Two-sided Cayley graphs

Journal of Graph Theory, 2015

We study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets L, R of a group G, we define the two-sided group digraph − → 2S(G; L, R) to have vertex set G, and an arc from x to y if and only if y = −1 xr for some ∈ L and r ∈ R. In common with Cayley graphs and digraphs, two-sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on L and R under which − → 2S(G; L, R) may be viewed as a simple graph of valency |L| • |R|, and we call such graphs two-sided group graphs. We also give sufficient conditions for two-sided group digraphs to be connected, vertex-transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8.

SIAM Journal on Discrete Mathematics, 2009

We study a class of Cayley graphs as models for interconnection networks. With focus on efficient communication we prove that for any graph in the class there exists a gossiping protocol which exhibits attractive features, and moreover we give an algorithm for constructing such a protocol. In particular, these hold for two important subclasses of graphs, namely, Cayley graphs admitting a complete rotation and Frobenius graphs of a certain type. For such Frobenius graphs, we obtain the minimum gossip time and give an optimal gossiping protocol under which messages are transmitted along shortest paths and each arc is used exactly once at each time step. Moreover, for such Frobenius graphs we construct an all-to-all routing which is a shortest path routing, arc-transitive, edge-and arc-uniform, and optimal for the edge-and arc-forwarding indices simultaneously.

Journal of Combinatorial Theory, Series B, 2020

A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. Such a graph Γ is called normal if H is normal in the full automorphism group of Γ, and normal edge-transitive if the normaliser of H in the full automorphism group of Γ is transitive on the edges of Γ. In this paper, we give a characterisation of normal edgetransitive bi-Cayley graphs, and in particular, we give a detailed description of 2-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic p-groups. We find that under certain conditions, 'normal edgetransitive' is the same as 'normal' for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most 6, and answer some open questions from the literature about 2-arc-transitive, half-arc-transitive and semisymmetric graphs.

Houston journal of mathematics

A. V. Kelarev and C. E. Praeger in [11] gave necessary and sufficient conditions for Cayley graphs of semigroups to be vertex-transitive. Also S. Fan and Y. Zeng in [4] gave a description of all vertex-transitive Cayley graphs of finite bands. In this paper we give similar descriptions for all vertex-transitive Cayley graphs of left groups. Also we extend some of the results to every direct product of a group and a band.

Graphs and Combinatorics, 2016

In this paper we work to classify which of the (n, k)-star graphs, denoted by S n,k , are Cayley graphs. Although the complete classification is left open, we derive infinite and non-trivial classes of both Cayley and non-Cayley graphs. We give a complete classification of the case when k = 2, showing that S n,2 is Cayley if and only if n is a prime power. We also give a sufficient condition for S n,3 to be Cayley and study other structural properties, such as demonstrating that S n,k always has a uniform shortest path routing.

2020

In this paper, we consider a generalization of Cayley graphs and digraphs (directed graphs) introduced by Iradmusa and Praeger. For non-empty subsets L,R of group G, two-sided group digraph −→ 2S(G;L,R) has been defined as a digraph having the vertex set G, and an arc from x to y if and only if y = l−1xr for some l ∈ L and r ∈ R. This article has strived to answer some open problems posed by Iradmusa and Praeger related to these graphs. Further, we determine sufficient conditions by which two-sided group graphs to be non-planar, and then we consider some specific cases on subsets L,R. We prove that the number of connected components of −→ 2S(G;L,R) is equal to the number of double cosets of the pair L,R when they are two subgroups of G.

Journal of Graph Theory, 1996

The Petersen graph on 10 vertices is the smallest example of a vertextransitive graph which is not a Cayley graph. In 1983, D. Marušič asked: for what values of n does there exist such a graph on n vertices? We give several new constructions of families of vertex-transitive graphs which are not Cayley graphs and complete the proof that, if n is divisible by p 2 for some prime p, then there is a vertex-transitive graph on n vertices which is not a Cayley graph unless n is p 2 , p 3 , or 12.

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.

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.

Discrete Mathematics, 2010

In this paper, we first give a characterization of Cayley graphs of rectangular groups. Then, vertex transitivity of Cayley graphs of rectangular groups is considered. Further, it is shown that Cayley graphs Cay(S, C) which are automorphism-vertex transitive, are in fact Cayley graphs of rectangular groups, if the subsemigroup generated by C is an orthodox semigroup. Finally, a characterization of vertex transitive graphs which are Cayley graphs of finite semigroups is concluded.