On middle cube graphs

We study a family of graphs related to the $n$-cube. The middle cube graph of parameter $k$ is the subgraph of $Q_{2k-1}$ induced by the set of vertices whose binary representation has either $k-1$ or $k$ number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors).


Introduction
The n-cube Q n , or n-dimensional hypercube, has been extensively studied. Nevertheless, many open questions remain. Harary et al. wrote a comprehensive survey on hypercube graphs [20]. Recall that the n-cube Q n has vertex set V = {0, 1} n and n-tuples representing vertices are adjacent if and only if they differ in exactly one coordinate. Then, Q n is an n-regular bipartite graph with 2 n vertices and it is natural to consider its vertex set as partitioned into n + 1 layers, the layer L k consisting of the n k vertices containing exactly k 1s, 0 ≤ k ≤ n. Seeing the vertices of Q n as the characteristic vector of subsets of [n] = {1, 2, . . . , n}, the vertices of layer L k correspond to the subsets of cardinality k, while the adjacencies correspond to the inclusion relation.
If n is odd, n = 2k − 1, the middle two layers L k and L k−1 of Q n have the same number n k = n k−1 of vertices. Then the middle cube graph, denoted by M Q k , is the graph induced by these two layers. It has been conjectured by Dejter, Erdős, Havel [21] among others, that M Q k is Hamiltonian. It is known that the conjecture holds for n ≤ 16 (see Savage and Shields [26]), and it was almost solved by Robert Johnson [25].
In this paper we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors). In this context, Qiu and Das provided experimental results for eigenvalues of several interconnection networks for which no complete characterization were known (see [24, §3.2]).
Before proceeding with our study, we fix some basic definitions and notation used throughout the paper. We denote by G = (V, E) a (simple, connected and finite) graph with vertex set V an edge set E. The order of the graph G is n = |V | and its size is m = |E|. We label the vertices with the integers 1, 2, . . . , n. If i is adjacent to j, that is, ij ∈ E, we write i ∼ j or i (E) ∼ j. The distance between two vertices is denoted by dist(i, j). We also use the concepts of even distance and odd distance between vertices (see Bond and Delorme [6]), denoted by dist + and dist − , respectively. They are defined as the length of a shortest even (respectively, odd) walk between the corresponding vertices. The set of vertices which are -apart from vertex i, with respect to the usual distance, is Γ (i) = {j : dist(i, j) = }, so that the degree of vertex i is simply δ i := |Γ 1 (i)| ≡ |Γ(i)|. The eccentricity of a vertex is ecc(i) := max 1≤j≤n dist(i, j) and the diameter of the graph is D ≡ D(G) := max 1≤i≤n ecc(i). Given 0 ≤ ≤ D, the distance-graph G has the same vertex set as G and two vertices are adjacent in G if and only if they are at distance in G. An antipodal graph G is a connected graph of diameter D for which G D is a disjoint union of cliques. In this case, the folded graph of G is the graph G whose vertices are the maximal cliques of G D and two vertices are adjacent if their union contains and edge of G. If, moreover, all maximal cliques of G D have the same size r then G is also called an antipodal r-cover of G (double cover if r = 2, triple cover if r = 3, etc.).
Recall that a graph G with diameter D is distance-regular when, for all integers h, i, j (0 ≤ h, i, j ≤ D) and vertices u, v ∈ V with dist(u, v) = h, the numbers do not depend on u and v. In this case, such numbers are called the intersection parameters and, for notational convenience, we write c i = p i 1i−1 , b i = p i 1i+1 , and a i = p i 1i (see Brower et al. [7] and Fiol [11]).

