No abstract

The concept of Sombor index (SO) was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index SO: SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of vertex vi in G. Here, we present novel lower and upper bounds on the Sombor index of graphs by using some graph parameters. Moreover, we obtain several relations on Sombor index with the first and second Zagreb indices of graphs. Finally, we give some conclusions and propose future work.

No abstract

Let us consider the finite monogenic semigroup S M

The semi-direct product of arbitrary two monoids and a presentation for this product have received considerable attention, see for instance [12, 14, 15]. In [15], Wang defined a trivializer set of the Squier complex associated with this presentation. In this paper, as a main result, we discuss necessary and sufficient conditions for the standard presentation of the semi-direct product of any two monoids to be p-Cockcroft for any prime p or 0. Finally we present some applications of this main theorem.