Iterated line graphs with only negative eigenvalues $-2$, their complements and energy

TL;DR

This work addresses the classification of graphs whose least eigenvalues are all $-2$, focusing on iterated line graphs $\mathcal{L}^k(G)$ that exhibit this property for $k\ge 1$ and on the spectra and energy of their complements. Using equitable partitions and quotient-matrix techniques, the authors derive partial spectra and establish energy formulas, enabling the construction of large families of equienergetic graphs. They identify sufficient conditions on $G$ (e.g., $d_u+d_v\ge 6$ for each edge $uv$) that guarantee property $\rho$ for $\mathcal{L}^k(G)$ with explicit energy $\mathcal{E}(\mathcal{L}^k(G))=4(n_k-n_{k-1})$, and they analyze the spectra of the complements, showing exactly two positive eigenvalues and related energy relations. The results extend prior work on line graphs and their energies, providing a systematic framework for generating equienergetic pairs and deepening understanding of spectral properties under line-graph and complement operations.

Abstract

The graphs with all equal negative or positive eigenvalues are special kind in the spectral graph theory. In this article, several iterated line graphs $\mathcal{L}^k(G)$ with all equal negative eigenvalues $-2$ are characterized for $k\ge 1$ and their energy consequences are presented. Also, the spectra and the energy of complement of these graphs are obtained, interestingly they have exactly two positive eigenvalues with different multiplicities. Moreover, we characterize a large class of equienergetic graphs which generalize some of the existing results. There are two different quotient matrices defined for an equitable partition of $H$-join (generalized composition) of regular graphs to find the spectrum (partial) of adjacency matrix, Laplacian matrix and signless Laplacian matrix, it has been proved that these two quotient matrices give the same respective spectrum of graphs.

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