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

References

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