OOn Middle Cube Graphs
C. Dalf´o † , M.A. Fiol † , M. Mitjana ‡† Departament de Matem`atica Aplicada IV ‡ Departament de Matem`atica Aplicada IUniversitat Polit`ecnica de Catalunya { cdalfo,fiol } @[email protected] May 23, 2018
Abstract
We study a family of graphs related to the n -cube. The middle cube graph of parameter k is the subgraph of Q k − induced by the set of vertices whose binary representation haseither k − 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 ofthe 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 associatedeigenvectors). The n -cube Q n , or n -dimensional hypercube, has been extensively studied. Nevertheless, manyopen questions remain. Harary et al. wrote a comprehensive survey on hypercube graphs [20].Recall that the n -cube Q n has vertex set V = { , } n and n -tuples representing vertices areadjacent if and only if they differ in exactly one coordinate. Then, Q n is an n -regular bipartitegraph with 2 n vertices and it is natural to consider its vertex set as partitioned into n + 1layers, the layer L k consisting of the (cid:0) nk (cid:1) vertices containing exactly k ≤ k ≤ n . Seeing thevertices of Q n as the characteristic vector of subsets of [ n ] = { , , . . . , n } , the vertices of layer L k correspond to the subsets of cardinality k , while the adjacencies correspond to the inclusionrelation.If n is odd, n = 2 k −
1, the middle two layers L k and L k − of Q n have the same number (cid:0) nk (cid:1) = (cid:0) nk − (cid:1) of vertices. Then the middle cube graph, denoted by M Q k , is the graph induced bythese two layers. It has been conjectured by Dejter, Erd˝os, 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 ofthe theory of distance-regular graphs. In particular, we completely determine their spectra(eigenvalues and their multiplicities, and associated eigenvectors). In this context, Qiu and Dasprovided experimental results for eigenvalues of several interconnection networks for which nocomplete characterization were known (see [24, § 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 thevertices with the integers 1 , , . . . , n . If i is adjacent to j , that is, ij ∈ E , we write i ∼ j or1 a r X i v : . [ m a t h . C O ] A ug ( 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 (cid:96) -apart from vertex i ,with respect to the usual distance, is Γ (cid:96) ( i ) = { j : dist( i, j ) = (cid:96) } , so that the degree of vertex i issimply δ i := | Γ ( i ) | ≡ | Γ( i ) | . The eccentricity of a vertex is ecc( i ) := max ≤ j ≤ n dist( i, j ) and the diameter of the graph is D ≡ D ( G ) := max ≤ i ≤ n ecc( i ). Given 0 ≤ (cid:96) ≤ D , the distance - (cid:96) graph G (cid:96) has the same vertex set as G and two vertices are adjacent in G (cid:96) if and only if they are atdistance (cid:96) in G . An antipodal graph G is a connected graph of diameter D for which G D is adisjoint union of cliques. In this case, the folded graph of G is the graph G whose vertices arethe 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 p hij = |{ w ∈ V : dist( u, w ) = i, dist( w, v ) = j }| 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 i − , b i = p i i +1 , and a i = p i i (see Brower etal. [7] and Fiol [11]). The odd graph, independently introduced by Balaban et al. [2] and Biggs [3], is a family ofgraphs 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 graphsalthough they have, in general, a more involved structure.For k ≥ the odd graph O k has vertices representing the ( k − k −
1] = { , , . . . , k − } , and two vertices are adjacent if and only if they are disjoint. For example, O is the complete graph K , and O is the Petersen graph. In general, O k is a k -regular graphon n = (cid:0) k − k − (cid:1) vertices, diameter D = k − g = 3 if k = 2, g = 5 if k = 3, and g = 6if k > O k is a distance-regular graph with intersection parameters b j = k − (cid:20) j + 12 (cid:21) , c j = (cid:20) j + 12 (cid:21) (0 ≤ j ≤ k − . With respect to the spectrum, the distinct eigenvalues of O k are λ i = ( − i ( k − i ), 0 ≤ i ≤ k −
