Upper signed domination number

Let G=(V,E)G=(V,E) be a graph. A signed dominating function on G   is a function f:V→{-1,1}f:V→{-1,1} such that ∑u∈N[v]f(u)⩾1 for each v∈Vv∈V, where N[v]N[v] is the closed neighborhood of vv. The weight of a signed dominating function f   is ∑v∈Vf(v). A signed dominating function f is minimal if there exists no signed dominating function g   such that g≠fg≠f and g(v)⩽f(v)g(v)⩽f(v) for each v∈Vv∈V. The upper signed domination number of a graph G  , denoted by Γs(G)Γs(G), equals the maximum weight of a minimal signed dominating function of G  . In this paper, we establish an tight upper bound for Γs(G)Γs(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.