MDPI - Publisher of Open Access Journals

30 pages, 4214 KiB

Open AccessArticle

Spectral Polynomials and Spectra of Graphs Beyond Cubic and Icosahedral Symmetries: n-Octahedra, n-Cubes, Symmetric and Semi-Symmetric Graphs, Giant Fullerene Cages and Generalized Petersen Graphs

by Krishnan Balasubramanian

Symmetry 2025, 17(2), 247; https://doi.org/10.3390/sym17020247 - 7 Feb 2025

Viewed by 1211

Abstract

We report the results of our computations of the spectral polynomials and spectra of a number of graphs possessing automorphism symmetries beyond cubic and icosahedral symmetries. The spectral (characteristic) polynomials are computed in fully expanded forms. The coefficients of these polynomials contain a wealth of combinatorial information that finds a number of applications in many areas including nanomaterials, genetic networks, dynamic stereochemistry, chirality, and so forth. This study focuses on a number of symmetric and semi-symmetric graphs with automorphism groups of high order. In particular, Heawood, Coxeter, Pappus, Möbius–Kantor, Tutte–Coxeter, Desargues, Meringer, Dyck, n-octahedra, n-cubes, icosahedral fullerenes such as C₈₀(I_h), golden supergiant C₂₄₀(I_h), Archimedean (I_h), and generalized Petersen graphs up to 720 vertices, among others, have been studied. The spectral polynomials are computed in fully expanded forms as opposed to factored forms. Several applications of these polynomials are briefly discussed. Full article

(This article belongs to the Collection Feature Papers in Chemistry)

► Show Figures

Figure 1

20 pages, 583 KiB

Open AccessArticle

Topological Symmetry Groups of the Petersen Graphs

by Deion Elzie, Samir Fridhi, Blake Mellor, Daniel Silva and Robin T. Wilson

Symmetry 2023, 15(6), 1267; https://doi.org/10.3390/sym15061267 - 15 Jun 2023

Viewed by 2430

Abstract

The topological symmetry group of an embedding

Γ

of an abstract graph

γ

S^{3}

is the group of automorphisms of

γ

that can be realized by homeomorphisms of the pair

(S^{3}, Γ)

. These groups are motivated [...] Read more.

The topological symmetry group of an embedding

Γ

of an abstract graph

γ

S^{3}

is the group of automorphisms of

γ

that can be realized by homeomorphisms of the pair

(S^{3}, Γ)

. These groups are motivated by questions about the symmetries of molecules in space. The Petersen family of graphs is an important family of graphs for many problems in low-dimensional topology, so it is desirable to understand the possible groups of symmetries of their embeddings in space. In this paper, we find all the groups that can be realized as topological symmetry groups for each of the graphs in the Petersen family. Along the way, we also complete the classification of the realizable topological symmetry groups for

K_{3, 3}

. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Symmetry/Asymmetry)

► Show Figures

Figure 1

15 pages, 415 KiB

Open AccessReview

The Genus of a Graph: A Survey

by Liangxia Wan

Symmetry 2023, 15(2), 322; https://doi.org/10.3390/sym15020322 - 23 Jan 2023

Cited by 1 | Viewed by 3146

Abstract

The problem of determining the genus for a graph can be dated to the Map Color Conjecture proposed by Heawood in 1890. This was implied to be a Thread Problem by Hilbert and Cohn-Vossen. The conjecture was finally established by Ringel, Youngs, and many other mathematicians. Subsequently, the genera of some special graphs with symmetry were determined. The study of genus embeddings of graphs is closely related to other invariants of a graph. Specifically, the computational complexity is dependent on the genus of the underlying graph for certain well-known NP-hard problems. In this survey, main construction techniques and certain criteria are stated in the topic of the genus of a graph. Most graphs with a known genus are listed. A new theorem is shown that the method of joint trees of a graph is reasonable. Moreover, a formal set is introduced, and related results are obtained. Although a cubic graph of Hamilton cycle is asymmetric, it is interesting that a set of associate surfaces of all its joint trees is a formal set with symmetry. Full article

(This article belongs to the Special Issue Symmetry in Graph and Hypergraph Theory)

► Show Figures