A graph Γ is G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. If V(Γ) admits a nontrivial G-invariant partition B such that ...for blocks B,C∈B adjacent in the quotient graph ΓB of Γ relative to B, exactly one vertex of B has no neighbour in C, then Γ is called an almost multicover of ΓB. In this case an incidence structure with point set B arises naturally, and it is a (G,2)-point-transitive and G-block-transitive 2-design if in addition ΓB is a complete graph. In this paper we classify all G-symmetric graphs Γ such that (i) B has block size |B|≥3; (ii) ΓB is complete and almost multi-covered by Γ; (iii) the incidence structure involved is a linear space; and (iv) G contains a regular normal subgroup which is elementary abelian. This classification together with earlier results in Gardiner and Praeger (2018), Giulietti et al. (2013) and Fang et al. (2016) completes the classification of symmetric graphs satisfying (i) and (ii).
In this paper, we study the flag graph FG(P) of a regular abstract polytope P from two aspects of Cayley graphs: Hamiltonicity and Cayley index. We show that FG(P) has a Hamiltonian cycle, and ...introduce the Cayley index of P as the fraction |Aut(FG(P))|∕|Γ(P)|, where Γ(P) is the automorphism group of P. A new construction of arc-transitive tetravalent graphs will be described by means of regular abstract polyhedra of Cayley index larger than 1. In addition, polyhedra of type {p,q} such that p≤5 or q≤5 that have Cayley index larger than 1 are characterized.
A graph Γ is called G-symmetric if it admits G as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of G-symmetric graphs Γ with ...V(Γ) admitting a nontrivial G-invariant partition B such that there is exactly one edge of Γ between any two distinct blocks of B. This is achieved by giving a classification of (G,2)-point-transitive and G-block-transitive designs D together with G-orbits Ω on the flag set of D such that Gσ,L is transitive on L∖{σ} and L∩N={σ} for distinct (σ,L),(σ,N)∈Ω, where Gσ,L is the setwise stabilizer of L in the stabilizer Gσ of σ in G. Along the way we determine all imprimitive blocks of Gσ on V∖{σ} for every 2-transitive group G on a set V, where σ∈V.
Medial Symmetry Type Graphs Hubard, Isabel; Del Río Francos, María; Orbanić, Alen ...
The Electronic journal of combinatorics,
08/2013, Letnik:
20, Številka:
3
Journal Article
Recenzirano
Odprti dostop
A $k$-orbit map is a map with its automorphism group partitioning the set of flags into $k$ orbits. Recently $k$-orbit maps were studied by Orbanić, Pellicer and Weiss, for $k \leq 4$. In this paper ...we use symmetry type graphs to extend such study and classify all the types of $5$-orbit maps, as well as all self-dual, properly and improperly, symmetry type of $k$-orbit maps with $k\leq 7$. Moreover, we determine, for small values of $k$, all types of $k$-orbits maps that are medial maps. Self-dualities constitute an important tool in this quest.
A map on a closed surface is a two-cell embedding of a finite connected graph. Maps on surfaces are conveniently described by certain trivalent graphs, known as flag graphs. Flag graphs themselves ...may be considered as maps embedded in the same surface as the original graph. The flag graph is the underlying graph of the dual of the barycentric subdivision of the original map. Certain operations on maps can be defined by appropriate operations on flag graphs. Orientable surfaces may be given consistent orientations, and oriented maps can be described by a generating pair consisting of a permutation and an involution on the set of arcs (or darts) defining a partially directed arc graph. In this paper we describe how certain operations on maps can be described directly on oriented maps via arc graphs.
Edge even graceful labeling is a new type of labeling since it was introduced in 2017 by Elsonbaty and Daoud (Ars Combinatoria 130:79–96, 2017). In this paper, we obtained an edge even graceful ...labeling for some path-related graphs like Y- tree, the double star
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Flag graphs have been used in the past for describing maps on closed surfaces. In this paper we use them for the first time in mathematical chemistry for describing benzenoids and some other similar ...structures. Examples include catacondensed and pericondensed benzenoids. Several theorems are included. Symmetries of benzenoid systems, flag graphs, and symmetry type graphs are briefly discussed.
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