An Ubat-space is a nonempty set U together with a binary operation * satisfying:A g-group is a nonempty set G together with a binary operation * satisfying:there is e ∈ G such that g * e = e * g = g (we call e an identity); and (g3) for each g ∈ G, there exists h ∈ G such that g * h = h * g = e for some identity e described in (g2).In this paper, we present some important properties of the two algebraic structures (algebra).
In this paper, we introduced a new binary graph operation which we call acquiant vertex gluing. Moreover, we gave the domination number of the acquiant vertex gluing of some graphs.
Let G = (V, E) be a graph and k be a positive integer. A signed qRoman k-dominating function on G is a function f : V → {−1, 1, 2} with the following two properties: (1) f [u] q = v∈N (u) f (v) ≥ k for all u ∈ V ; and (2) for every v ∈ V with f (v) = −1, there exists w ∈ V with f (w) = 2 such that vw ∈ E. The weight of a signed qRoman k-dominating function is w (f) = v∈V f (v). The signed qRoman kdomination number of G, denoted by γ k qR (G), is the weight of a minimum signed qRoman k-dominating function on G. In this paper, we introduced the signed qRoman k-dominating function and gave the signed qRoman k-domination number of paths, cycles, the join and corona of some graphs.
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