scholarly journals Consistent Cycles in $1\over2$-Arc-Transitive Graphs

10.37236/94 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Marko Boben ◽  
Štefko Miklavič ◽  
Primož Potočnik
Keyword(s):  
Directed Cycle ◽  
Vertex Transitive ◽  
Edge Transitive ◽  
Transitive Graphs

A directed cycle $C$ of a graph is called $1\over k$-consistent if there exists an automorphism of the graph which acts as a $k$-step rotation of $C$. These cycles have previously been considered by several authors in the context of arc-transitive graphs. In this paper we extend these results to the case of graphs which are vertex-transitive, edge-transitive but not arc-transitive.

Vertex and Edge Transitive, but not 1-Transitive, Graphs

1970 ◽  
Vol 13 (2) ◽  
pp. 231-237 ◽  
Author(s):  
I. Z. Bouwer
Keyword(s):  
Undirected Graph ◽  
The Other ◽  
Vertex Transitive ◽  
Edge Transitive ◽  
Transitive Graphs

A (simple, undirected) graphGisvertex transitiveif for any two vertices ofGthere is an automorphism ofGthat maps one to the other. Similarly,Gisedge transitiveif for any two edges [a,b] and [c,d] ofGthere is an automorphism ofGsuch that {c,d} = {f(a),f(b)}. A 1-pathofGis an ordered pair (a,b) of (distinct) verticesaandbofG, such thataandbare joined by an edge.Gis 1-transitiveif for any two 1-paths (a,b) and (c,d) ofGthere is an automorphismfofGsuch thatc=f(a) andd=f(b). A graph isregular of valency dif each of its vertices is incident with exactlydof its edges.

scholarly journals CUBIC EDGE-TRANSITIVE GRAPHS OF ORDER 8p2

2008 ◽  
Vol 77 (2) ◽  
pp. 315-323 ◽  
Author(s):  
MEHDI ALAEIYAN ◽  
MOHSEN GHASEMI
Keyword(s):  
Undirected Graph ◽  
Cubic Graphs ◽  
Transitive Graph ◽  
Symmetric Graphs ◽  
Vertex Transitive ◽  
Regular Line ◽  
Edge Transitive ◽  
Transitive Graphs

AbstractA simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime. It was shown by Folkman [J. Folkman, ‘Regular line-symmetric graphs’, J. Combin. Theory3 (1967), 215–232] that a regular edge-transitive graph of order 2p or 2p2 is necessarily vertex-transitive. In this paper an extension of his result in the case of cubic graphs is given. It is proved that every cubic edge-transitive graph of order 8p2 is vertex-transitive.

Trivalent Non-symmetric Vertex-Transitive Graphs of Order at Most 150

2008 ◽  
Vol 15 (03) ◽  
pp. 379-390 ◽  
Author(s):  
Xuesong Ma ◽  
Ruji Wang
Keyword(s):  
Automorphism Group ◽  
Bipartite Graphs ◽  
Type Ii ◽  
Symmetric Graph ◽  
Trivalent Graph ◽  
Block Graphs ◽  
Group A ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut (X) acts transitively on the vertices and the vertex-stabilizer Av of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

Properties of Classes of Random Graphs

1994 ◽  
Vol 3 (4) ◽  
pp. 435-454 ◽  
Author(s):  
Neal Brand ◽  
Steve Jackson
Keyword(s):  
Random Graphs ◽  
Higher Order ◽  
Order Property ◽  
First Order ◽  
New Class ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Almost All ◽  
Transitive Graphs

In [11] it is shown that the theory of almost all graphs is first-order complete. Furthermore, in [3] a collection of first-order axioms are given from which any first-order property or its negation can be deduced. Here we show that almost all Steinhaus graphs satisfy the axioms of almost all graphs and conclude that a first-order property is true for almost all graphs if and only if it is true for almost all Steinhaus graphs. We also show that certain classes of subgraphs of vertex transitive graphs are first-order complete. Finally, we give a new class of higher-order axioms from which it follows that large subgraphs of specified type exist in almost all graphs.

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