Pendant paths and integral generalized sun graphs
Abstract
A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. We prove that every simple graph having a pendant path with at least three edges has an eigenvalue in (1,2cos(π/9)] and one in [−2cos(π/9),−1), and hence is not integral. This settles a conjecture of Braga, Del-Vecchio and Rodrigues (2021) on integral generalized sun graphs. The argument is matrix-theoretic: adjoining a terminal path on three new coordinates to an arbitrary real symmetric matrix produces the same spectral obstruction, and the positive interval above is optimal in this generality. We then disprove a second conjecture of the same authors, which asserts that the cycle of an integral generalized sun graph other than a cycle has length divisible by four. The graph obtained from a hexagon by attaching $6,6,12,6,6$ pendant vertices to five of its six vertices is integral and has $42$ vertices. We show that it is the smallest member of an infinite family governed by the Pell equation x2−2k2=−7, and we compute in closed form the characteristic polynomial of the analogous graphs on an arbitrary even cycle. Integrality within this family forces the cycle to be a square or a hexagon, and the square case yields a second infinite family governed by the Pell equation k2−2c2=1.
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BibTeX
@misc{braga2026pendant,
title = {{Pendant paths and integral generalized sun graphs}},
author = {Rodrigo O. Braga and Jean Carlo Moraes and Matheus C. Santos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28754},
eprint = {2609.28754},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28754},
}
RIS
TY - UNPB
TI - Pendant paths and integral generalized sun graphs
AU - Rodrigo O. Braga
AU - Jean Carlo Moraes
AU - Matheus C. Santos
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28754
UR - https://arxiv.org/abs/2609.28754
AB - A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. We prove that every simple graph having a pendant path with at least three edges has an eigenvalue in (1,2cos(π/9)] and one in [−2cos(π/9),−1), and hence is not integral. This settles a conjecture of Braga, Del-Vecchio and Rodrigues (2021) on integral generalized sun graphs. The argument is matrix-theoretic: adjoining a terminal path on three new coordinates to an arbitrary real symmetric matrix produces the same spectral obstruction, and the positive interval above is optimal in this generality. We then disprove a second conjecture of the same authors, which asserts that the cycle of an integral generalized sun graph other than a cycle has length divisible by four. The graph obtained from a hexagon by attaching $6,6,12,6,6$ pendant vertices to five of its six vertices is integral and has $42$ vertices. We show that it is the smallest member of an infinite family governed by the Pell equation x^2−2k^2=−7, and we compute in closed form the characteristic polynomial of the analogous graphs on an arbitrary even cycle. Integrality within this family forces the cycle to be a square or a hexagon, and the square case yields a second infinite family governed by the Pell equation k^2−2c^2=1.
ER -