The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States

We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss the problem of separability by pointing out that separability of density matrices of graphs does not always depend on the labelling of the vertices. We consider graphs with a tensor product structure and simple cases for which combinatorial properties are linked to the entanglement of the state. We calculate the concurrence of all graphs on four vertices representing entangled states. It turns out that for these graphs the value of the concurrence is exactly fractional. Received July 28, 2004     

The Laplacian spectra of graphs with a tree structure

Trees are very common in the theory and applications of combinatorics. In this article, we consider graphs whose underlying structure is a tree, except that its vertices are graphs in their own right and where adjacent graphs (vertices) are linked by taking their join. We study the spectral properties of the Laplacian matrices of such graphs. It turns out that in order to capture known spectral properties of the Laplacian matrices of trees, it is necessary to consider the Laplacians of vertex-weighted graphs. We focus on the second smallest eigenvalue of such Laplacians and on the properties of their corresponding eigenvector. We characterize the second smallest eigenvalue in terms of the Perron branches of a tree. Finally, we show that our results are applicable to advancing the solution to the problem of whether there exists a graph on n vertices whose Laplacian has the integer eigenvalues 0, 1, …, n ? 1.     

On graphs whose Laplacian matrices have distinct integer eigenvalues

In this paper, we investigate graphs for which the corresponding Laplacian matrix has distinct integer eigenvalues. We define the set S_i,n to be the set of all integers from 0 to n, excluding i. If there exists a graph whose Laplacian matrix has this set as its eigenvalues, we say that this set is Laplacian realizable. We investigate the sets S_i,n that are Laplacian realizable, and the structures of the graphs whose Laplacian matrix has such a set as its eigenvalues. We characterize those i < n such that S_i,n is Laplacian realizable, and show that for certain values of i, the set S_i,n is realized by a unique graph. Finally, we conjecture that S_n,n is not Laplacian realizable for n ≥ 2 and show that the conjecture holds for certain values of n. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Bounds for the Least Laplacian Eigenvalue of a Signed Graph

A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.     

Fractional revival and association schemes

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose–Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.     

Torsion formulas for signed graphs

Following our recent exposition on the algebraic foundations of signed graphs, we introduce bond (circuit) basis matrices for the tension (flow) lattices of signed graphs, and compute the torsions of such matrices and Laplacians. We present closed formulas for the torsions of the incidence matrix, the Laplacian, bond basis matrices, and circuit basis matrices. These formulas show that the torsions of all such matrices are powers of 2, and so imply that the matroids of signed graphs are representable over any field of characteristic not 2. A notable feature of using torsion is that the Matrix-Tree formula for ordinary graphs and Zaslavsky’s formula for unbalanced signed graphs are unified into one Matrix-Basis formula in terms of the torsion of its Laplacian matrix, rather than in terms of its determinant, which vanishes for an ordinary graph unless one row is deleted from the incidence matrix.     

Theorems on partitioned matrices revisited and their applications to graph spectra

Some old results about spectra of partitioned matrices due to Goddard and Schneider or Haynsworth are re-proved. A new result is given for the spectrum of a block-stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. The result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation. The concept of a reduced graph matrix is introduced. The question of when n-2 is the second largest signless Laplacian eigenvalue of a connected graph of order n is treated. A recent conjecture posed by Tam, Fan and Zhou on graphs that maximize the signless Laplacian spectral radius over all (not necessarily connected) graphs with given numbers of vertices and edges is refuted. The Laplacian spectrum of a (degree) maximal graph is reconsidered.     

Entries of the group inverse of the Laplacian matrix for generalized Johnson graphs

In this paper, we use graph theoretic properties of generalized Johnson graphs to compute the entries of the group inverse of Laplacian matrices for generalized Johnson graphs. We then use these entries to compute the Zenger function for the group inverse of Laplacian matrices of generalized Johnson graphs.     

Strongly self-dual graphs

We present a class of graphs whose adjacency matrices are nonsingular with integral inverses, denoted h-graphs. If the h-graphs G and H with adjacency matrices M(G) and M(H) satisfy M(G)^-1=SM(H)S, where S is a signature matrix, we refer to H as the dual of G. The dual is a type of graph inverse. If the h-graph G is isomorphic to its dual via a particular isomorphism, we refer to G as strongly self-dual. We investigate the structural and spectral properties of strongly self-dual graphs, with a particular emphasis on identifying when such a graph has 1 as an eigenvalue.     

On the construction of cospectral nonisomorphic bipartite graphs

In this article, we construct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices using the notion of partitioned tensor products. This extends the construction of Ji, Gong, and Wang [9]. Our proof of the cospectrality of adjacency matrices simplifies the proof of the bipartite case of Godsil and McKay's construction [4], and shows that the corresponding normalized Laplacian matrices are also cospectral. We partially characterize the isomorphism in Godsil and McKay's construction, and generalize Ji et al.'s characterization of the isomorphism to biregular bipartite graphs. The essential idea in characterizing the isomorphism uses Hammack's cancellation law as opposed to Hall's marriage theorem used by Ji et al.     

