Some properties on the lexicographic product of graphs obtained by monogenic semigroups Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

Abstract

In (Das et al. in J. Inequal. Appl. 2013:44, 2013), a new graph [InlineEquation not available: see fulltext.] on monogenic semigroups [InlineEquation not available: see fulltext.] (with zero) having elements [InlineEquation not available: see fulltext.] was recently defined. The vertices are the non-zero elements [InlineEquation not available: see fulltext.] and, for [InlineEquation not available: see fulltext.], any two distinct vertices [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.] are adjacent if [InlineEquation not available: see fulltext.] in [InlineEquation not available: see fulltext.]. As a continuing study, in an unpublished work, some well-known indices (first Zagreb index, second Zagreb index, Randić index, geometric-arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over [InlineEquation not available: see fulltext.] were investigated by the same authors of this paper. In the light of the above references, our main aim in this paper is to extend these studies to the lexicographic product over [InlineEquation not available: see fulltext.]. In detail, we investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two (not necessarily different) graphs [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]. MSC: 05C10, 05C12, 06A07, 15A18, 15A36.