A classic result of Fáry states that every planar graph can be drawn in the plane without crossings using only straight line segments. Harborth et al. conjecture that every planar graph has such a drawing where every edge length is integral. Biedl proves that every planar graph of maximum degree 4 that is not 4-regular has such a straight-line embedding, but the techniques are insufficient for ...

The bi-Cayley graph of a finite group G with respect to a subset S ⊆ G, which is denoted by BCay(G,S), is the graph with vertex set G× {1, 2} and edge set {{(x, 1), (sx, 2)} | x ∈ G, s ∈ S}. A finite group G is called a bi-Cayley integral group if for any subset S of G, BCay(G,S) is a graph with integer eigenvalues. In this paper we prove that a finite group G is a bi-Cayley integral group if a...

Given a set of integers S; G(S) = (S; E) is a graph, where the edge uv exists if and only if u+ v∈ S. A graph G = (V; E) is an integral sum graph or ISG if there exists a set S ⊂ Z such that G=G(S). This set is called a labeling of G. The main results of this paper concern regular ISGs. It is proved that all 2-regular graphs with the exception of C4 are integral sum graphs and that for every po...

The idea of integral sum graphs was introduced by Harary (1994). A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u) + f(v) = f(w) for some node w in G. A tree is said to be a generalized star if it can be obtained from a star by extending each edge to...

A graph G is called integral or Laplacian integral if all the eigenvalues of the adjacency matrix A(G) or the Laplacian matrix Lap(G) = D(G)−A(G) of G are integers, where D(G) denote the diagonal matrix of the vertex degrees of G. Let Kn,n+1 ≡ Kn+1,n and K1,p[(p−1)Kp] denote the (n+1)-regular graph with 4n+2 vertices and the p-regular graph with p2 + 1 vertices, respectively. In this paper, we ...

The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L functions on three-dimensional hyperbolic space. To ‘evaluate’ such a ...

The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is identical to the adjacency matrix of a circulant graph is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus...