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

Abstract

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