A bicirculant is a graph admitting an automorphism with two cycles of equal length in its cycle decomposition. A graph is said to be arc-transitive if its automorphism group acts transitively on the set of its arcs. All cubic and tetravalent arc-transitive bicirculants are known, and this paper gives a complete classification of connected pentavalent arc-transitive bicirculants. In particular, it is shown that, with the exception of seven particular graphs, a connected pentavalent bicirculant is arc-transitive if and only if it is isomorphic to a Cayley graph Cay(D 2n , {b, ba, ba r +1 , ba r 2 +r +1 , ba r 3 +r 2 +r +1 }) on the dihedral group D 2n = a, b | a n = b 2 = baba = 1 , where r ∈ Z * n such that r 4 + r 3 + r 2 + r + 1 ≡ 0 (mod n).

Given natural numbers n ≥ 3 and 1≤ a, r ≤ n−1, the rose window graph R n (a, r) is a quartic graph with vertex set {x i |i ∈ Z n }∪{y i |i ∈ Z n } and edge set {{x i ,

A graph X is said to be distance-balanced if for any edge uv of X, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. A graph X is said to be strongly distance-balanced if for any edge uv of X and any integer k, the number of vertices at distance k from u and at distance k + 1 from v is equal to the number of vertices at distance k + 1 from u and at distance k from v. Obviously, being distance-balanced is metrically a weaker condition than being strongly distance-balanced. In this paper, a connection between symmetry properties of graphs and the metric property of being (strongly) distance-balanced is explored. In particular, it is proved that every vertex-transitive graph is strongly distance-balanced.A graph is said to be semisymmetric if its automorphism group acts transitively on its edge set, but does not act transitively on its vertex set. An infinite family of semisymmetric graphs, which are not distance-balanced, is constructed.Finally, we give a complete classification of strongly distance-balanced graphs for the following infinite families of generalized Petersen graphs: GP(n, 2), GP(5k +1, k), GP(3k ± 3, k), and GP(2k + 2, k). IntroductionLet X be a graph with diameter d, and let V (X) and E(X) denote the vertex set and the edge set of X, respectively. For u, v ∈ V (X), we let d(u, v) denote the minimal path-length distance between u and v. We say that X is distance-balanced ifholds for an arbitrary pair of adjacent vertices u and v of X. These graphs were, at least implicitly, first studied by Handa [8] who considered distance-balanced partial cubes. The term itself, however, is due to Jerebic, Klavžar and Rall [11] who studied distance-balanced graphs in the framework of various kinds of graph products.

As a generalization of undirected strongly regular graphs, a digraph X without loops, of valency k and order v is said to be a (v, k, μ, λ,t)-directed strongly regular graph whenever for any vertex u of X there are t undirected edges having u as an endvertex and for every two different vertices u and w of X the number of paths of length 2 starting at u and ending at w is λ or μ depending only on whether uw is an arc of X or not. An m-Cayley digraph of a group H is a digraph admitting a semiregular group of automorphisms having m orbits, all of equal length, isomorphic to H. In this paper, the structure of directed strongly regular 2-Cayley graphs of cyclic groups is investigated. In particular, the arithmetic conditions on parameters v, k, μ, λ, and t are given.Also, several infinite families of directed strongly regular graphs which are also 2-Cayley digraphs of abelian groups are constructed.

A graph is said to be cyclically k-edge-connected, if at least k edges must be removed to disconnect it into two components, each containing a cycle. Such a set of k edges is called a cyclic-k-edge cutset and it is called a trivial cyclic-k-edge cutset if at least one of the resulting two components induces a single k-cycle.It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclically 5-edge-connected. In this article it is shown that a fullerene F containing a nontrivial cyclic-5-edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is shown that F has a Hamilton cycle, and as a consequence at least 15 · 2 n/20−1/2 perfect matchings, where n is the order of F.