Upper signed domination number

Let image be a graph. A signed dominating function on G is a function image such that image for each image, where image is the closed neighborhood of image. The weight of a signed dominating function f is image. A signed dominating function f is minimal if there exists no signed dominating function g such that image and image for each image. The upper signed domination number of a graph G, denoted by image, equals the maximum weight of a minimal signed dominating function of G. In this paper, we establish an tight upper bound for image 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.