Arc-transitive cyclic covers of symmetric graphs with a specific prime valency and order twice a prime have been studied by nearly ten papers in the literature. In this paper, we will give a general ...characterization of arc-transitive cyclic covers of graphs with any prime valency and order twice a prime.
In this paper, we classify antipodal distance-regular graphs of diameter three that admit an arc-transitive action of SU3(q). In particular, we find a new infinite family of distance-regular ...antipodal r-covers of a complete graph on q3+1 vertices, where q is odd and r is any divisor of q+1 such that (q+1)∕r is odd. Further, we find several new constructions of arc-transitive antipodal distance-regular graphs of diameter three in case λ=μ.
A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. ...In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is G‐symmetric if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by G, then Γ is an imprimitive G‐symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.
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.
A graph-theoretic environment is used to study the connection between imprimitivity and semiregularity, two concepts arising naturally in the context of permutation groups. Among other, it is shown ...that a connected arc-transitive graph admitting a nontrivial automorphism with two orbits of odd length, together with an imprimitivity block system consisting of blocks of size 2, orthogonal to these two orbits, is either the canonical double cover of an arc-transitive circulant or the wreath product of an arc-transitive circulant with the empty graph
K
¯
2
on two vertices.
We classify finite primitive permutation groups having a suborbit of length 5. As a corollary, we obtain a classification of finite vertex-primitive graphs of valency 5. In the process, we also ...classify finite almost simple groups that have a maximal subgroup isomorphic to Alt(5) or Sym(5).
A graph is called arc-transitive (or symmetric) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ...ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer k, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is kp for some prime p, and that if d≥4, then there exist only finitely many connected d-valent 2-arc-transitive graphs whose order is kp or kp2 for some prime p. We also prove that there are infinitely many (even) values of k for which there are only finitely many connected 3-valent symmetric graphs of order kp where p is prime.
LDPC codes constructed from cubic symmetric graphs Crnković, Dean; Rukavina, Sanja; Šimac, Marina
Applicable algebra in engineering, communication and computing,
11/2022, Letnik:
33, Številka:
5
Journal Article
Recenzirano
Odprti dostop
Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from ...cubic symmetric graphs. The codes constructed are (3, 3)-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the codes obtained and present bounds for the code parameters, the dimension and the minimum distance. Furthermore, we give an expression for the variance of the syndrome weight of the codes constructed. Information on the LDPC codes constructed from bipartite cubic symmetric graphs with less than 200 vertices is presented as well. Some of the codes constructed are optimal, and some have an additional property of being self-orthogonal or linear codes with complementary dual (LCD codes).
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