1, with multiplicities m ( λ i ) = (cid:18) k − i (cid:19) − (cid:18) k − i − (cid:19) = k − ik (cid:18) ki (cid:19) . Let G = ( V, E ) be a graph of order n , with vertex set V = { , , . . . , n } . Its bipartite double graph (cid:101) G = ( (cid:101) V , (cid:101) E ) is the graph with the duplicated vertex set (cid:101) V = { , , . . . , n, (cid:48) , (cid:48) , . . . , n (cid:48) } ,2 Figure 1: The path P and its bipartite double graph.and adjacencies induced from the adjacencies in G as follows: i ( E ) ∼ j ⇒ (cid:40) i ( (cid:101) E ) ∼ j (cid:48) , and j ( (cid:101) E ) ∼ i (cid:48) . (1)Thus, the edge set of (cid:101) G is (cid:101) E = { ij (cid:48) | ij ∈ E } .From the definition, it follows that (cid:101) G is a bipartite graph with stable subsets V = { , , . . . , n } and V = { (cid:48) , (cid:48) , . . . , n (cid:48) } . For example, if G is a bipartite graph, then its bipartite double graph (cid:101) G consists of two non-connected copies of G (see Fig. 1).The bipartite double graph (cid:101) G has an involutive automorphism without fixed edges, whichinterchanges vertices i and i (cid:48) . On the other hand, the map from (cid:101) G onto G defined by i (cid:48) (cid:55)→ i, i (cid:55)→ i is a 2-fold covering.If G is a δ -regular graph, then (cid:101) G also is. Moreover, if the degree sequence of the orig-inal graph G is δ = ( δ , δ , . . . , δ n ), the degree sequence for its bipartite double graph is (cid:101) δ = ( δ , δ , . . . , δ n , δ , δ , . . . , δ n ).The distance between vertices in the bipartite double graph (cid:101) G can be given in terms of theeven and odd distances in G . Namely,dist (cid:101) G ( i, j ) = dist + G ( i, j )dist (cid:101) G ( i, j (cid:48) ) = dist − G ( i, j ) . Note that always dist − G ( i, j ) > i = j . Actually, (cid:101) G is connected if and only if G isconnected and non-bipartite.More precisely, it was proved by Bond and Delorme [6] that if G is a non-bipartite graphwith diameter D , then its bipartite double graph (cid:101) G has diameter (cid:101) D ≤ D + 1, and (cid:101) D = 2 D + 1if and only if for some vertex i ∈ V the subgraph induced by the vertices at distance less than D from i , G ≤ D − ( i ), is bipartite.In Figs. 2-5, we can see the bipartite double graph of three different graphs. The cycle C and Petersen graph both have diameter D = 2, and their bipartite double graphs have diameter (cid:101) D = 2 D + 1 = 5, while in the first example (Fig. 2) (cid:101) G has diameter (cid:101) D = 3 < D + 1.The extended bipartite double graph (cid:98) G of a graph G is obtained from its bipartite doublegraph by adding edges ( i, i (cid:48) ) for each i ∈ V . Note that when G is bipartite, then (cid:98) G is the directproduct G (cid:3) K . Let us now recall a useful result from spectral graph theory. For any graph, it is known that thecomponents 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 = λ v means that3
234 1 22‘ 3 3‘4‘4 1‘
Figure 2: Graph G has diameter 2 and (cid:101) G has diameter 3. Figure 3: C and its bipartite double graph.the sum of the charges of the neighbors of vertex i is λ times the charge of vertex i :( Av ) i = (cid:88) i ( E ) ∼ j v j = λv i . In what follows we compute the eigenvalues of the bipartite double graph (cid:101) G and the extendedbipartite double graph (cid:98) G as functions of the eigenvalues of a non-bipartite graph G . We alsoshow how to obtain the eigenvalues together with the corresponding eigenvectors of (cid:101) G and (cid:98) G .First, we recall the following technical result, due to Silvester [27], on the determinant ofsome 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 det F ( M ) = det F (cid:0) det R ( M ) (cid:1) . For example, if M = (cid:18) A BC D (cid:19) , where A , B , C , D are n × n matrices over F which commutewith each other, then Theorem 2.1 reads det F ( M ) = det F ( AD − BC ) . (2)Now we can use the above theorem to find the characteristic polynomial of the bipartitedouble and the extended bipartite double graphs. Theorem 2.2
Let G be a graph on n vertices, with the adjacency matrix A and characteristicpolynomial φ G ( x ) . Then, the characteristic polynomials of (cid:101) G and (cid:98) G are, respectively, φ (cid:101) G ( x ) = ( − n φ G ( x ) φ G ( − x ) , (3) φ (cid:98) G ( x ) = ( − n φ G ( x − φ G ( − x − . (4)4
245 3 15‘ 4‘3‘ 2‘12 34 5‘
Figure 4: C and C as another view of its bipartite double graph.