A characterization of complete matroid base graphs

This study grew from an attempt to give a local analysis of matroid base graphs. A neighborhood-preserving covering of graphs p:G → H is one such that p restricted to every neighborhood in G is an isomorphism. This concept arises naturally when considering graphs with a prescribed set of local properties. A characterization is given of all connected graphs with two local properties: (a) there is a pair of adjacent points, the intersection of whose neighborhoods does not contain three mutually nonadjacent points; (b) the intersection of the neigh-borhoods of points two apart is a 4-cycle. Such graphs have neighborhoods of the form K_n × K_m for fixed n, m and are either complete matroid base graphs or are their images under neighborhood-preserving coverings. If n ≠ m, the graph is unique; if n = m, there are n ? 3 such images which are nontrivial. These examples prove that no set of properties of bounded diameter can characterize matroid base graphs.     

On the Laplacian Eigenvalues of Signed Graphs

A signed graph is a graph with a sign attached to each edge. This article extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the largest Laplacian eigenvalue of a signed graph is investigated, which generalizes the corresponding results on the largest Laplacian eigenvalue of a graph.     

Graph isomorphism and equality of simplices

We show that the graph isomorphism problem is equivalent to the problem of recognizing equal simplices in ? ⁿ . This result can lead to new methods in the graph isomorphism problem based on geometrical properties of simplices. In particular, relations between several well-known classes of invariants of graphs and geometrical invariants of simplices are established.     

On the characteristic polynomial of a special class of graphs and spectra of balanced trees

Let H be a simple graph with n vertices and G be a sequence of n rooted graphs G₁,G₂,…,G_n. Godsil and McKay [C.D. Godsil, B.D. McKay, A new graph product and its spectrum, Bull. Austral. Math. Soc. 18 (1978) 21-28] defined the rooted product H(G), of H by G by identifying the root vertex of G_i with the ith vertex of H, and determined the characteristic polynomial of H(G). In this paper we prove a general result on the determinants of some special matrices and, as a corollary, determine the characteristic polynomials of adjacency and Laplacian matrices of H(G).Rojo and Soto [O. Rojo, R. Soto, The spectra of the adjacency matrix and Laplacian matrix for some balanced trees, Linear Algebra Appl. 403 (2005) 97-117] computed the characteristic polynomials and the spectrum of adjacency and Laplacian matrices of a class of balanced trees. As an application of our results, we obtain their conclusions by a simple method.     

Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.     

Self-clique graphs and matrix permutations

The clique graph of a graph is the intersection graph of its (maximal) cliques. A graph is self-clique when it is isomorphic with its clique graph, and is clique-Helly when its cliques satisfy the Helly property. We prove that a graph is clique-Helly and self-clique if and only if it admits a quasi-symmetric clique matrix, that is, a clique matrix whose families of row and column vectors are identical. We also give a characterization of such graphs in terms of vertex-clique duality. We describe new classes of self-clique and 2-self-clique graphs. Further, we consider some problems on permuted matrices (matrices obtained by permuting the rows and/or columns of a given matrix). We prove that deciding whether a (0,1)-matrix admits a symmetric (quasi-symmetric) permuted matrix is graph (hypergraph) isomorphism complete. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 178–192, 2003     

Commutativity of the adjacency matrices of graphs

We say that two graphs G₁ and G₂ with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers n such that the complete bipartite graph K_n,n is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most n−1 linearly independent commuting adjacency matrices of size n; and if this bound occurs, then there exists a Hadamard matrix of order n. Finally, we determine the centralizers of some families of graphs.     

Constructing integral uniform flows in symmetric networks with application to the edge-forwarding index problem

We study the integral uniform (multicommodity) flow problem in a graph G and construct a fractional solution whose properties are invariant under the action of a group of automorphisms Γ<Aut(G). The fractional solution is shown to be close to an integral solution (depending on properties of Γ), and in particular becomes an integral solution for a class of graphs containing Cayley graphs. As an application we estimate asymptotically (up to additive error terms) the edge congestion of an optimal integral uniform flow (edge forwarding index) in the cube-connected cycles and the butterfly. Moreover, we derive the best-known lower bound on the crossing number of a butterfly.     

Infinite quantum graphs

Infinite quantum graphs with δ-interactions at vertices are studied without any assumptions on the lengths of edges of the underlying metric graphs. A connection between spectral properties of a quantum graph and a certain discrete Laplacian given on a graph with infinitely many vertices and edges is established. In particular, it is shown that these operators are self-adjoint, lower semibounded, nonnegative, discrete, etc. only simultaneously.