of Ihara’s Theorem. Define S to be the matrix

so S is the endpoint incidence operator. Define T to be the matrix given by

so T is the starting point incidence operator. We will also define to be the matrix giving the reversal operator that switches a directed edge with its opposite. That is,

Now, a straightforward computation verifies that

and

Then from Lemma 3 and (1) we obtain

where u is chosen so that the matrix is inverivle.

Observe that, so that, so

. Thus, applying (2) and (3), the above becomes

where the last step is obtained by observing that. This is the desired equality for our choice of u. This is a polynomial of finite degree in u, and there are infinitely many u that make invertible, so the equality holds for all u. □

Applying the results of the previous section to regular graphs yields a different proof of the results from [1] on the mixing rate of non-backtracking random walks on regular graphs.

Let G be a regular graph where each vertex has degree d. Then choosing

for all x yields gives us that is the transition probability matrix for the non-backtracking random walk on G. We remark that, from the previous

section, we have, , , and.

Therefore, the decomposition in (11) becomes

Noting that can be thought of as block diagonal with m blocks of the form

, then taking determinants, we find that

and hence

where the product ranges over all the eigenvalues of the adjacency matrix A for. As remarked previously, the left hand side is the characteristic polynomial of evaluated at, so from this we obtain the spectrum of.

Theorem 5. Let G be a d-regular graph with m edges and n vertices, and let be the transition probability matrix for a non-backtracking random walk as defined above. Then the eigenvalues of are

where ranges over the eigenvalues of the adjacency matrix A, and each have multiplicity.

From this we obtain the result from [1] .

Corollary 1. Let G be a non-bipartite, connected d-regular graph on n vertices for, and let and denote the mixing rates of simple and non-backtracking random walk on G, respectively. Let be the second largest eigenvalue of the adjacency matrix of G in absolute value.

If, then

If and, then

Proof. We remark that the expression is precisely the ex-

pression derived by Alon et al. in [1] for the mixing rate of a non-backtracking random walk on a regular graph, and we may proceed with the analysis of the convergence rate in the same way they do. The convergence rate is given by the second largest eigenvalue of, which will be obtained setting to be the second largest eigenvalue of A. Let be this eigenvalue.

For we have

So. Since is the second largest eigenvalue of the transition

probability matrix P for the usual walk, the first case follows.

For, is complex, and we obtain

so.

We remark that in this case that, a classic result of Nilli [13] related to

the Alon-Boppana Theorem implies that we are never too far below this bound. Indeed, the result states that if G is d-regular with diameter at least, then

. With the restriction that, then the diameter is at

least, and so, and the second case follows. □

A graph G is called -biregular if it is bipartite and each vertex in one part of the bipartition has degree c, and each vertex of the other part has degree d. In the weighted

Ihara’s Theorem, we have, so in the case where G is -biregular, then we have. So since is a multiple of the

identity, as with regular graphs, in the decomposition (11), the term can be taken to the other side of the equation. Note that is diagonal with

if u has degree c, or if

u has degree d. Then is diagonal with entry

or Hence

the decomposition (11) becomes

where is the adjacency matrix of G.

Note that is similar to a block diagonal matrix with blocks of the form

, so taking the determinant above we obtain

so

We will look at the matrix. Suppose the first part in the

bipartition of G has size r, and the second part has size s, where without loss of generality,. By row reduction, this has the same determinant as the matrix

which is

Now, the above determinant is given by the product of the eigenvalues of the matrix. Observe that if is an eigenvalue of the adjacency matrix A, then is an eigenvalue of. Therefore, in all we have

where the product ranges over the largest eigenvalues of A (or in other words, ranges of the s eigenvalues of). Therefore the characteristic polynomial is given by

Thus we can explicitly obtain the eigenvalues of.

Theorem 6. Let G be a -biregular graph, let the part with degree c have size r, and the part with degree d have size s, and assume without loss of generality that. Suppose G has n vertices and m edges. Then the eigenvalues of the non-backtracking transition probability matrix defined above are

as well as the 4 roots of the polynomial

for each value of ranging over the s positive eigenvalues of the adjacency matrix A. These roots are

We can now give a version of Corollary 1 for -biregular graphs.

Corollary 2. Let G be a -biregular graph with. Let be the square of the second largest eigenvalue of the transition probability matrix P for a random walk on G, and let be the square of the second largest modulus of an eigenvalue of. Let be the second largest eigenvalue of the adjacency matrix of G. Then we have the following cases.

If, then

If and both c and d are, then

Proof. We need to compare the eigenvalues of to the eigenvalues of

. Note that for an eigenvalue of A, we have

which implies and. Then observe

so the eigenvalues of P are where ranges over the eigenvalues of A. Note that the largest eigenvalue of A is.

Let equal the expression (12), and consider the following cases.

If, then is real. Direct computation verifies that, evaluating the expression (12) at yields and for in this range. Therefore, in this case the eigenvalue of always has smaller absolute value than the corresponding eigenvalue of P, implying. The lower bound follows from (12) ignoring the square root inside. Thus the first case follows.

If, then is complex, and direct computation shows

so

A version of the Alon-Boppana Theorem exists for -biregular graphs as well, proven by Feng and Li in [14] (see also [15] ).

Theorem 7. [14] Let G be a -biregular graph, and let be the second largest eigenvalue of the adjacency matrix A of G. Then

where the diameter of G is greater than.

Observe that certainly the diameter is at least, so that the condition on the degrees and Theorem 7 imply that

As, so this gives the result for the second case. □