Vertex-minimal graphs with dihedral symmetry I

Let $D_{2n}$ denote the dihedral group of order $2n$, where $n\ge 3$. In this article, we build upon the findings of Haggard and McCarthy who, for certain values of $n$, produced a vertex-minimal graph with dihedral symmetry. Specifically, Haggard considered the situation when $\frac{n}{2}$ or $n$ is a prime power, and McCarthy investigated the case when $n$ is not divisible by $2$, $3$ or $5$. In this article, we assume $n$ is not divisible by $4$ and construct a vertex-minimal graph whose automorphism group is isomorphic to $D_{2n}$.     

Fulleroids with dihedral symmetry

Fulleroids are cubic convex polyhedra with faces of size 5 or greater. They are suitable as models of hypothetical all-carbon molecules. In this paper, sufficient and necessary conditions for existence of fulleroids of dihedral symmetry types and with pentagonal and n-gonal faces only depending on number n are presented. Either infinite series of examples are found to prove existence, or nonexistence is proved using symmetry invariants.     

On the fixing sets of dihedral groups

The fixing number of a graph $\Gamma$ is the minimum number of labeled vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of $\Gamma$. The fixing set of a finite group $G$ is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to $G$. Previously, authors have studied the fixing sets of both abelian groups and symmetric groups. In this article, we determine the fixing set of the dihedral group.     

Vertex-minimal planar graphs with cyclic 2-group symmetry

Journal of Algebraic Combinatorics - For the positive integer m, let $${mathbb {Z}}_m$$ denote the cyclic group of order m. Vertex-minimal planar graphs with prescribed automorphism group...     

On dihedral flows in embedded graphs

Let be a multigraph with for each vertex a cyclic order of the edges incident with it. For , let be the dihedral group of order . Define . Goodall et al in 2016 asked whether admits a nowhere-identity -flow if and only if it admits a nowhere-identity -flow with (a “nowhere-identity dihedral -flow”). We give counterexamples to this statement and provide general obstructions. Furthermore, the complexity of deciding the existence of nowhere-identity -flows is discussed. Lastly, graphs in which the equivalence of the existence of flows as above is true are described. We focus particularly on cubic graphs.     

A note on the automorphism groups of cubic Cayley graphs of finite simple groups

For a finite group G, a Cayley graph on G is said to be normal if . In this note, we prove that connected cubic non-symmetric Cayley graphs of the ten finite non-abelian simple groups G in the list of non-normal candidates given in [X.G. Fang, C.H. Li, J. Wang, M.Y. Xu, On cubic Cayley graphs of finite simple groups, Discrete Math. 244 (2002) 67-75] are normal.     

Enumeration of cubic Cayley graphs on dihedral groups

Let p be an odd prime, and D_2p =<a, b|a^p = b² = 1, bab = a^-1 the dihedral group of order 2p. In this paper, we completely classify the cubic Cayley graphs on D_2p up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on D_2p are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on D_2p by using Gauss' celebrated law of quadratic reciprocity.     

Symmetric colorings of the dihedral group

Let G be a finite group and let r∈?. An r-coloring of G is any mapping χ:G→{1,…,r}. Colorings χ and ψ are equivalent if there exists g∈G such that χ(xg^?1) = ψ(x) for every x∈G. A coloring χ is symmetric if there exists g∈G such that χ(gx^?1g) = χ(x) for every x∈G. Let S_r(G) denote the number of symmetric r-colorings of G and s_r(G) the number of equivalence classes of symmetric r-colorings of G. We count S_r(G) and s_r(G) in the case where G is the dihedral group D_n.     

Edge-primitive Cayley graphs on abelian groups and dihedral groups

A graph is called edge-primitive if its automorphism group acts primitively on its edge set. In 1973, Weiss (1973) determined all edge-primitive graphs of valency three, and recently Guo et al. (2013,2015) classified edge-primitive graphs of valencies four and five. In this paper, we determine all edge-primitive Cayley graphs on abelian groups and dihedral groups.     

Note on calculating wiener numbers of molecular graphs with symmetry

The authors provided a simple method for calculating Wiener numbers of moleculargraphs with symmetry in 1997. This paper intends to further improve on it and simplifies thecalculation of the Wiener numbers of the molecular graphs.     

Nowhere-zero 3-flows in Cayley graphs on generalized dihedral group and generalized quaternion group

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow. In this paper, we show that this conjecture is true for Cayley graph on generalized dihedral groups and generalized quaternion groups, which generalizes the result of F. Yang and X. Li [Inform. Process. Lett., 2011, 111: 416–419]. We also generalizes an early result of M. Nánásiová and M. ?koviera [J. Algebraic Combin., 2009, 30: 103–110].     

Yetter–Drinfeld Modules over the Hopf–Ore Extension of the Group Algebra of Dihedral Group

Let k be an algebraically closed field of characteristic zero, and D _n be the dihedral group of order 2n, where n is a positive even integer. In this paper, we investigate Yetter-Drinfeld modules over the Hopf-Ore extension A(n, 0) of kD _n . We describe the structures and properties of simple Yetter-Drinfeld modules over A(n, 0), and classify all simple Yetter-Drinfeld modules over A(n, 0).     

On the action of a group on a graph, II

This paper is a continuation of the survey by the author (V.I. Trofimov, On the action of a group on a graph, Acta Appl. Math. 29 (1992) 161–170) on some results concerning groups of automorphisms of locally finite vertex-symmetric graphs.