On the smallest eigenvalues of the line graphs of some trees
Akihiro Munemasa, Yoshio Sano, and Tetsuji Taniguchi
TL;DR
This paper investigates the smallest eigenvalues of line graphs derived from generalized Bethe trees, revealing an infinite family sharing the same smallest eigenvalue, thus extending known results about coronas of complete graphs.
Contribution
It introduces a new family of line graphs of generalized Bethe trees with identical smallest eigenvalues, generalizing previous work on coronas of complete graphs.
Findings
Identified an infinite family of line graphs with the same smallest eigenvalue.
Extended the class of graphs known to share eigenvalue properties.
Connected properties of Bethe trees with spectral graph theory.
Abstract
In this paper, we study the characteristic polynomials of the line graphs of generalized Bethe trees. We give an infinite family of such graphs sharing the same smallest eigenvalue. Our family generalizes the family of coronas of complete graphs discovered by Cvetkovi\'c and Stevanovi\'c.
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