$RD_α$-Spectra of Joined Union Graphs with Applications to Power Graphs of Finite Groups
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| Format: | Preprint |
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2026
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| _version_ | 1866910132666892288 |
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| author | Singh, Aditya Singh, Yogendra Tiwari, Anand Kumar |
| author_facet | Singh, Aditya Singh, Yogendra Tiwari, Anand Kumar |
| contents | The \emph{generalized reciprocal distance matrix} of a graph $\mathscr{G}$, denoted by $RD_α(\mathscr{G})$, is defined as $RD_α(\mathscr{G})=α\,RT_r(\mathscr{G})+(1-α)\,RD(\mathscr{G}), \, α\in[0,1],$ where $RT_r(\mathscr{G})$ represents the diagonal matrix of reciprocal vertex transmissions, and $RD(\mathscr{G})$ is the Harary (reciprocal distance) matrix of $\mathscr{G}$. In this paper, we investigate the $RD_α$-spectrum of graphs obtained through the joined union operation. We derive explicit formulas for the characteristic polynomial of $RD_α(\mathscr{G})$ when $\mathscr{G}$ 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 $RD_α$-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 $pq$. |
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arxiv_https___arxiv_org_abs_2604_14195 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $RD_α$-Spectra of Joined Union Graphs with Applications to Power Graphs of Finite Groups Singh, Aditya Singh, Yogendra Tiwari, Anand Kumar Combinatorics Spectral Theory 15A18, 05C25, 05C50, 05C12 The \emph{generalized reciprocal distance matrix} of a graph $\mathscr{G}$, denoted by $RD_α(\mathscr{G})$, is defined as $RD_α(\mathscr{G})=α\,RT_r(\mathscr{G})+(1-α)\,RD(\mathscr{G}), \, α\in[0,1],$ where $RT_r(\mathscr{G})$ represents the diagonal matrix of reciprocal vertex transmissions, and $RD(\mathscr{G})$ is the Harary (reciprocal distance) matrix of $\mathscr{G}$. In this paper, we investigate the $RD_α$-spectrum of graphs obtained through the joined union operation. We derive explicit formulas for the characteristic polynomial of $RD_α(\mathscr{G})$ when $\mathscr{G}$ 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 $RD_α$-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 $pq$. |
| title | $RD_α$-Spectra of Joined Union Graphs with Applications to Power Graphs of Finite Groups |
| topic | Combinatorics Spectral Theory 15A18, 05C25, 05C50, 05C12 |
| url | https://arxiv.org/abs/2604.14195 |