In this paper we answer the question of when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. The network is described by a circulant graph G, which is characterized by its circulant adjacency matrix A. Formally, we say that there exists a perfect state transfer (PST) between vertices a, b ∈ V (G) if |F (τ)ab| = 1, for some positive real number τ ...

Cyclic timetabling for public transportation companies is usually modeled by the periodic event scheduling problem. To deduce a mixed-integer programming formulation, artificial integer variables have to be introduced. There are many ways to define these integer variables. We show that the minimal number of integer variables required to encode an instance is achieved by introducing an integer v...

An undirected graph is called integral, if all of its eigenvalues are integers. Let Γ = Zm1 ⊗· · ·⊗Zmr be an abelian group represented as the direct product of cyclic groups Zmi of order mi such that all greatest common divisors gcd(mi,mj) ≤ 2 for i 6= j. We prove that a Cayley graph Cay(Γ, S) over Γ is integral, if and only if S ⊆ Γ belongs to the the Boolean algebra B(Γ) generated by the subg...

A graph is integral if the spectrum (of its adjacency matrix) consists entirely of integers. The problem of determining all non-regular bipartite integral graphs with maximum degree four which do not have ±1 as eigenvalues was posed in K.T. Balińska, S.K. Simić, K.T. Zwierzyński: Which nonregular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues? Discrete Math., 2...

A graph-theoretic method, simpler than existing ones, is used to characterize the minimal set of monomial generators for the integral closure of any polynomial ring generated by quadratic monomials. The toric ideal of relations between these generators is generated by a set of graphically described binomials. The spectra of the original ring and of its integral closure turn out to be canonicall...

A graph is said to be Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. Using combinatorial and matrix-theoretic techniques, we identify, up to isomorphism, the 21 connected Laplacian integral graphs of maximum degree 3 on at least 6 vertices.

The evaluation of a relativistic spin network for the classical case of the Lie group is given by an integral formula over copies of SU(2). For the graph determined by a 4-simplex this gives the evaluation as an integral over a space of geometries for a 4-simplex.

On the level of Lie algebras, the Kontsevich integral of a knot (a graph-valued invariant) becomes the colored Jones function (a power series invariant). Rozansky conjectured and the authors proved a Rationality Conjecture for the Kontsevich integral. In this note, we explain how the Rationality Conjecture is related to Lie groups.

In this paper, we define duplication corona, duplication neighborhood corona and duplication edge corona of two graphs. We compute their adjacency spectrum, Laplacian spectrum and signless Laplacian. As an application, our results enable us to construct infinitely many pairs of cospectral graphs and also integral graphs.