An independent dominating set of a graph, also known as maximal set, is S pairwise non-adjacent vertices such that every vertex not in adjacent to some S. We prove for Δ=4 or Δ≥6, connected n-vertex graph maximum degree at most Δ has an size (1−Δ⌊Δ2/4⌋+Δ)(n−1)+1. In addition, we characterize all graphs having the equality and show other have (1−Δ⌊Δ2/4⌋+Δ)n.