  Let $G = (V(G), E(G))$ be a graph. The maximum cardinality of a set $M_k
\subseteq E(G)$ such that $M_k$ contains exactly $k$-pairs of adjacent edges of
$G$ is called the $k$-nearly edge independence number of $G$, and is denoted by
$\alpha'_k(G)$. In this paper we study $\alpha_1'(G)$. In particular, we prove
a tight lower (resp. upper) bound on $\alpha_1(G)$ 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.