The odd graphs
The odd graph, independently introduced by Balaban et al. [2] and Biggs [3], is a family of graphs that has been studied by many authors (see [4,5,18]). More recently, Fiol et al. [16] introduced the twisted odd graphs, which share some interesting properties with the odd graphs although they have, in general, a more involved structure.
The odd graph O k is a distance-regular graph with intersection parameters  and adjacencies induced from the adjacencies in G as follows:

The bipartite double graph
From the definition, it follows that G is a bipartite graph with stable subsets V 1 = {1, 2, . . . , n} and V 2 = {1 , 2 , . . . , n }. For example, if G is a bipartite graph, then its bipartite double graph G consists of two non-connected copies of G (see Fig. 1).
The bipartite double graph G has an involutive automorphism without fixed edges, which interchanges vertices i and i . On the other hand, the map from G onto G defined by i → i, i → i is a 2-fold covering.
The distance between vertices in the bipartite double graph G can be given in terms of the even and odd distances in G. Namely, Note that always dist − G (i, j) > 0 even if i = j. Actually, G is connected if and only if G is connected and non-bipartite.
More precisely, it was proved by Bond and Delorme [6] that if G is a non-bipartite graph with diameter D, then its bipartite double graph G has diameter D ≤ 2D + 1, and D = 2D + 1 if and only if for some vertex i ∈ V the subgraph induced by the vertices at distance less than D from i, G ≤D−1 (i), is bipartite.
In Figs. 2-5, we can see the bipartite double graph of three different graphs. The cycle C 5 and Petersen graph both have diameter D = 2, and their bipartite double graphs have diameter D = 2D + 1 = 5, while in the first example (Fig. 2) G has diameter D = 3 < 2D + 1.
The extended bipartite double graph G of a graph G is obtained from its bipartite double graph by adding edges (i, i ) for each i ∈ V . Note that when G is bipartite, then G is the direct product G K 2 .

Spectral properties of the bipartite double graph
Let us now recall a useful result from spectral graph theory. For any graph, it is known that the components of its eigenvalues can be seen as charges on each vertex (see Fiol and Mitjana [17] and Godsil [19]). Let G = (V, E) be a graph with adjacency matrix A and λ-eigenvector v. Then, the charge of vertex i ∈ V is the entry v i of v, and the equation Av = λv means that In what follows we compute the eigenvalues of the bipartite double graph G and the extended bipartite double graph G as functions of the eigenvalues of a non-bipartite graph G. We also show how to obtain the eigenvalues together with the corresponding eigenvectors of G and G.
First, we recall the following technical result, due to Silvester [27], on the determinant of some block matrices: Theorem 2.1 Let F be a field and let R be a commutative subring of F n×n , the set of all n × n matrices over F . Let M ∈ R m×m , then Now we can use the above theorem to find the characteristic polynomial of the bipartite double and the extended bipartite double graphs.
Theorem 2.2 Let G be a graph on n vertices, with the adjacency matrix A and characteristic polynomial φ G (x). Then, the characteristic polynomials of G and G are, respectively,  Thus, by (2), the characteristic polynomial of G is whereas, the characteristic polynomial of G is As a consequence, we have the following corollary:

Corollary 2.3 Given a graph G with spectrum
where the superscripts denote multiplicities, then the spectra of G and G are, respectively, P roof. Just note that, by (3) and (4), for each root λ of φ G (x), µ = ±λ are roots of φ G (x), whereas µ = ±(1 + λ) are roots of φ G (x).
Note that the spectra of G and G are symmetric, as expected, because both G and G are bipartite graphs.
In the next theorem we are concerned with the eigenvectors of G and G, in terms of the eigenvectors of G. The computations also give an alternative derivation of the above spectra.
• u + is a λ-eigenvector of G and a (1 + λ)-eigenvector of G; • u − is a (−λ)-eigenvector of G and a (−1 − λ)-eigenvector of G. P roof. In order to show that u + is a λ-eigenvector of G, we distinguish two cases: • For a given vertex i, 1 ≤ i ≤ n, all its adjacent vertices are of type j , with i • For a given vertex i , 1 ≤ i ≤ n, all its adjacent vertices are of type j, with i By a similar reasoning with u − , we obtain Therefore, u − is a (−λ)-eigenvector of the bipartite double graph G.
In the same way, we can prove that u + and u − are eigenvectors of G with respective eigenvalues 1 + λ and −1 − λ.
Notice that, for every linearly independent eigenvectors v 1 and v 2 of G, we get the linearly independent eigenvectors u ± 1 and u ± 2 of G. As a consequence, the geometric multiplicity of eigenvalue λ of G coincides with the geometric multiplicities of the eigenvalues λ and −λ of G, and 1 + λ and −1 − λ of G.

