On the spectra and spectral radii of token graphs

Let $G$ be a graph on $n$ vertices. The $k$-token graph (or symmetric $k$-th power) of $G$, denoted by $F_k(G)$ has as vertices the ${n\choose k}$ $k$-subsets of vertices from $G$, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in $G$. In particular, $F_k(K_n)$ is the Johnson graph $J(n,k)$, which is a distance-regular graph used in coding theory. In this paper, we present some results concerning the (adjacency and Laplacian) spectrum of $F_k(G)$ in terms of the spectrum of $G$. For instance, when $G$ is walk-regular, an exact value for the spectral radius $\rho$ (or maximum eigenvalue) of $F_k(G)$ is obtained. When $G$ is distance-regular, other eigenvalues of its $2$-token graph are derived using the theory of equitable partitions. A generalization of Aldous' spectral gap conjecture (which is now a theorem) is proposed.


Introduction
For a (simple and connected) graph G = (V, E) with adjacency matrix A, its local spectrum at vertex u plays a role similar to the (standard adjacency) spectrum when the graph is 'seen' from vertex u.For instance, the local spectra of G, for every u ∈ V , were used by Fiol and Garriga [15] to prove the so-called 'spectral excess theorem', which gives a quasi spectral characterization of distance-regular graphs.In turn, this result was the crucial tool for the discovery, by van Dam and Koolen [11], of the first known family of non-vertex-transitive distance-regular graphs with unbounded diameter.Besides, Fiol, Garriga, and Yebra [17] used the local spectra to define the local predistance polynomials, which were used to characterize a general kind of local distance-regularity (intended for not necessarily regular graphs).
One of the most important parameters in spectral graph theory is the index or spectral radius of a graph, which corresponds to the largest eigenvalue of its adjacency matrix.This parameter has special relevance in the study of many integer-valued graph invariants, such as the diameter, the radius, the domination number, the matching number, the clique number, the independence number, the chromatic number, or the sequence of vertex degrees.In turn, this leads to studying the structure of graphs having an extremal spectral radius and fixed values of some of such parameters.See Brualdi, Carmona, Van den Driessche, Kirkland, and Stevanović [5,Cap. 3].
In this work, we use some information given by the local spectra to obtain new results about the spectral radius of an ample family of graphs, which are known as token graphs or symmetric k-th powers, defined as follows.For a given integer k, with 1 ≤ k ≤ n (where n is the order of G), the k-token graph F k (G) of G is the graph whose vertex set V (F k (G)) consists of the n k k-subsets of vertices of G, and two vertices A and B of F k (G) are adjacent whenever their symmetric difference A △ B is a pair {a, b} such that a ∈ A, b ∈ B, and {a, b} ∈ E(G).In Figure 1, we show the 2-token graph of the cycle C 9 on 9 vertices.In particular, if k = 1, then F 1 (G) ∼ = G; and if G is the complete graph K n , then F k (K n ) ∼ = J(n, k), where J(n, k) denotes the Johnson graph, see Fabila-Monroy, Flores-Peñaloza, Huemer, Hurtado, Urrutia, and Wood [13].
The name 'token graph' also comes from the observation in [13], that vertices of F k (G) correspond to configurations of k indistinguishable tokens placed at distinct vertices of G, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex.Such graphs are also called symmetric k-th power of a graph in Audenaert, Godsil, Royle, and Rudolph [2]; and n-tuple vertex graphs in Alavi, Lick, and Liu [1].They have applications in physics; a connection between symmetric powers of graphs and the exchange of Hamiltonian operators in quantum mechanics is given in [2].Our interest is in relation to the graph isomorphism problem.It is well known that there are cospectral non-isomorphic graphs, where often the spectrum of the adjacency matrix of a graph is used.For instance, Rudolph [27] showed that there are cospectral non-isomorphic graphs that can be distinguished by the adjacency spectra of their 2-token graphs, and he also gave an example for the Laplacian spectrum.Audenaert, Godsil, Royle, and Rudolph [2] also proved that 2-token graphs of strongly regular graphs with the same parameters are cospectral and also derived bounds on the (adjacency and Laplacian) eigenvalues of F 2 (G) for general graphs.For more information, see again [2] or [13].The vertices inducing a circumference (in dashed line) of radius r ℓ , with ℓ = 1, 2, 3, 4 and What can be said about the spectrum of F k (G)?The three main results that we want to recall are the following.
Theorem 1.2 (Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, Zaragoza Martínez [9]).For any graph G on n vertices, the Laplacian spectrum of its h-token graph is contained in the Laplacian spectrum of its k-token graph for every and both bounds are tight.
In fact, the lower bound in (1) was first proved by Dalfó, Fiol, and Messegué in [26,.
In this paper, we mainly derive new results about the spectral radius of token graphs, and it is organized as follows.The next section begins with some basic concepts, definitions, and results.More precisely, we recall some known results about the local spectra and derive the basic tools for computing the spectral radius.In Section 3, we introduce the new concepts of k-algebraic connectivity and k-spectral radius.There, we study some of their properties and propose a generalization of Aldou's spectral gap conjecture, already a theorem (see Caputo, Liggett, and Richthammer [7]).In Section 4, we give both lower and upper bounds for the spectral radius of a token graph, which are shown to be asymptotically tight.In the same section, we present some infinite families in which the exact values of the spectral radius are obtained.Finally, in the last section, we deal with the case of distance-regular and strongly regular graphs, where two results are presented in the form of Audenaert, Godsil, Royle, and Rudolph's result [2], and Lew's result [23].