23 4 56 7 8 91012‘ 3‘4‘5‘ Figure 5: Petersen’s graph and its bipartite double graph.
P roof.
From the definitions of (cid:101) G and (cid:98) G , their adjacency matrices are, respectively, (cid:101) A = (cid:18) O AA O (cid:19) and (cid:98) A = (cid:18) O A + IA + I O (cid:19) . Thus, by (2), the characteristic polynomial of (cid:101) G is φ (cid:101) G ( x ) = det( x I n − (cid:101) A ) = det (cid:18) x I n − A − A x I n (cid:19) = det( x I n − A )= det( x I n − A ) det( x I n + A ) = ( − n φ G ( x ) φ G ( − x ) , whereas, the characteristic polynomial of (cid:98) G is φ (cid:98) G ( x ) = det( x I n − (cid:98) A ) = det (cid:18) x I n − A − I n − A − I n x I n (cid:19) = det (cid:0) x I n − ( A + I n ) (cid:1) = det (cid:0) x I n − ( A + I n ) (cid:1) det (cid:0) x I n + ( A + I n ) (cid:1) = det (cid:0) ( x − I n − A (cid:1) ( − n det (cid:0) − ( x + 1) I n − A (cid:1) = ( − n φ G ( x − φ G ( − x − . (cid:3) As a consequence, we have the following corollary:5 orollary 2.3
Given a graph G with spectrum sp G = { λ m , λ m , . . . , λ m d d } , where the superscripts denote multiplicities, then the spectra of (cid:101) G and (cid:98) G are, respectively, sp (cid:101) G = {± λ m , ± λ m , . . . , ± λ m d d } , sp (cid:98) G = {± (1 + λ ) m , ± (1 + λ ) m , . . . , ± (1 + λ d ) m d } . P roof.
Just note that, by (3) and (4), for each root λ of φ G ( x ), µ = ± λ are roots of φ (cid:101) G ( x ), whereas µ = ± (1 + λ ) are roots of φ (cid:98) G ( x ). (cid:3) Note that the spectra of (cid:101) G and (cid:98) G are symmetric, as expected, because both (cid:101) G and (cid:98) G arebipartite graphs.In the next theorem we are concerned with the eigenvectors of (cid:101) G and (cid:98) G , in terms of theeigenvectors of G . The computations also give an alternative derivation of the above spectra. Theorem 2.4
Let G be a graph and v a λ -eigenvector of G . Let us consider the vector u + with components u + i = u + i (cid:48) = v i , and u − , with components u − i = v i and u − i (cid:48) = − v i , ≤ i, i (cid:48) ≤ n .Then, • u + is a λ -eigenvector of (cid:101) G and a (1 + λ ) -eigenvector of (cid:98) G ; • u − is a ( − λ ) -eigenvector of (cid:101) G and a ( − − λ ) -eigenvector of (cid:98) G . P roof.
