Some graph parameters on the strong product of monogenic semigroup graphs

Abstract

In Das et al. (2013), it has been defined a new algebraic graph on monogenic semigroups. Our main scope in this study, is to extend this study over the special algebraic graphs to the strong product. In detail, we will determinate some important graph parameters (diameter, girth, radius, maximum degree, minimum degree, chromatic number, clique number and domination number) for the strong product of any two monogenic semigroup graphs.

Some new properties on N semigroups

“…If it satisfies 𝑥𝑦 = 𝑦𝑥, then 𝑆 is said to be (𝐶) −inversive semigroup. If 𝑃 and 𝑄 are two semigroups having a common homomorphic image 𝑌, then the spined product of 𝑃 and 𝑄 with respect to 𝑌 is 𝑆 = {(𝑎, 𝑏) ∈ 𝑃 × 𝑄 | 𝜑(𝑎) = 𝜓(𝑏)}, where 𝜑: 𝑃 → 𝑌 and 𝜓: 𝑄 → 𝑌 are homomorphisms onto 𝑌 [1]. Let us notate the spined product by × 𝑆 .…”

“…When a new semigroup appears, it is necessary to investigate the properties of this semigroup. For example, in [13] and [14] the authors considered a new product and gave some properties on a special product of semigroups (and monoids) [1]. In this paper we more deeply investigate the place  in the literature as a continuation of the work [11].…”

“…Then the set 𝑀 𝑅 × 𝑀 𝐶 defines the semigroup 𝑀 0 [𝑆 {0} , 𝐺 0 ; 𝑀 𝑅 , 𝑀 𝐶 ; 𝑃, 𝑃′] using the operation given in (1). For simplicity, we denote this new semigroup by .…”

“…it is well known that third line of operarion ⋆ defines the general situation. For this reason, we consider the form (𝑟, 𝑐) which is the most common form among the others and clearly including other cases in(1). Therefore we have…”

Sum Connectivity Index Under the Cartesian and Strong Products Graph of Monogenic Semigroup

“…For instance, the solution of various product graphs through the monogenic semigroup was obtained by Aydm et al [5] Wiener index over the dot product and Seda [6] Sombor index over the Cartesian and Tensor products. Akgunes [7] demonstrated some importance graph parameter used strong product to solve the monogenic semigroups.…”