Graphs and their spectra
Let G be a (simple and connected) graph with vertex set V (G) = {1, 2, . . ., n} and edge set E(G).Let G have adjacency matrix A, and spectrum Thus, by the Perron-Frobenius theorem, G has spectral radius ρ(G) = θ 0 .
Let L = D − A be the Laplacian matrix of G, with eigenvalues Recall that λ 2 is the algebraic connectivity, and D is the diagonal matrix whose diagonal entries are the vertex degrees of G.

The local spectra of a graph
Let G have different eigenvalues θ 0 > • • • > θ d , with respective multiplicities m 0 , . . ., m d .If U i is the n × m i matrix whose columns are the orthonormal eigenvectors of θ i , the matrix i , for i = 0, 1, . . ., d, is the (principal) idempotent of A and represents the orthogonal projection of R n onto the eigenspace Ker(A − θ i I).The (u-)local multiplicities of the eigenvalue θ i are defined as where e u is the n-dimensional vector with a 1 in the u-th entry and zeros elsewhere.In particular, m u (θ 0 ) = v 2 u > 0, where v is the corresponding normalized Perron eigenvector, that is, the eigenvector that can be chosen to have strictly positive components.Although the local multiplicities are, of course, not necessarily integers, they have nice properties when the graph is studied from a vertex, so justifying their name.Thus, they satisfy uu of closed walks of length ℓ rooted at vertex u can be computed as (see Fiol, Garriga, and Yebra [17, Corollary 2.2]).By picking up the eigenvalues θ i with non-null local multiplicities, The eccentricity of a vertex u satisfies an upper bound similar to that satisfied by the diameter of G in terms of its distinct eigenvalues.More precisely, In coding theory, d u corresponds to the so-called 'dual degree' of the trivial code {u}.For more information, see Fiol, Garriga, and Yebra [17].
We use the following lemma to prove the results of Section 4. Notice that this is just a reformulation of the power method in terms of the number of walks given by (2).
be the number of ℓ-walks starting from (any fixed) vertex u, and let a (ℓ) uu be the number of closed ℓ-walks rooted at u.Then, where ' sup' denotes the supremum.

Regular partitions and their spectra
Dealing with a regular partition is a useful method in graph theory, as it allows us to obtain some information about a graph considering a smaller version of it (the so-called 'quotient graph').Besides, if we consider a graph G and a group of automorphisms Γ, then a partition of the vertices of the graph into orbits by Γ is a regular partition.
Let G = (V, E) be a graph with vertex set V = V (G), adjacency matrix A, and Laplacian matrix L. A partition π of its vertex set V into r cells C 1 , C 2 , . . ., C r is called regular (or equitable) whenever, for any i, j = 1, . . ., r, the intersection numbers b ij (u) = |G(u)∩C j | (where u ∈ C i , and G(u) is the set of vertices that are neighbors of the vertex u) do not depend on the vertex u but only on the cells C i and C j .In this case, such numbers are simply written as b ij , and the r × r matrices are, respectively, referred to as the quotient matrix and quotient Laplacian matrix of G with respect to π.In turn, these matrices correspond to the quotient (weighted) directed graph G/π, whose vertices representing the r cells, and there is an arc with weight b ij from vertex C i to vertex C j if and only if b ij = 0. Of course, if b ii > 0, for some i = 1, . . ., r, the quotient graph (or digraph) G/π has loops.Given a partition π of V with r cells, let S be the characteristic matrix of π, that is, the n × r times matrix whose columns are the characteristic vectors of the cells of π.Then, π is a regular partition if and only if AS = SQ A or LS = SQ L .Moreover, Thus, there is a strong analogy with similar results satisfied by the Laplacian matrices of the h-token graph and k-token graph of G for h ≤ k.More precisely, we have the following statements (Those numbered with 1 can be found in Godsil [19,20].The statements in 2 are derived similarly to those in number 1 when we use the quotient Laplacian matrix, defined in (4), instead of the standard quotient matrix.Finally, the statements in 3 follow from the results given by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete in [9].): 1.If π is a regular partition with characteristic matrix S and quotient matrix (c) The column space (and its orthogonal complement) of S is A-invariant.
(d) The characteristic polynomial of Q A divides the characteristic polynomial of A. Thus, sp Q A ⊆ sp A.
(e) If v is an eigenvector of Q A with eigenvalue λ, then Sv is an eigenvector of A with the same eigenvalue.(The eigenvector v of Q A 'lifts' to an eigenvector of A.) (f) If v is an eigenvector of A with eigenvalue λ and S ⊤ v = 0, then S ⊤ v is an eigenvector of Q A with the same eigenvalue.
2. If π is a regular partition with characteristic matrix S and quotient Laplacian matrix (c) The column space (and its orthogonal complement) of S is L-invariant.
(d) The characteristic polynomial of Q L divides the characteristic polynomial of L. Thus, sp Q L ⊆ sp L.
(e) If v is an eigenvector of Q L with eigenvalue λ, then Sv is an eigenvector of L with the same eigenvalue.
(f) If v is an eigenvector of L with eigenvalue λ and S ⊤ v = 0, then S ⊤ v is an eigenvector of Q L with the same eigenvalue.

Let
) be, respectively, the Laplacian matrices of the h-token and k-token graphs of G, for h ≤ k, and let S b be the (k, h)-binomial matrix.This is a n k × n h matrix whose rows are indexed by the k-subsets of A ⊂ [n], and its columns are indexed by the h-subsets of X ⊂ [n], with entries

Walk-regular graphs
Let a (ℓ) u denote the number of closed walks of length ℓ rooted at vertex u, that is, a If these numbers only depend on ℓ, for each ℓ ≥ 0, then G is called walk-regular, a concept introduced by Godsil and McKay in [21].
Notice that, as a (2) u = δ u , the degree of vertex u, a walk-regular graph is necessarily regular.
Moreover, a graph G is called spectrally regular when all vertices have the same local spectrum: sp u G = sp v G for any u, v ∈ V .The following result (in Delorme and Tillich [12], Fiol and Garriga [16], and also Godsil and McKay [21]) provide some characterizations of such graphs.Lemma 2.2 ([12], [16], [21]).Let G = (V, E) be a graph.The following statements are equivalent.
(i) G is walk-regular.
(ii) G is spectrally regular.
(iii) The spectra of the vertex-deleted subgraphs are all equal: sp (G \ u) = sp (G \ v) for any u, v ∈ V .
3 The k-algebraic connectivity and k-spectral radius In this section, we always consider the Laplacian spectrum.Let G be a graph on n vertices, and Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martínez [9], it is known that the Laplacian spectra of the token graphs of G satisfy Let denote α(G) and ρ(G) the algebraic connectivity (see Fiedler [18]) and the spectral radius of a graph G, respectively.Then, from (5), we have The concepts of algebraic connectivity and spectral radius, together with ( 5)-( 7), suggest the following definitions.Notice that, since n k > n k−1 for 1 ≤ k ≤ ⌊n/2⌋, the parameters α k and ρ k always exist.
, and so on.
The equalities in (iii) come from the fact that the Johnson graph J(n, k) has different Laplacian eigenvalues From what is known about token graphs, we can suggest some conjectures and state some results, as follows.
Notice that, from (7), if this conjecture holds, then ρ k (G) = ρ(F k (G)) for any k ≤ n/2.Lemma 3.4.For any graph G and its complementary graph G, the k-algebraic connectivity and k-spectral radius of G satisfy Proof.It was proved in [9] that each eigenvalue of J(n, k) is the sum of one eigenvalue of F k (G) and one eigenvalue of F k (G).Then, from (5), the eigenvalues of sp F k (G) \ sp F k−1 (G) and sp F k (G) \ sp F k−1 (G) must be paired in such a way that their sums are all equal to the 'new' eigenvalue of sp In particular, this happens with the minimum eigenvalue of sp Moreover, α(G) = n − ρ(G), as it is well known.Corollary 3.5.For any graph G on n vertices, From Lemma 3.4 and Proposition 5.2, we get the following result, which will be proved in Section 5.

