In this paper, the structure of directed strongly regular 2-Cayley graphs of cyclic groups is investigated. In particular, the arithmetic conditions on parametersv,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

In 1969, Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of its easy formulation, no major breakthrough has been achieved thus far, and the problem is now commonly accepted to be very hard. The same holds for the special subclass of Cayley graphs where the existence of Hamilton cycles has been conjectured. In 2007, Glover and Marušič proved that a cubic Cayley graph on a finite (2, s, 3)-generated group G = ( a, x| a^2 = x^s = (ax)^3 = 1, \dots ) has a Hamilton path when |G| is congruent to 0 modulo 4, and has a Hamilton cycle when |G| is congruent to 2 modulo 4. The Hamilton cycle was constructed, combining the theory of Cayley maps with classical results on cyclic stability in cubic graphs, as the contractible boundary of a tree of faces in the corresponding Cayley map. With a generalization of these methods, Glover, Kutnar and Marušič in 2009 resolved the case when, apart from |G|, also s is congruent to 0 modulo 4. In this article, with a further extension of the above "tree of faces" approach, a Hamilton cycle is shown to exist whenever |G| is congruent to 0 modulo 4 and s is odd.

COBISS.SI-ID: 1024390740

For a simple graph G with n vertices and m edges, let M1 and M2 denote the ﬁrst and the second Zagreb index of G. The inequality M1/n ≤ M2/m in the case of trees has been proved ﬁrst by Vukičević and Graovac, and a new proof has been found recently by Andova, Cohen and Škrekovski In this paper we improve this inequality by showing that, if G is not a star, then nM2 − mM1 ≥ 2(n − 3) + (Δ − 1)(Δ − 2), where Δ is the maximum vertex degree in G.

COBISS.SI-ID: 1024425556