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 ...

Vacuum energy and other spectral functions of Laplace-type differential operators have been studied approximately by classical-path constructions and more fundamentally by boundary integral equations. As the first step in a program of elucidating the connections between these approaches and improving the resulting calculations, I show here how the known solutions for Kirchhoff quantum graphs em...

As a generalisation of the stable matching problem Bäıou and Balinski [3] defined the stable allocation problem for bipartite graphs, where both the edges and the vertices may have capacities. They constructed a so-called inductive algorithm, that always finds a stable allocation in strongly polynomial time. Here, we generalise their algorithm for non-bipartite graphs with integral capacities. ...

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...

Let Γ be a finite, additive group, S ⊆ Γ, 0 6∈ S, − S = {−s : s ∈ S} = S. The undirected Cayley graph Cay(Γ, S) has vertex set Γ and edge set {{a, b} : a, b ∈ Γ, a − b ∈ S}. A graph is called integral, if all of its eigenvalues are integers. For an abelian group Γ we show that Cay(Γ, S) is integral, if S belongs to the Boolean algebra B(Γ) generated by the subgroups of Γ. The converse is proven...

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.