The spectral radius of token graphs
In contrast with the previous section, in this section, we always consider the spectral radius of the adjacency matrix of a (connected) graph.Consider a graph G with spectral radius ρ(G) and vertex-connectivity κ (that is, the minimum number of vertices whose suppression either disconnects the graph or results in a singleton).By taking the spectral radii of its U -deleted subgraphs, with U ⊂ V and |U | = k < κ, we define the following two parameters: If G is distance-regular with degree δ, it is known that it has vertex-connectivity κ(G) = δ (see Brouwer and Koolen [4]).Moreover, Dalfó, Van Dam, and Fiol [8] showed that sp(G \ U ) only depends on the distances in G between the vertices of U .Thus, for every k ≤ δ − 1, the computation of ρ k M (G) and ρ k m (G) can be drastically reduced by considering only the subsets U with different 'distance-pattern' between vertices.For instance, if G has diameter D, In general, by using interlacing (see Haemers [22] or Fiol [14]), we have the following result.
Lemma 4.1.Let G be a graph with n vertices, vertex-connectivity κ, and eigenvalues From the above results, Lemma 2.1, and the bounds for the spectral radius of graph perturbations obtained in Dalfó, Garriga, and Fiol [10] and Nikiforov [24], we obtain the following main result.(i) The spectral radius of the k-token graph F k (G) satisfies (ii) If G is a graph of order n and diameter D, the spectral radius of the k-token graph Proof.(i) To prove the upper bound in (10) the key idea is the following: Given some integer ℓ large enough, every ℓ-walk W in F k (G) from a given vertex A can be seen as ℓ i -walks in G for i = 1, . . ., k such that k i=1 ℓ i = ℓ.Each step of W corresponds to one step given by a token, say '1', whereas the remaining tokens are 'still'.Thus, the move of token '1' is done in G \ U for some vertex subset U , with |U | = k − 1, and the same holds for every token.Thus, for ℓ great enough, W induces a number of walks of order ℓ ℓ 1 ,ℓ 2 ,...,ℓ k since, in each step, one of the k tokens can be moved.For large ℓ, the number of  1: Spectral radii of the 2-token graphs of the cycles C n with respect to spectral radii of the paths graphs P n−1 .
5 The case of distance-regular graphs In this section, we adopt the terminology of Brouwer, Cohen, and Neumaier [3] for distanceregular graphs.Furthermore, since we examine both the adjacency and Laplacian spectra, we denote their respective spectral radii as ρ A and ρ L .In the following result, consider that G is a distance-regular graph with degree δ = b 0 , diameter d, intersection array or intersection matrix where a i = δ − b i − c i , for i = 1, . . ., d.
Lemma 5.1.Let F 2 (G) be the 2-token graph of a distance-regular graph G with degree δ = b 0 , diameter d, and intersection array ι(G) as in (14).Then, F 2 = F 2 (G) has a regular partition π with quotient matrix and quotient Laplacian matrix Then, has a regular partition π with quotient matrix and quotient Laplacian matrix where c i + a i + b i = δ, for i = 0, 1, . . ., d.
Proof.Let us consider the partition π with classes (and the same equalities hold when we interchange u and v), the vertex {u, v} ∈ C i is adjacent to 2c i−1 , 2a i , and 2b i+1 vertices in C i−1 , C i , and C i+1 , respectively.Thus, since this holds for every vertex in C i , the partition π is regular with the quotient matrix in (16).From this and (4), we obtain the Laplacian quotient matrix of (17).
The following result shows that the quotient matrices of a regular partition can be used to find the spectral radius or Laplacian spectral radius of the 2-token graph of G. Proposition 5.2.Let G be a distance-regular graph with adjacency and Laplacian matrices A and L. Let F 2 = F 2 (G) be its 2-token graph with adjacency and Laplacian matrices A(F 2 ) and L(F 2 ) with respective spectral radii ρ A (F 2 ) and ρ L (F 2 ).Let A(F 2 /π) and L(F 2 /π) be the quotient matrices in (16) and (17) with respective spectral radii ρ A (F 2 /π) and ρ L (F 2 /π).Then, the following holds: Proof.(a) This follows from Lemma 5.1 and the known fact that if G is a graph with regular partition π, then the spectral radius of G is equal to the spectral radius of G/π.(The reason is that the Perron vector of G/π lifts to the Perron vector of G.) (b) In this case, we know that the eigenvector v of the spectral radius ρ L (F 2 /π) satisfies v ⊤ 1 = 0.Then, we only can conclude that ρ L (F 2 ) ≥ ρ L (F 2 /π).If G is bipartite, all the Other consequences of Lemma 5.1 and Proposition 5.2 are the following.First, from Theorem 1.1, we got the following result.
Corollary 5.4.Let F be the family of all distance-regular graphs with diameter d and the same parameters (or intersection array).Then, every graph G ∈ F has 2-token graph F 2 = F 2 (G) with the d (adjacency or Laplacian) eigenvalues of A(F 2 /π) or L(F 2 /π) given in ( 16)-( 17)).In particular, F 2 has spectral radii ρ A (F 2 ) = ρ A (F 2 /π), and ρ Thus, the natural question is if, as in the case of strongly regular graphs (see Godsil [19,20]), all distance-regular graphs with the same parameters are also cospectral (with respect to the adjacency or Laplacian matrix).In fact, notice that since every graph G ∈ F has the same eigenvalues, the spectrum of its k-token graph F k (G) also has such eigenvalues (Theorem 1.2).Moreover, from Theorem 1.3 and the interlacing theorem (see Haemers [22]), we get the following consequence.Proof.Apart from the factor 2, the matrix A(F 2 /π) in ( 16) is a principal d × d submatrix of the intersection matrix B in (15).Then, the result follows by using interlacing (see Haemers [22]).

