A graph is symmetric or arc-transitive if its automorphism group acts transitively on vertices, edges and arcs. Let p, q be odd primes with p, q ≥ 5 and X a q-valent symmetric graph of order 4p. In this paper, we proved that X K4p with 4p-1=q, X K2p,2p-2pK2 with 2p-1=q, the quotient graph of X is isomorphic to Kp,p and p=q, or K2p and 2p-1=q.

A family of sets is said to be symmetric if its automorphism group is transitive, and 3-wise intersecting if any three sets in the family have nonempty intersection. Frankl conjectured in 1981 that if A is a symmetric 3-wise intersecting family of subsets of {1, 2, . . . , n}, then |A| = o(2). Here, we give a short proof of Frankl’s conjecture using a sharp threshold result of Friedgut and Kalai.

The Johnson graph J(v, k) has, as vertices, the k-subsets of a v-set V and as edges the pairs of k-subsets with intersection of size k − 1. We introduce the notion of a neighbour-transitive code in J(v, k). This is a vertex subset Γ such that the subgroup G of graph automorphisms leaving Γ invariant is transitive on both the set Γ of ‘codewords’ and also the set of ‘neighbours’ of Γ, which are ...

A boolean function f(x1, ..., xn) is weakly symmetric if it is invariant under a transitive permutation group on its variables. A boolean function f(x1, ..., xn) is elusive if we have to check all x1,..., xn to determine the output of f(x1, ..., xn) in the worst-case. It is conjectured that every nontrivial monotone weakly symmetric boolean function is elusive, which has been open for a long ti...

A graph X is k-arc-transitive if its automorphism group acts transitively on the set of it-arcs of X. A circulant is a Cayley graph of a cyclic group. A classification of 2-arc-transitive circulants is given.

Slovenija Abstract A vertex-transitive graph is said to be 1 2-transitive if its automor-phism group is vertex and edge but not arc-transitive. Some recent results on 1 2-transitive graphs are given, a few open problems proposed , and possible directions in the research of the structure of these graphs discussed. 1 In the beginning Throughout this paper graphs are simple and, unless otherwise s...

Chordal graphs are important in the structural and algorithmic graph theory. A digraph analogue of chordal was introduced by Haskin Rose 1973 but has not been subject active studies until recently when a characterization semicomplete digraphs terms forbidden subdigraphs found Meister Telle. Locally digraphs, quasi-transitive extended amongst most popular generalizations digraphs. We extend subd...

It is shown that each subgroup of odd index in an alternating group degree at least 10 has all insoluble composition factors to be alternating. A classification then given 2-arc-transitive graphs order admitting or a symmetric group. This the second series papers aiming towards order.

We present a symmetric LDPC code with constant rate and constant distance (i.e. good LDPC code) that its constraint space is generated by the orbit of one constant weight constraint under a group action. Our construction provides the first symmetric LDPC good codes. This solves the main open problem raised by Kaufman and Wigderson in [4].

of finite symmetric groups.