Partial cubes are graphs isometrically embeddable into hypercubes. In this paper it is proved that every cubic, vertex-transitive partial cube is isomorphic to one of the following graphs: K2ƒC2n, for some n ≥ 2, generalized Petersen graph G(10, 3), permutahedron, truncated cuboctahedron, or truncated icosidodecahedron. This complete classification of cubic, vertex-transitive partial cubes, a f...

a recursive-circulant $g(n; d)$ is defined to be acirculant graph with $n$ vertices and jumps of powers of $d$.$g(n; d)$ is vertex-transitive, and has some strong hamiltonianproperties. $g(n; d)$ has a recursive structure when $n = cd^m$,$1 leq c < d $ [10]. in this paper, we will find the automorphismgroup of some classes of recursive-circulant graphs. in particular, wewill find that the autom...

In line with symmetrical graphs such as Cayley graphs and vertex transitive graphs, we introduce a new class of symmetrical graphs called diametrically uniform graphs. The class of diametrically uniform graphs includes vertex transitive graphs and hence Cayley graphs. A tree t-spanner of graph G is a spanning tree T in which the distance between every pair of vertices is at most t times their d...

An innnite familiy of 1 2-transitive graphs of valency 4 with alternating cycles of length 4 is given. Besides, an innnite family of 1 2-transitive graphs of valency 4 with vertex stabilizer isomorphic to Z 2 Z 2 , is constructed.

We call an undirected graph X half-transitive if the automorphism group Aut X of X acts transitively on the vertex set and edge set but not on the set of ordered pairs of adjacent vertices of X. In this paper we determine all half-transitive graphs of order p3 and degree 4, where p is an odd prime; namely, we prove that all such graphs are Cayley graphs on the non-Abelian group of order p3 and ...

We show that an action of SL(2, p), p ≥ 7 an odd prime such that 4 6 | (p − 1), has exactly two orbital digraphs Γ1, Γ2, such that Aut(Γi) admits a complete block system B of p + 1 blocks of size 2, i = 1, 2, with the following properties: the action of Aut(Γi) on the blocks of B is nonsolvable, doubly-transitive, but not a symmetric group, and the subgroup of Aut(Γi) that fixes each block of B...

It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency.

We consider economical representations for the path information in a directed graph. A directed graph G is said to be a transitive reduction of the directed graph G provided that (i) G has a directed path from vertex u to vertex v if and only if G has a directed path from vertex u to vertex v, and (ii) there is no graph with fewer arcs than G satisfying condition (i). Though directed graphs wit...

For a positive integer n, does there exist a vertex-transitive graph r on n vertices which is not a Cayley graph, or, equivalently, a graph r on n vertices such that Aut F is transitive on vertices but none of its subgroups are regular on vertices? Previous work (by Alspach and Parsons, Frucht, Graver and Watkins, MaruSic and Scapellato, and McKay and the second author) has produced answers to ...

We will discuss three ways to bound the chromatic number on a Cayley graph. (a) If the connection set contains information about a smaller graph, then these two graphs are related. Using this information, we will show that Cayley graphs cannot have chromatic number three. (b) We will prove a general statement that all vertex-transitive maximal triangle-free graphs on n vertices with valency gre...