Platonic Graphs Research Articles

On families of 2-nearly Platonic graphs

A 2 -nearly Platonic graph of type ( k | d ) is a k -regular planar graph with f faces, f − 2 of which are of size d and the remaining two are of sizes d 1 , d 2 , both different from d . Such a graph is called balanced if d 1 = d 2 . We show that all connected 2 -nearly Platonic graphs are necessarily balanced. This proves a recent conjecture by Keith, Froncek, and Kreher.

Finding the Maximum Independent Sets of Platonic Graphs Using Rydberg Atoms

A topology-preserving transformation of a Rydberg-atom system from a 3D surface to a plane is experimentally demonstrated using quantum wires, providing a feasible route toward large-scale quantum simulations.

2-nearly Platonic graphs

2-nearly Platonic graphs

A note on nearly Platonic graphs with connectivity one

A k -regular planar graph G is nearly Platonic when all faces but one are of the same degree while the remaining face is of a different degree. We show that no such graphs with connectivity one can exist. This complements a recent result by Keith, Froncek, and Kreher on non-existence of 2-connected nearly Platonic graphs.

Plane integral drawings of the platonic solid graphs with triangle faces

Plane integral drawings of the platonic solid graphs with triangle faces

Trivalent expanders, $(\Delta – Y)$-transformation, and hyperbolic surfaces

We construct a new family of trivalent expanders tessellating hyperbolic surfaces with large isometry groups. These graphs are obtained from a family of Cayley graphs of nilpotent groups via (Delta – Y) -transformations. We study combinatorial, topological and spectral properties of our trivalent graphs and their associated hyperbolic surfaces. We compare this family with Platonic graphs and their associated hyperbolic surfaces and see that they are generally very different with only one hyperbolic surface in the intersection. Finally, we provide a number theory free proof of the Ramanujan property for Platonic graphs and a special family of subgraphs.

Drawing Graphs on Few Circles and Few Spheres

Given a drawing of a graph, its visual complexity is defined as the number of geometrical entities in the drawing, for example, the number of segments in a straight-line drawing or the number of arcs in a circular-arc drawing (in 2D). Recently, Chaplick et al. [GD 2016] introduced a different measure for the visual complexity, the affine cover number, which is the minimum number of lines (or planes) that together cover a crossing-free straight-line drawing of a graph $G$ in 2D (3D). In this paper, we introduce the spherical cover number, which is the minimum number of circles (or spheres) that together cover a crossing-free circular-arc drawing in 2D (or 3D). It turns out that spherical covers are sometimes significantly smaller than affine covers. For complete, complete bipartite, and platonic graphs, we analyze their spherical cover numbers and compare them to their affine cover numbers as well as their segment and arc numbers. We also link the spherical cover number to other graph parameters such as treewidth and linear arboricity.

Perfect 3-Colorings of the Platonic Graph

Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte. A perfect m-coloring of a graph G with m colors is a partition of the vertex set of G into m parts $$A_1, \ldots,A_m$$ such that, for all $$i,j\in \lbrace 1,\ldots ,m\rbrace$$ , every vertex of $$A_i$$ is adjacent to the same number of vertices, namely $$a_{ij}$$ vertices, of $$A_j$$ . The matrix $$A=(a_{ij})_{i,j\in \lbrace 1,\ldots ,m\rbrace }$$ is called the parameter matrix. We study the perfect 3-colorings (also known as the equitable partitions into three parts) of the Platonic graphs. In particular, we classify all the realizable parameter matrices of perfect 3-colorings for the Platonic graphs.

Experimental test of quantum correlations from Platonic graphs

Great effort has been made in the investigation of contextual correlations between compatible observables due to their both fundamental and practical importance. The graph-theoretic approach to correlate events has been proved to be an effective method in the characterization of quantum contextuality, which implies that quantum violations of noncontextual inequalities derived in the noncontextual hidden-variable models should be achievable. Finding experimentally more friendly and theoretically more powerful noncontextual inequalities associated with specific graphs is of particular interest. Here we consider Platonic graphs to vindicate the quantum maximum predicted by graph theory and test the quantum violation against the mixedness of the state. Among these solids we refer particularly to the icosahedron to build the experiment, as it gives rise to the largest quantum-classical difference. The contextual correlations are demonstrated on quantum four-dimensional states encoded in the spatial modes of single photons generated from a defect in a bulk silicon carbide. Our results shed new light on the conflict between quantum and classical physics and may promote deep understanding of the connection between quantum theory, graph theory, and operator theory.

