Nordhaus-Gaddum inequalities for the number of connected induced subgraphs in graphs

Let η(G) be the number of connected induced subgraphs in a graph G, and Ḡ complement G. We prove that + η(Ḡ) is minimum, among all n-vertex graphs, if only G has no path on four vertices. Since star Sn with maximum degree n − 1 unique tree diameter 2, minimum trees, while shown to achieved by whose sequence (⌈n/2⌉, ⌊n/2⌋, 1, . , 1). Furthermore, we every order ≥ 5 must have at most 3, cut vertex property also connected. In both cases trees graphs same order, find then minimum.As corollaries our results, characterise given vertices satisfying |V (G)| = |E(G)| minimises η(Ḡ).