In order to show that u + is a λ -eigenvector of (cid:101) G , we distinguish two cases: • For a given vertex i , 1 ≤ i ≤ n , all its adjacent vertices are of type j (cid:48) , with i ( E ) ∼ j . Then( Au + ) i = (cid:88) j (cid:48) ( (cid:101) E ) ∼ i u + j (cid:48) = (cid:88) j ( E ) ∼ i v j = λv i = λu + i . • For a given vertex i (cid:48) , 1 ≤ i ≤ n , all its adjacent vertices are of type j , with i ( E ) ∼ j . Then( Au + ) i (cid:48) = (cid:88) j ( (cid:101) E ) ∼ i (cid:48) u + j = (cid:88) j ( E ) ∼ i v j = λv i = λu + i . By a similar reasoning with u − , we obtain( Au − ) i = (cid:88) j (cid:48) ( (cid:101) E ) ∼ i u − j (cid:48) = − (cid:88) j ( E ) ∼ i v j = − λu − i and ( Au − ) i (cid:48) = (cid:88) j ( (cid:101) E ) ∼ i (cid:48) u − j = (cid:88) j ( E ) ∼ i v j = − λu − i (cid:48) . Therefore, u − is a ( − λ )-eigenvector of the bipartite double graph (cid:101) G .In the same way, we can prove that u + and u − are eigenvectors of (cid:98) G with respectiveeigenvalues 1 + λ and − − λ . (cid:3) Notice that, for every linearly independent eigenvectors v and v of G , we get the linearlyindependent eigenvectors u ± and u ± of (cid:101) G . As a consequence, the geometric multiplicity ofeigenvalue λ of G coincides with the geometric multiplicities of the eigenvalues λ and − λ of (cid:101) G ,and 1 + λ and − − λ of (cid:98) G . 6 The middle cube graphs
For k ≥ n = 2 k −
1, the middle cube graph
M Q k is the subgraph of the n -cube Q n inducedby the vertices whose binary representations have either k − k number of 1s. Then, M Q k has order 2 (cid:0) nk (cid:1) and is k -regular, since a vertex with k − k zeroes, so it is adjacent to k vertices with k k
1s has k adjacent vertices with k − M Q k is a bipartite graph with stable sets V and V constituted bythe 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 = 2 k − M Q k is the bipartite double graph of O k Notice that, if A and B are both subsets of [2 k − A ⊂ B if and only if A and B are disjoint.Moreover, if | B | = k then | B | = k −
1. This gives the following result.
Proposition 3.1
The middle cube graph
M Q k is isomorphic to (cid:101) O k , the bipartite double graphof O k . P roof.
The mapping from (cid:101) O k to M Q k defined by: f : V [ (cid:101) O k ] → V [ M Q k ] u (cid:55)→ uu (cid:48) (cid:55)→ u is clearly bijective. Moreover, according to the definition of bipartite double graph in Eq.(1), if u and v (cid:48) are two vertices of (cid:101) O k , then u ∼ v (cid:48) ⇔ u ∩ v = ∅ ⇔ u ⊂ v , which is equivalent to say that if u ∼ v (cid:48) , in (cid:101) O k , then f ( u ) = u ∼ v = f ( v (cid:48) ), in M Q k . (cid:3) For example, the middle cube graph
M Q contains vertices with one or two 1s in their binaryrepresentation. The adjacencies give simply a 6-cycle (see Fig. 6), which is isomorphic to (cid:101) O .As another example, M Q has 20 vertices because there are (cid:0) (cid:1) = 10 vertices with two 1s, and (cid:0) (cid:1) = 10 vertices with three 1s in their binary representation (see Fig. 7). Compare the Figs. 5and 7 in order to realize the isomorphism between the definitions of M Q and (cid:101) O .It is known that (cid:101) O k is a bipartite 2-antipodal distance-regular graph. See Biggs [5] andBrower et al. [7] for more details. The spectrum of the hypercube Q k − contains all the eigenvalues (including multiplicities) ofthe middle cube M Q k : sp M Q k ⊆ sp Q k − . According to the result of Corollary 2.3, the spectrum of the middle cube graph
M Q k (cid:39) (cid:101) O k can be obtained from the spectrum of the odd graph O k . The distinct eigenvalues of M Q k are θ + i = ( − i ( k − i ) and θ − i = − θ + i , 0 ≤ i ≤ k −
1, with multiplicities m ( θ + i ) = m ( θ − i ) = k − ik (cid:18) ki (cid:19) . (5)7igure 6: The middle cube graph M Q as a subgraph of Q or as the bipartite double graph of O = K . Figure 7: The middle cube graph
M Q .For example, sp M Q = {± , ± } , sp M Q = {± , ± , ± } , sp M Q = {± , ± , ± , ± } , sp M Q = {± , ± , ± , ± , ± } . The middle cube graph is a distance-regular graph. For instance, the distance polynomials of
M Q k are p ( x ) = 1 ,p ( x ) = x,p ( x ) = x − ,p ( x ) = 12 ( x − x ) ,p ( x ) = 14 ( x − x + 12) ,p ( x ) = 112 ( x − x + 22 x ) .
