$RD_\alpha$-Spectra of Joined Union Graphs with Applications to Power Graphs of Finite Groups
Aditya Singh, Yogendra Singh, Anand Kumar Tiwari

TL;DR
This paper studies the $RD_\alpha$-spectra of joined union graphs, especially power graphs of finite groups, deriving explicit formulas and spectral properties for various group classes.
Contribution
It provides explicit formulas for the $RD_\alpha$-spectra of joined union graphs and applies these results to power graphs of finite groups, including dihedral and quaternion groups.
Findings
Derived closed-form characteristic polynomials for $RD_\alpha$-spectra of joined union graphs.
Established spectral formulas for power graphs of dihedral and quaternion groups.
Connected spectral properties to structural features of finite groups.
Abstract
The \emph{generalized reciprocal distance matrix} of a graph , denoted by , is defined as where represents the diagonal matrix of reciprocal vertex transmissions, and is the Harary (reciprocal distance) matrix of . In this paper, we investigate the -spectrum of graphs obtained through the joined union operation. We derive explicit formulas for the characteristic polynomial of when is formed as a joined union of regular graphs. These results provide closed-form expressions for the corresponding spectra of several important graph classes. Moreover, we show that the power graphs of the dihedral group and the generalized quaternion group …
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