On the distance spectral gap and construction of D-equienergetic graphs
Haritha T, Chithra A.
TL;DR
This work investigates the distance spectral gap $\delta_{D^G}=\rho_1-\rho_2$ of connected graphs, establishing bounds in terms of the sum of squared distance eigenvalues $S_D(G)$ and relating $\delta_{D^G}$ to the distance Laplacian for transmission-regular graphs. It provides upper and lower bounds for the distance energy $E_D(G)$ via spectral data and presents constructions of non-$D$-cospectral but $D$-equienergetic graphs, including new families with diameter 3 and 4, achieved through subdivision-joins and cycle partitions. The results offer tools to compare distance spectra, characterize extremal cases, and generate large classes of equienergetic graphs with controlled diameter, contributing to the broader study of distance-based graph spectra. The constructions yield nontrivial examples that separate $D$-cospectrality from $D$-equienergeticity, enriching the landscape of distance spectral graph theory.
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 $δ_{D^G} = ρ_1 - ρ_2$, where $ρ_1$ and $ρ_2$ 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 $4$.