Strongly regular graphs
Let G be a (connected) strongly regular graph on n vertices, which is a distance-regular graph with diameter 2. Let G have parameters (n, d, a, c), that is, G is d-regular (with b 0 = d), a 1 = a, and c 2 = c.Then, its intersection matrix is Then, the 2-token graph F 2 = F 2 (G) has a regular partition π with quotient matrix Such a regular partition was given by Audenaert, Godsil, Royle, and Rudolph in [2], and noted that the adjacency eigenvalues of A(F 2 /π) are They also commented that the positive eigenvalue θ 1 has a positive eigenvector (Perron vector) and, so, it corresponds to the (adjacency) spectral radius ρ A (F 2 (G)).
(a) L k S b = S b L h .(b) L h = (S ⊤ b S b ) −1 S ⊤ b L k S b .(c) The column space (and its orthogonal complement) of S b is L k -invariant.(d) The characteristic polynomial of L h divides the characteristic polynomial of L k .Thus, sp L h ⊆ sp L k .(e) If v is an eigenvector of L h with eigenvalue λ, then S b v is an eigenvector of L k with eigenvalue λ.(f) If v is an eigenvector of L k with eigenvalue λ and S ⊤ b v = 0, then S ⊤ b v is an eigenvector of L h with the same eigenvalue.

Definition 3 . 1 .
Given a graph G on n vertices and an integer k such that 1 ≤ k ≤ ⌊n/2⌋, the k-algebraic connectivity α k = α k (G) and the k-spectral radius ρ k = ρ k (G) of G are, respectively, the minimum and maximum eigenvalues of the multiset sp F k (G)\sp F k−1 (G).

Theorem 4 . 2 .
Let G be a graph with spectral radius ρ(G) and vertex-connectivity κ > 1.Given an integer k, with 1 ≤ k < κ, let ρ k M (G) and ρ k m (G) be the maximum and minimum of the spectral radii of the U -deleted subgraphs of G, where |U | = k.

Figure 2 :
Figure 2: The Heawood graph is the point-line incidence graph of the Fano plane.