A Family of Tetravalent Half-transitive Graphs

1994

Abstract The Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.

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.

Graphs and Combinatorics, 2002

In connection with the classification problem for non-Cayley tetravalent metacirculant graphs, three families of special tetravalent metacirculant graphs, denoted by U 1 , U 2 and U 3 , have been defined . It has also been shown [11] that any non-Cayley tetravalent metacirculant graph is isomorphic to a union of disjoint copies of a non-Cayley graph in one of the families U 1 , U 2 or U 3 . A natural question raised from the result is whether all graphs in these families are non-Cayley. We have proved recently in [12] that every graph in U 2 is non-Cayley. In this paper, we show that every graph in U 1 is also a connected non-Cayley graph and find an infinite class of connected non-Cayley graphs in the family U 3 .

Indian Journal of Pure and Applied Mathematics, 2020

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s + 1)-arcs. Let p be a prime. In this article a complete classification of tetravalent s-transitive graphs of order 5p 2 is given.

Bulletin of the Australian Mathematical Society, 1999

An approach to analysing the family of Cayley graphs for a finite group G is given which identifies normal edge-transitive Cayley graphs as a sub-family of central importance. These are the Cayley graphs for G for which a subgroup of automorphisms exists which both normalises G and acts transitively on edges. It is shown that, for a nontrivial group G, each normal edge-transitive Cayley graph for G has at least one homomorphic image which is a normal edge-transitive Cayley graph for a characteristically simple quotient group of G. Moreover, given a normal edge-transitive Cayley graph ΓH for a quotient group G/H, necessary and sufficient conditions are obtained for the existence of a normal edge-transitive Cayley graph Γ for G which has ΓH as a homomorphic image, and a method for obtaining all such graphs Γ is given.

Journal of Algebraic Combinatorics, 2013

We investigate connected normal 2-geodesic transitive Cayley graphs Cay(T , S). We first prove that if Cay(T , S) is neither cyclic nor K 4[2] , then a \ {1} S for all a ∈ S. Next, as an application, we give a reduction theorem proving that each graph in this family which is neither a complete multipartite graph nor a bipartite 2arc transitive graph, has a normal quotient that is either a complete graph or a Cayley graph in the family for a characteristically simple group. Finally we classify complete multipartite graphs in the family.

Iraqi journal of science, 2023

𝑆 -1 ⊆ 𝑆. Suppose that 𝐶𝑎𝑦(𝐺, 𝑆) is the Cayley graph whose vertices are all elements of 𝐺 and two vertices 𝑥 and 𝑦 are adjacent if and only if 𝑥𝑦 -1 ∈ 𝑆. In this paper,we introduce the generalized Cayley graph denoted by 𝐶𝑎𝑦 𝑚 (𝐺, 𝑆) which is a graph with a vertex set consisting of all column matrices 𝑋 𝑚 in which all components are in 𝐺 and two vertices 𝑋 𝑚 and 𝑌 𝑚 are adjacent if and only if 𝑋 𝑚 [(𝑌 𝑚 ) -1 ] 𝑡 ∈ 𝑀(𝑆), where 𝑌 𝑚 -1 is a column matrix that each entry is the inverse of the similar entry of 𝑌 𝑚 and 𝑀(𝑆) is 𝑚 × 𝑚 matrix with all entries in 𝑆 , [𝑌 -1 ] 𝑡 is the transpose of 𝑌 -1 and 𝑚 ≥ 1 and m∈ 𝑁. We aim to provide some basic properties of the new graph and determine the structure of 𝐶𝑎𝑦 𝑚 (𝐺, 𝑆) when 𝐶𝑎𝑦(𝐺, 𝑆) is a complete graph 𝐾 𝑛 for every 𝑚 ≥ 2, n ≥ 3 and n, m∈ 𝑁 .

European Journal of Combinatorics, 2003

We investigate Cayley graphs of semigroups and show that they sometimes enjoy properties analogous to those of the Cayley graphs of groups.

Journal of Combinatorial Theory, Series B, 1999

In this paper we study finite, connected, 4-valent graphs X which admit an action of a group G which is transitive on vertices and edges, but not transitive on the arcs of X. Such a graph X is said to be (G, 1Â2)-transitive. The group G induces an orientation of the edges of X, and a certain class of cycles of X (called alternating cycles) determined by the group G is identified as having an important influence on the structure of X. The alternating cycles are those in which consecutive edges have opposite orientations. It is shown that X is a cover of a finite, connected, 4-valent, (G, 1Â2)-transitive graph for which the alternating cycles have one of three additional special properties, namely they are tightly attached, loosely attached, or antipodally attached. We give examples with each of these special attachment properties, and moreover we complete the classification (begun in a separate paper by the first author) of the tightly attached examples.

… ASI Series C …, 1997