No abstract

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O'Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T G Aut(T ) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to 3 D 4 (q), then T is line-transitive, where q is a power of the prime p.

This article is a contribution to the study of the automorphism groups of 4 − (v, k, λ) designs. Let S = (P, B) be a non-trivial 4 − (q 2 + 1, k, 4) design, where q = 2 2n+1 for some positive integer n ≥ 1, and G ≤ Aut(S) acts block-transitively on S. If the socle of G is isomorphic to the simple groups of Lie type Sz(q), then G is not flag-transitive.

The Merrifield-Simmons indexi(G)of a graphGis defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, that is, the number of independent vertex sets ofGThe Hosoya indexz(G)of a graphGis defined as the total number of independent edge subsets, that is, the total number of its matchings. ByC(n,k,λ)we denote the set of graphs withnvertices,kcycles, the length of every cycle isλ, and all the edges not on the cycles are pendant edges which are attached to the same vertex. In this paper, we investigate the Merrifield-Simmons indexi(G)and the Hosoya indexz(G)for a graphGinC(n,k,λ).

This article is a contribution to the study of the automorphism groups of ( ) v k − 4 , ,λ designs. Let 2 2, then G is not flag-transitive.