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 α'ₖ(G). In this paper we study α₁'(G). In particular, we prove a tight lower (resp. upper) bound on α₁(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.