We study some properties of the Cayley graph of the R.Thompson’s group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is ...

We characterize Cayley graphs of abelian groupswhich admit a nowhere-zero 3-flow. In particular, we prove that every k-valent Cayley graph of an abelian group, where k 4, admits a nowhere-zero

A digraph Γ is called n-Cayley digraph over a group G, if there exists a semiregular subgroup RG of Aut(Γ) isomorphic to G with n orbits. In this paper, we represent the adjacency matrix of Γ as a diagonal block matrix in terms of irreducible representations of G and determine its characteristic polynomial. As corollaries of this result we find: the spectrum of semi-Cayley graphs over abelian g...

The connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertex-transitive graphs. First, the connective constant decreases strictly when the graph is replaced by a nontrivial quotient graph. Second, the connective constant increases strictly when a quasitransitiv...

A graph is vertex-transitive if its automorphism group acts transitively on its vertices. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the cubic vertextransitive non-Cayley graphs of order 8p are classified for each prime p. It follows from this classification that there are two sporadic and two infini...

Cayley graph is a graph constructed out of a group Γ and a generating set A ⊆ Γ. When Γ = Zn, the corresponding Cayley graph is called as a circulant graph and denoted by Cir(n, A). In this paper, we attempt to find the total domination number of Cir(n, A) for a particular generating set A of Zn and a minimum total dominating set of Cir(n, A). Further, it is proved that Cir(n, A) is 2-total exc...

Let S be a set of transpositions generating the symmetric group Sn (n ≥ 5). The transposition graph of S is defined to be the graph with vertex set {1, . . . , n}, and with vertices i and j being adjacent in T (S) whenever (i, j) ∈ S. In the present note, it is proved that two transposition graphs are isomorphic if and only if the corresponding two Cayley graphs are isomorphic. It is also prove...

Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when the domain bounded by a graph of a function defined on Rn is also homogeneou...

In this work we study different properties of graph symmetry. The first, is vertex transitivity, which informally says, that each vertex has the same global view of the graph. We give a simple one-sided error testing algorithm for vertex transitivity in the dense-graphs model, whose query complexity is Θ̃(n). In addition, we prove that any one-sided error testing algorithm for vertex transitivit...

The classical Lovász conjecture says that every connected Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log2 |G|, such that the corresponding Cayley graph contains a Hamiltonian cycle. We also present an explicit construction...