On structural properties and adjacency spectrum of coprime graph of integers

Let [Formula: see text] denote the coprime graph having vertex set [Formula: see text] with any two vertices [Formula: see text] being adjacent if and only if [Formula: see text]. In this paper, we first study some structural properties of [Formula: see text]. We study the vertex connectivity and crossing number of the coprime graph of integers. We discover a lower constraint on the multiplicity of [Formula: see text], which appears as an eigenvalue in the adjacency matrix of [Formula: see text]. We demonstrate our findings with a variety of cases. We also show that the adjacency matrix of [Formula: see text] is singular, i.e. has determinant [Formula: see text]. Furthermore, we give a lower bound on the multiplicity of [Formula: see text], which appears as an eigenvalue in the adjacency matrix of [Formula: see text]. Finally, we establish that the greatest eigenvalue of the adjacency matrix of [Formula: see text] is always above [Formula: see text].