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.
Let G = (V, E) be a graph of order 2n. If A ⊆ V and hAi ∼= hV \Ai, then A is said to be isospectral. If for every n-element subset A of V we have hAi ∼= hV \Ai, then we say that G is spectral-equipartite. In [1], Igor Shparlinski communicated with Bibak et al., proposing a full characterization of spectral-equipartite graphs. In this paper, we gave a characterization of disconnected spectral-equipartite graphs. Moreover, we introduced the concept eccentricity-equipartite graphs.
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