A distance-transitive graph (DTG) is a graph in which for every two ordered pairs of vertices (u,v) and (u',v') such that the distance between u and v is equal to the distance between u' and v' there exists an automorphism mapping u to u' and v to v'. A semiregular element of a permutation group is a non-identity element having all cycles of equal length in its cycle decomposition. The paper gives a complete answer to a problem posed by a world-class scientist M. Guidici during a workshop in 2008 in Banff, Canada. More specifically, it is shown that every DTG admits a semiregular automorphism.

COBISS.SI-ID: 1024085332

Bannai and Ito defined association scheme theory as doing ''group theory without groups'', thus raising a basic question as to which results about permutation groups are, in fact, results about association schemes? By considering transitive permutation groups in a wider setting of association schemes, it is shown that one such result is a generalization from odd primes p to arbitrary prime powers p^n, of the classical theorem of Wielandt about primitive permutation groups of degree 2p^n, p ) 2 a prime, being of rank 3. The paper is published in the esteemed general scientific mathematical journal Trans. Americ. Math. Soc. that ranks in A' (ARRS methodology).

COBISS.SI-ID: 1024198996

The existence of semiregular automorphisms in fullerenes is discussed. In particular, the family of fullerene graphs is described via the existence of semiregular automorphisms in their automorphism groups.

COBISS.SI-ID: 4178202

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,\mu,\lambda, and t are given. Also, several infinite families of directed strongly regular graphs which are also 2-Cayley digraphs of abelian groups are constructed.

COBISS.SI-ID: 1024426836

This discussion is published in the esteemed general scientific mathematical journal Proc. Lond. Math. Soc. that ranks in A' (ARRS methodology). It solves the hamiltonicity problem for cubic Cayley graphs on groups with respect to genereting sets consisting of an involution, a non-involution of odd order and the inverse of this non-involution.

COBISS.SI-ID: 1024390740