On the non-incident quasi-total graph and its complement

A novel graph is derived from an undirected simple graph [Formula: see text] whose vertex set and edge set are [Formula: see text] and [Formula: see text], respectively. This new graph of [Formula: see text] is constructed with vertex set [Formula: see text] in such a way that any two vertices are adjacent if and only if they correspond to two non-adjacent vertices of [Formula: see text] or two adjacent edges of [Formula: see text] or a vertex and an edge not incident to it in [Formula: see text]. The resulting graph is denoted as [Formula: see text]. This paper depicts various graph-theoretic properties of [Formula: see text] and its complement, examining their structural characteristics and interrelationships with [Formula: see text], providing a broader theoretical framework with potential applications across diverse fields.