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.

An Edge but not Vertex Transitive Cubic Graph*

1968 ◽  
Vol 11 (4) ◽  
pp. 533-535 ◽  
Author(s):  
I. Z. Bouwer
Keyword(s):  
Undirected Graph ◽  
The Other ◽  
Cubic Graph ◽  
Vertex Transitive ◽  
Multiple Edges ◽  
Edge Transitive

Let G be an undirected graph, without loops or multiple edges. An automorphism of G is a permutation of the vertices of G that preserves adjacency. G is vertex transitive if, given any two vertices of G, there is an automorphism of the graph that maps one to the other. Similarly, G is edge transitive if for any two edges (a, b) and (c, d) of G there exists an automorphism f of G such that {c, d} = {f(a), f(b)}. A graph is regular of degree d if each vertex belongs to exactly d 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.

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.

scholarly journals Cayley Graphs Without a Bounded Eigenbasis

2020 ◽  
Author(s):  
Ashwin Sah ◽  
Mehtaab Sawhney ◽  
Yufei Zhao
Keyword(s):  
Cayley Graph ◽  
Orthonormal Basis ◽  
Cayley Graphs ◽  
The Other ◽  
Transitive Graph ◽  
Classic Result ◽  
Other Hand ◽  
Small Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graph

Abstract Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.

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.

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