A Computational Analysis of Spectral Properties and Energies of Distance-Based Matrices for Selected Hydrocarbons

This paper extends the spectral analysis of distance-based matrices associated with chemical graph structures of order n, focusing on the distance matrix D(G), the distance Laplacian DL(G), and the distance signless Laplacian DQ(G). We investigate the spectral integrality of these matrices for selected acyclic and cyclic hydrocarbon molecular graphs by examining whether their corresponding spectra consist entirely of integers. In addition, we compute and compare the associated distance energies, namely, the distance energy ED, the distance Laplacian energy EDL, and the distance signless Laplacian energy EDQ to explore their structural significance. Using computational tools, we present numerical results and graphical comparisons that reveal meaningful relationships among these energies. In particular, our analysis establishes the conjecture in the form of a strict inequality EDL>EDQ>ED. These findings demonstrate that the distance Laplacian energy is more sensitive to molecular structural variations, highlighting its effectiveness as a discriminative molecular descriptor in chemical graph theory.