Upper signed domination number

Let G=(V,E) be a graph. A signed dominating function on G is a function f:V→{-1,1} such that for each v∈V, where Nv] is the closed neighborhood of v. The weight of a signed dominating function f is . A signed dominating function f is minimal if there exists no signed dominating function g such that g≠f and g(v)?f(v) for each v∈V. The upper signed domination number of a graph G, denoted by Γ_s(G), equals the maximum weight of a minimal signed dominating function of G. In this paper, we establish an tight upper bound for Γ_s(G) in terms of minimum degree and maximum degree. Our result is a generalization of those for regular graphs and nearly regular graphs obtained in O. Favaron, Signed domination in regular graphs, Discrete Math. 158 (1996) 287-293] and C.X. Wang, J.Z. Mao, Some more remarks on domination in cubic graphs, Discrete Math. 237 (2001) 193-197], respectively.