PreprintCC BY 4.0arXiv
The real eigenvalues of the non-backtracking Laplacian
Abstract
Let G be a simple graph on N nodes and M edges, with minimum degree at least $2$. We prove that the non-backtracking Laplacian of G has at least 2M−2N+2c+2b real eigenvalues, counted with algebraic multiplicity, where c is the number of connected components of G and b is the number of bipartite ones. More precisely, at least M−N+2c of them lie in [0,1) and at least M−N+2b lie in (1,2]. We show that the bound 2M−2N+2c+2b is sharp, and that a connected d-regular graph attains it if and only if it is strictly Ramanujan, i.e., every adjacency eigenvalue μ≠± d satisfies |μ|<2√(d−1). For completeness, we also show that the non-backtracking Laplacian has at most 2M−2c−2b real eigenvalues.
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BibTeX
@misc{mulas2026real,
title = {{The real eigenvalues of the non-backtracking Laplacian}},
author = {Raffaella Mulas and Leo Torres},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02194},
eprint = {2611.02194},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02194},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The real eigenvalues of the non-backtracking Laplacian
AU - Raffaella Mulas
AU - Leo Torres
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02194
UR - https://arxiv.org/abs/2611.02194
AB - Let G be a simple graph on N nodes and M edges, with minimum degree at least $2$. We prove that the non-backtracking Laplacian of G has at least 2M−2N+2c+2b real eigenvalues, counted with algebraic multiplicity, where c is the number of connected components of G and b is the number of bipartite ones. More precisely, at least M−N+2c of them lie in [0,1) and at least M−N+2b lie in (1,2]. We show that the bound 2M−2N+2c+2b is sharp, and that a connected d-regular graph attains it if and only if it is strictly Ramanujan, i.e., every adjacency eigenvalue μ≠± d satisfies |μ|<2√(d−1). For completeness, we also show that the non-backtracking Laplacian has at most 2M−2c−2b real eigenvalues.
ER -