Common Neighbourhood Eccentric Dominating and Its Energy of Graphs

A. Riyaz Ur Rehman; A. Mohamed Ismayil; Ismail Naci Cangul

doi:https://doi.org/10.14445/22315373/IJMTT-V69I9P504

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 69 | Issue 9 | Year 2023 | Article Id. IJMTT-V69I9P504 | DOI : https://doi.org/10.14445/22315373/IJMTT-V69I9P504

Common Neighbourhood Eccentric Dominating and Its Energy of Graphs

A. Riyaz Ur Rehman, A. Mohamed Ismayil, Ismail Naci Cangul

Received	Revised	Accepted	Published
20 Jul 2023	21 Aug 2023	08 Sep 2023	30 Sep 2023

Citation :

A. Riyaz Ur Rehman, A. Mohamed Ismayil, Ismail Naci Cangul, "Common Neighbourhood Eccentric Dominating and Its Energy of Graphs," International Journal of Mathematics Trends and Technology (IJMTT), vol. 69, no. 9, pp. 23-37, 2023. Crossref, https://doi.org/10.14445/22315373/IJMTT-V69I9P504

Abstract

In this paper, we introduce the concept of common neighbourhood eccentric domination in graphs. An eccentric dominating set 𝐷 is called a common neighbourhood eccentric dominating set (CNED-set) if for all 𝑣𝑗∈𝑉−𝐷, there exists a vertex 𝑣𝑖∈𝐷 such that (𝑣𝑖,𝑣𝑗)∈𝐸(𝐺) and Γ(𝑣𝑖,𝑣𝑗)≥1. We calculate the common neighbourhood eccentric domination number for some standard graphs, and some results are stated and proved. The minimum common neighbourhood eccentric dominating energy 𝕄𝑐𝑛𝑒𝑑(𝐺) is the sum of the eigenvalues obtained from the minimum common neighbourhood eccentric dominating 𝑛×𝑛 matrix 𝕄𝑐𝑛𝑒𝑑(𝐺)=(𝑚𝑖𝑗). 𝕄𝑐𝑛𝑒𝑑(𝐺) of standard graphs are computed. New properties, upper and lower bounds for 𝕄𝑐𝑛𝑒𝑑(𝐺) are established.

Keywords

Common neighbourhood, Eccentricity, Domination, Minimum common neighbourhood eccentric dominating set, Eigenvalues, Energy.

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