On Structural Properties and Adjacency Spectrum of Coprime Graph of Integers

TL;DR

This paper investigates the structural and spectral properties of the coprime graph of integers, revealing insights into its connectivity, eigenvalues, and matrix singularity, with implications for understanding its graph-theoretic and algebraic features.

Contribution

It provides new results on the adjacency spectrum, vertex connectivity, crossing number, and eigenvalue multiplicities of the coprime graph of integers, including bounds and singularity conditions.

Findings

Adjacency matrix of TCG_n is singular with determinant 0.

Greatest eigenvalue of TCG_n's adjacency matrix exceeds 2.

Lower bounds on multiplicities of eigenvalues -1 and 0.

Abstract

Let $TCG_n$ denote the coprime graph having vertex set $\{1,2,\ldots,n\}$ with any two vertices $i,j$ being adjacent if and only if $\gcd(i,j)=1$. In this article, we first study some structural properties of $TCG_n$. We study the vertex connectivity and crossing number of the coprime graph of integers. We discover a lower constraint on the multiplicity of $-1$, which appears as an eigenvalue in the adjacency matrix of $TCG_n$. We demonstrate our findings with a variety of cases. We also show that the adjacency matrix of $TCG_n$ is singular, i.e. has determinant $0$. Furthermore, we give a lower bound on the multiplicity of $0$, which appears as an eigenvalue in the adjacency matrix of $TCG_n$. Finally, we establish that the greatest eigenvalue of the adjacency matrix of $TCG_n$ is always above $2.$