Classifying cubic edge-transitive graphs of order 8p (original) (raw)

ON THE CUBIC EDGE-TRANSITIVE GRAPHS OF ORDER 58p258p258p2

Journal of the Indonesian Mathematical Society, 2015

A graph is called edge-transitive, if its full automorphism group acts transitively on its edge set. In this paper, we inquire the existence of connected edge-transitive cubic graphs of order 58p 2 for each prime p. It is shown that only for p = 29, there exists a unique edge-transitive cubic graph of order 58p 2 .

Semisymmetric Cubic Graphs of Order 4p

2009

An regular graph is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. In this paper we prove that for every prime p(6= 5), there is no semisymmetric cubic graph of order 4pn, where n ≥ 1. 2000 Mathematics Subject Classification: 05C25, 20B25.

Cubic semisymmetric graphs of order

Discrete Mathematics, 2010

A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. By Folkman [J. Folkman, Regular line-symmetric graphs, J. Combin Theory 3 (1967) 215-232], there is no semisymmetric graph of order 2p or 2p 2 for a prime p and by Malnič, et al. [A. Malnič, D. Marušič, C.Q. Wang, Cubic edge-transitive graphs of order 2p 3 , Discrete Math. 274 (2004) 187-198], there exists a unique cubic semisymmetric graph of order 2p 3 , the so-called Gray graph of order 54. In this paper it is shown that a connected cubic semisymmetric graph of order 6p 3 exists if and only if p − 1 is divisible by 3. There are exactly two such graphs for a given order, which are constructed explicitly.

On cubic non-Cayley vertex-transitive graphs

Journal of Graph Theory, 2012

In 1983, the second author [D. Marušič, Ars Combinatoria 16B (1983), 297-302] asked for which positive integers n there exists a non-Cayley vertex-transitive graph on n vertices. (The term non-Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265-269] asked to determine the smallest valency ϑ(n) among valencies of non-Cayley vertex-transitive graphs of order n. As cycles are clearly Cayley graphs, ϑ(n) ≥ 3 for any non-Cayley number n. In this paper a goal is set to determine those non-Cayley numbers n for which ϑ(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non-Cayley vertex-transitive Contract grant sponsors: Agencija za raziskovalno dejavnost Republike Slovenije, research program P1-0285 (to K. K., D. M., C. Z.); Agencija za raziskovalno dejavnost Republike Slovenije, proj. mladi raziskovalci (to C. Z.).

A more detailed classification of symmetric cubic graphs

2000

A graph is symmetric if its automorphism group acts transitively on the arcs of , and s-regular if its automorphism group acts regularly on the set of s-arcs of . Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. Djokoviÿc and Miller (1980) proved that there are seven types of arc-transitive

Semisymmetric cubic graphs of order 16p 2

Proceedings - Mathematical Sciences, 2010

An undirected graph without isolated vertices is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. In this paper, we inquire the existence of connected semisymmetric cubic graphs of order 16p 2 . It is shown that for every odd prime p, there exists a semisymmetric cubic graph of order 16p 2 and its structure is explicitly specified by giving the corresponding voltage rules generating the covering projections.

Semisymmetric cubic graphs of orders 36p, 36p2

Filomat, 2013

A cubic graph is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. The semisymmetric cubic graphs of orders 6p and 6p 2 were classified in (Com. in Algebra, 28 (6) (2000) 2685-2715) and (Science in China Ser. A Mathematics, 47 (2004) No.1 1-17), respectively. In this paper we first classify all connected cubic semisymmetric graphs of order 36p for each prime p and also classify all connected cubic semisymmetric graphs of order 36p 2 , where p 5 and 7 is a prime.

Tetravalent vertex-transitive graphs of order 6p6p6p

2022

A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the tetravalent vertex-transitive non-Cayley graphs of order 6p are classified for each prime p. 1. Introduction In this paper we consider undirected finite connected graphs without loops or multiple edges. For a graph X we use V (X), E(X), A(X) and Aut(X) to denote its vertex set, edge set, arc set and its full automorphism group, respectively. For u, v ∈ V (X), u ∼ v represents that u is adjacent to v, and is denoted by {u, v} the edge incident to u and v in X, and N X (u) is the neighborhood of u in X, that is, the set of vertices adjacent to u in X. A graph X is said to be G-vertex-transitive, G-edge-transitive and G-arctransitive (or G-symmetric) if G ≤ Aut(X) acts transitively on V (X), E(X) and A(X), respectively. In the special case, if G = Aut(X) then X is said to be vertex-transitive, edge-transitive and arc-transitive (or symmetric). An s-arc in a graph X is an ordered (s + 1)-tuple (v 0 , v 1 , • • • , v s) of vertices of X such that v i−1 is adjacent to v i for 1 ≤ i ≤ s, and v i−1 = v i+1 for 1 ≤ i ≤ s; in other words, a directed walk of length s which never includes a backtracking. A graph X is said to be s-arc-transitive if Aut(X) is transitive on the set of s-arcs in X. A subgroup of the automorphism group of a graph X is said to be s-regular if it acts regularly on the set of s-arcs of X. Recall that a permutation group G acting on a set Ω is called semiregular if the stabilizer of α ∈ G, G α = 1 for all α ∈ G and is called regular if it is semiregular and transitive.

CUBIC SYMMETRIC GRAPHS OF ORDERS 36p AND 36p2

A graph is symmetric, if its automorphism group is transitive on the set of its arcs. In this paper, we classify all the connected cubic symmetric graphs of order 36p and 36p 2 , for each prime p, of which the proof depends on the classification of finite simple groups.

A CLASSIFICATION OF SEMISYMMETRIC CUBIC GRAPHS OF ORDER 28p&sup2

Journal of the Indonesian Mathematical Society, 2010

A graph is said to be semisymmetric if its full automorphism group actstransitively on its edge set but not on its vertex set. In this paper, we prove thatthere is only one semisymmetric cubic graph of order 28p2, where p is a prime.DOI : http://dx.doi.org/10.22342/jims.16.2.38.139-143