The 1-nearly edge independence number of a graph

Let G = (V (G), E (G) ) be a graph. The maximum cardinality of a set Mₖ E (G) such that Mₖ contains exactly k-pairs of adjacent edges of G is called the k-nearly edge independence number of G, and is denoted by 'ₖ (G). In this paper we study ₁' (G). In particular, we prove a tight lower (resp. upper) bound on ₁ (G) if G is a graph with given number of vertices. Furthermore, we present a characterisation of the general (resp. connected) graphs with given number of vertices and smallest 1-nearly edge independence number. Lastly, we pose an open problem for further exploration of this study.