$R{D}_{\alpha }$-Spectra of Joined Union Graphs with Applications to Power Graphs of Finite Groups

Abstract

The \emph{generalized reciprocal distance matrix} of a graph $\mathcal{G}$, denoted by $R{D}_{\alpha }(\mathcal{G})$, is defined as $R{D}_{\alpha }(\mathcal{G})=\alpha R{T}_r(\mathcal{G})+(1-\alpha )RD(\mathcal{G}),\alpha \in [0,1],$ where $R{T}_r(\mathcal{G})$ represents the diagonal matrix of reciprocal vertex transmissions, and $RD(\mathcal{G})$ is the Harary (reciprocal distance) matrix of $\mathcal{G}$. In this paper, we investigate the $R{D}_{\alpha }$-spectrum of graphs obtained through the joined union operation. We derive explicit formulas for the characteristic polynomial of $R{D}_{\alpha }(\mathcal{G})$ when $\mathcal{G}$ 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 $D_{2n}$ and the generalized quaternion group $Q_{4n}$ admit representations as joined union graphs. Using this structural characterization, we determine the $R{D}_{\alpha }$-spectra of power graphs arising from various classes of finite groups, including cyclic groups ${\mathbb{Z}}_n$, dihedral groups $D_{2n}$, generalized quaternion groups $Q_{4n}$, elementary abelian $p$-groups, and certain non-abelian groups of order $pq$.

19 pages