The middle cube graphs
For k ≥ 1 and n = 2k −1, the middle cube graph M Q k is the subgraph of the n-cube Q n induced by the vertices whose binary representations have either k − 1 or k number of 1s. Then, M Q k has order 2 n k and is k-regular, since a vertex with k − 1 1s has k zeroes, so it is adjacent to k vertices with k 1s, and similarly a vertex with k 1s has k adjacent vertices with k − 1 1s (see Figs. 6 and 7).
The middle cube graph M Q k is a bipartite graph with stable sets V 0 and V 1 constituted by the vertices whose corresponding binary string has, respectively, even or odd Hamming weight, that is, number of 1s. The diameter of the middle cube graph M Q k is D = 2k − 1.

M Q
For example, the middle cube graph M Q 2 contains vertices with one or two 1s in their binary representation. The adjacencies give simply a 6-cycle (see Fig. 6), which is isomorphic to O 2 . As another example, M Q 3 has 20 vertices because there are 5 2 = 10 vertices with two 1s, and 5 3 = 10 vertices with three 1s in their binary representation (see Fig. 7). Compare the Figs. 5 and 7 in order to realize the isomorphism between the definitions of M Q 3 and O 3 . It is known that O k is a bipartite 2-antipodal distance-regular graph. See Biggs [5] and Brower et al. [7] for more details.

Spectral properties
The spectrum of the hypercube Q 2k−1 contains all the eigenvalues (including multiplicities) of the middle cube M Q k : According to the result of Corollary 2.3, the spectrum of the middle cube graph M Q k O k can be obtained from the spectrum of the odd graph O k . The distinct eigenvalues of M Q k are As the sum of the distance polynomials is the Hoffman polynomial [22], we have The eigenvalues of the M Q 3 are λ 0 = 3 and the zeroes of polynomial (6): and their multiplicities, m(λ i ), can be computed using the highest degree polynomial p 2k−1 , according to the result by Fiol [11]: Of course, this expression yields the same result as Eq. (5).

Middle cube graphs as boundary graphs
Let G be a graph with diameter D and distinct eigenvalues ev G = {λ 0 , λ 1 , . . . , λ d }, where λ 0 > λ 1 > · · · > λ d . A classical result states that D ≤ d (see, for instance, Biggs [5]). Other results related to the diameter D and some (or all) different eigenvalues have been given by Alon and Milman [1], Chung [8], van Dam and Haemers [9], Delorme and Solé [10], and Mohar [23], among others. Fiol et al. [12,14,15] showed that many of these results can be stated with the following common framework: If the value of a certain polynomial P at λ 0 is large enough, then the diameter is at most the degree of P . More precisely, it was shown that optimal results arise when P is the so-called k-alternating polynomial, which in the case of degree d−1 is characterized by P (λ i ) = (−1) i+1 , 1 ≤ i ≤ d, and satisfies P ( In particular, when G is a regular graph on n vertices, the following implication holds: This result suggested the study of the so-called boundary graphs [13,15], characterized by d i=1 π 0 π i = n. Fiol et al. [13] showed that extremal (D = d) boundary graphs, where each vertex has maximum eccentricity, are 2-antipodal distance-regular graphs. As we show in the next result, this is the case of the middle cube graphs M Q k where the antipodal pairs of vertices are (x; x), with x = x 0 x 1 . . . x 2k−1 and x = x 0 x 1 . . . x 2k−1 . that is, λ i = k − i, λ k+i = −(i + 1), 0 ≤ i < k. Now, according to Eq. (7), we have to prove that 2k−1 i=0 π 0 π i = 2 2k−1 k . Computing π i , for 0 ≤ i ≤ 2k − 1, we get This implies giving exactly the multiplicities of the corresponding eigenvalues, as found in Eq. 5. By summing up we get and where we have used Eq. (9) changing k by k − 1. Thus, replacing the above values in Eq. (8), we get the result.