The Mean Integrity of Paths and Cycles

Abstract

Let \(G\) be a graph with \(n\) vertices. The mean integrity of \(G\) is defined as follows:\(J(G) = min_{P \subseteq V} \{|P| + \tilde{m}(G – P)\},\) where \(\tilde{m}(G – P) = \frac{1}{n-|P|}\sum_{v \in G – P} n_v\) and \(n_v\) is the size of the component containing \(v\). The main result of this article is a formula for the mean integrity of a path \(P_n\) of \(n\) vertices. A corollary of this formula establishes the mean integrity of a cycle \(C_n\) of \(n\) vertices.