Perfect $2$-colorings of the Platonic graphs

In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and the icosahedral graph.

Unimodular graphs and Eisenstein sums

Motivated in part by combinatorial applications to certain sum-product phenomena, we introduce unimodular graphs over finite fields and, more generally, over finite valuation rings. We compute the spectrum of the unimodular graphs, by using Eisenstein sums associated with unramified extensions of such rings. We derive an estimate for the number of solutions to the restricted dot product equation $$a\cdot b=r$$a·b=r over a finite valuation ring. Furthermore, our spectral analysis leads to the exact value of the isoperimetric constant for half of the unimodular graphs. We also compute the spectrum of Platonic graphs over finite valuation rings, and products of such rings--e.g., $$\mathbb {Z}/(N)$$Z/(N). In particular, we deduce an improved lower bound for the isoperimetric constant of the Platonic graph over $$\mathbb {Z}/(N)$$Z/(N).

Isoperimetric Numbers of Regular Graphs of High Degree with Applications to Arithmetic Riemann Surfaces

We derive upper and lower bounds on the isoperimetric numbers and bisection widths of a large class of regular graphs of high degree. Our methods are combinatorial and do not require a knowledge of the eigenvalue spectrum. We apply these bounds to random regular graphs of high degree and the Platonic graphs over the rings $\mathbb{Z}_n$. In the latter case we show that these graphs are generally non-Ramanujan for composite $n$ and we also give sharp asymptotic bounds for the isoperimetric numbers. We conclude by giving bounds on the Cheeger constants of arithmetic Riemann surfaces. For a large class of these surfaces these bounds are an improvement over the known asymptotic bounds.

The spectrum of Platonic graphs over finite fields

We give a decomposition theorem for Platonic graphs over finite fields and use this to determine the spectrum of these graphs. We also derive estimates for the isoperimetric numbers of the graphs.

Hamiltonian paths on Platonic graphs

We develop a combinatorial method to show that the dodecahedron graph has, up to rotation and reflection, a unique Hamiltonian cycle. Platonic graphs with this property are called topologically uniquely Hamiltonian. The same method is used to demonstrate topologically distinct Hamiltonian cycles on the icosahedron graph and to show that a regular graph embeddable on the 2‐holed torus is topologically uniquely Hamiltonian.

Cubic inflation, mirror graphs, regular maps, and partial cubes

Partial cubes are, by definition, isometric subgraphs of hypercubes. Cubic inflation is an operation that transforms a 2-cell embedded graph G into a cubic graph embedded in the same surface; its result can be described as the dual of the barycentric subdivision of G. New concepts of mirror and pre-mirror graphs are also introduced. They give rise to a characterization of Platonic graphs (i) as pre-mirror graphs and (ii) as planar graphs of minimum degree at least three whose cubic inflation is a mirror graph. Using cubic inflation we find five new prime cubic partial cubes.

Cheeger constants of Platonic graphs

The Platonic graphs πn arise in several contexts, most simply as a quotient of certain Cayley graphs associated to the projective special linear groups. We show that when n=p is prime, πn can be viewed as a complete multigraph in which each vertex is itself a wheel on n+1 vertices. We prove a similar structure theorem for the case of an arbitrary prime power. These theorems are then used to obtain new upper bounds on the Cheeger constants of these graphs. These results lead immediately to similar results for Cayley graphs of the group PSL(2,Zn).

Minimum Integral Drawings of the Platonic Graphs

(1994). Minimum Integral Drawings of the Platonic Graphs. Mathematics Magazine: Vol. 67, No. 5, pp. 355-358.

Fluctuation theory for the Ehrenfest urn via electric networks

Using the electric network approach, we give very simple derivations for the expected first passage from the origin to the opposite vertex in thed-cube (i.e. the Ehrenfest urn model) and the Platonic graphs.

Fluctuation theory for the Ehrenfest urn via electric networks

Using the electric network approach, we give very simple derivations for the expected first passage from the origin to the opposite vertex in the d-cube (i.e. the Ehrenfest urn model) and the Platonic graphs.