8s the sum of the distance polynomials is the Hoffman polynomial [22], we have (cid:88) i =0 p i ( x ) = 112 ( x − x − x + 3)( x + 2)( x + 1) . (6)The eigenvalues of the M Q are λ = 3 and the zeroes of polynomial (6):ev M Q = { , , , − , − , − } , and their multiplicities, m ( λ i ), can be computed using the highest degree polynomial p k − ,according to the result by Fiol [11]: m ( λ i ) = φ p k − ( λ ) φ i p k − ( λ i ) , ≤ i ≤ k − , where φ i = (cid:81) k − j =0 , j (cid:54) = i ( λ i − λ j ). Of course, this expression yields the same result as Eq. (5).Namely, m ( λ i ) = m ( λ k − − i ) = m ( θ ± i ), 0 ≤ i ≤ k − p (3) = p (1) = p ( −
1) = 1 and p (2) = p ( −
1) = p ( −
3) = −
1. Moreover, φ = − φ = 240, φ = − φ = −
60, and φ = − φ = 48.Then, m ( λ ) = m ( λ ) = m ( θ ± ) = 1 ,m ( λ ) = m ( λ ) = m ( θ ± ) = 4 ,m ( λ ) = m ( λ ) = m ( θ ± ) = 5 . Let G be a graph with diameter D and distinct eigenvalues ev G = { λ , λ , . . . , λ d } , where λ > λ > · · · > λ d . A classical result states that D ≤ d (see, for instance, Biggs [5]). Otherresults related to the diameter D and some (or all) different eigenvalues have been given by Alonand Milman [1], Chung [8], van Dam and Haemers [9], Delorme and Sol´e [10], and Mohar [23],among others. Fiol et al. [12, 14, 15] showed that many of these results can be stated with thefollowing common framework: If the value of a certain polynomial P at λ is large enough, thenthe diameter is at most the degree of P . More precisely, it was shown that optimal results arisewhen P is the so-called k -alternating polynomial , which in the case of degree d − P ( λ i ) = ( − i +1 , ≤ i ≤ d , and satisfies P ( λ ) = (cid:80) di =1 π π i , where π i = (cid:81) dj =0 ,j (cid:54) = i | λ i − λ j | . Inparticular, when G is a regular graph on n vertices, the following implication holds: P ( λ ) + 1 = n (cid:88) i =0 π π i > n ⇒ D ≤ d − . This result suggested the study of the so-called boundary graphs [13, 15], characterized by d (cid:88) i =1 π π i = n. (7)Fiol et al. [13] showed that extremal ( D = d ) boundary graphs, where each vertex has maximumeccentricity, are 2-antipodal distance-regular graphs. As we show in the next result, this is thecase of the middle cube graphs M Q k where the antipodal pairs of vertices are ( x ; x ), with x = x x . . . x k − and x = x x . . . x k − . Proposition 3.2
The middle cube graph
M Q k is a boundary graph. roof. Recall that the eigenvalues of
M Q k areev M Q k − = { k, k − , . . . , , − , . . . , − k } , that is, λ i = k − i, λ k + i = − ( i + 1), 0 ≤ i < k . Now, according to Eq. (7), we have to provethat (cid:80) k − i =0 π π i = 2 (cid:0) k − k (cid:1) . Computing π i , for 0 ≤ i ≤ k −
1, we get π i = i !(2 k − i )! k − i = π k − ( i +1) , for 0 ≤ i < k. This implies π π i = π π k − ( i +1) = (2 k )! k ( k − i ) i ! (2 k − i )! = k − ik (cid:18) ki (cid:19) , for 0 ≤ i < k, giving exactly the multiplicities of the corresponding eigenvalues, as found in Eq. 5. By summingup we get k − (cid:88) i =0 π π i = 2 k − (cid:88) i =0 π π i = 2 (cid:32) k − (cid:88) i =0 (cid:18) ki (cid:19) − k − (cid:88) i =1 ik (cid:18) ki (cid:19)(cid:33) . (8)But k − (cid:88) i =0 (cid:18) ki (cid:19) = 12 (cid:18) k − (cid:18) kk (cid:19)(cid:19) = 2 k − − (cid:18) k − k (cid:19) , (9)and k − (cid:88) i =1 ik (cid:18) ki (cid:19) = 2 k − (cid:88) i =0 (cid:18) k − i (cid:19) = 2 k − − (cid:18) k − k − (cid:19) , where we have used Eq. (9) changing k by k −
1. Thus, replacing the above values in Eq. (8),we get the result. (cid:3)
Acknowledgements
Research supported by the
Ministerio de Ciencia e Innovaci´on , Spain, and the
European Re-gional Development Fund under project MTM2011-28800-C02-01, and the
Catalan ResearchCouncil under project 2009SGR1387.
References [1] N. Alon and V. Milman, λ , Isoperimetric inequalities for graphs and super-concentrators, J. Combinat. Theory Ser. B (1985) 73–88.[2] A.T. Balaban, D. Farcussiu, R. Banica, Graphs of multiple 1 , Rev. Roum. Chim. (1966) 1205–1227.[3] N. Biggs, An edge coloring problem, Amer. Math. Montly (1972) 1018–1020.[4] N. Biggs, Some odd graph theory, Ann. New York Acad. Sci. (1979) 71–81.[5] N. Biggs,
Algebraic Graph Theory , Cambridge University Press, Cambridge (1974), secondedition (1993). 106] I. Bond, C. Delorme, New large bipartite graphs with given degree and diameter,
Ars Com-bin. (1988) 123–132.[7] A.E. Brouwer, A. M. Cohen, A. Neumaier,
Distance-Regular Graphs , Springer, Berlin (1989).[8] F.R.K. Chung, Diameter and eigenvalues,
J. Amer. Math. Soc. (1989) 187–196.[9] E.R. van Dam, W.H. Haemmers, Eigenvalues and the diameter of graphs, Linear and Mul-tilinear Algebra (1995) 33–44.[10] C. Delorme, P. Sol´e, Diameter, covering index, covering radius and eigenvalues, EuropeanJ. Combin. (1991) 95–108.[11] M.A. Fiol, Algebraic characterizations of distance-regular graphs, Discrete Math. (2002) 111–129.[12] M.A. Fiol, E. Garriga, J.L.A. Yebra, On a class of polynomials and its relation with thespectra and diameter of graphs,
J. Combin. Theory Ser. B (1996) 48–61.[13] M.A. Fiol, E. Garriga, J.L.A. Yebra, From regular boundary graphs to antipodal distance-regular graphs, J. Graph Theory (1998) 123–140.[14] M.A. Fiol, E. Garriga, J.L.A. Yebra, Boundary graphs: The limit case of a spectral prop-erty, Discrete Math. (2001) 155–173.[15] M.A. Fiol, E. Garriga, J.L.A. Yebra, Boundary graphs: The limit case of a spectral property(II),
Discrete Math. (1998) 101–111.[16] M. A. Fiol, E. Garriga, J.L.A. Yebra, On twisted odd graphs,
Combin. Probab. Comput. (2000) 227–240.[17] M.A. Fiol, M. Mitjana, The spectra of some families of digraphs, Linear Algebra Appl. (2007), no. 1, 109–118.[18] C. D. Godsil, More odd graph theory,
Discrete Math. (1980) 205–207.[19] C. D. Godsil, Algebraic Combinatorics , Chapman and Hall, London/New York (1993).[20] F. Harary, J.P. Hayes, H.J. Wu, A survey of the theory of hypercube graphs,
Comp. Math.Appl. (1988), no. 4, 277–289.[21] I. Havel, Semipaths in directed cubes, in M. Fiedler (Ed.), Graphs and other CombinatorialTopics , Teunebner–Texte Math., Teubner, Leipzig (1983).[22] A. J. Hoffman, On the polynomial of a graph,
Amer. Math. Monthly (1963) 30–36.[23] B. Mohar, Eigenvalues, diameter and mean distance in graphs, Graphs Combin. TheorySer. B (1996) 179–205.[24] K. Qiu, S.K. Das, Interconnexion Networks and Their Eigenvalues, in Proc. of 2002 In-ternational Sumposiym on Parallel Architectures, Algorithms and Networks, ISPAN’02 , pp.163–168.[25] J. Robert Johnson, Long cycles in the middle two layers of the discrete cube,
J. Combin.Theory Ser. A (2004) 255–271. 1126] C.D. Savage, I. Shields, A Hamilton path heuristic with applications to the middle twolevels problem,
Congr. Numer. (1999) 161–178.[27] J.R. Silvester, Determinants of block matrices,
Maths Gazette84