On the distance spectral gap and construction of D-equienergetic graphs

TL;DR

This paper investigates the spectral properties of the distance matrix in graphs, providing bounds for the spectral gap and energy, and introduces new families of graphs with specific spectral characteristics.

Contribution

It offers new bounds on the distance spectral gap and constructs novel non D-cospectral D-energetic graphs with small diameters.

Findings

Derived bounds for the distance spectral gap.

Established bounds for distance eigenvalues and energy.

Constructed new families of D-energetic graphs with diameters 3 and 4.

Abstract

Let $D(G)$ denote the distance matrix of a connected graph $G$ with $n$ vertices. The distance spectral gap of a graph $G$ is defined as $\delta_{D^G} = \rho_1 - \rho_2$, where $\rho_1$ and $\rho_2$ represent the largest and second largest eigenvalues of $D(G)$, respectively. For a $k$-transmission regular graph $G$, the second smallest eigenvalue of the distance Laplacian matrix equals the distance spectral gap of $G$. In this article, we obtain some upper and lower bounds for the distance spectral gap of a graph in terms of the sum of squares of its distance eigenvalues. Additionally, we provide some bounds for the distance eigenvalues and distance energy of graphs. Furthermore, we construct new families of non $D$-cospectral $D$-equienergetic graphs with diameters of $3$ and $4$.