Zagreb indices of subgroup generating bipartite graph
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866909453895335936 |
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| author | Das, Shrabani Erfanian, Ahmad Nath, Rajat Kanti |
| author_facet | Das, Shrabani Erfanian, Ahmad Nath, Rajat Kanti |
| contents | Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is partitioned into two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. In this paper, we deduce expressions for first and second Zagreb indices of $\mathcal{B}(G)$ and obtain a condition such that $\mathcal{B}(G)$ satisfy Hansen-Vuki{č}evi{ć} conjecture [Hansen, P. and Vuki{č}evi{ć}, D. Comparing the Zagreb indices, {\em Croatica Chemica Acta}, \textbf{80}(2), 165-168, 2007]. It is shown that $\mathcal{B}(G)$ satisfies Hansen-Vuki{č}evi{ć} conjecture if $G$ is a cyclic group of order $2p, 2p^2, 4p$, $4p^2$ and $p^n$; dihedral group of order $2p$ and $2p^2$; and dicyclic group of order $4p$ and $4p^2$ for any prime $p$. While computing Zagreb indices of $\mathcal{B}(G)$ we have computed $°_{\mathcal{B}(G)}(H)$ for all $H \in L(G)$ for the above mentioned groups. Using these information we also compute Randic Connectivity index, Atom-Bond Connectivity index, Geometric-Arithmetic index, Harmonic index and Sum-Connectivity index of $\mathcal{B}(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06124 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zagreb indices of subgroup generating bipartite graph Das, Shrabani Erfanian, Ahmad Nath, Rajat Kanti Group Theory Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. The subgroup generating bipartite graph $\mathcal{B}(G)$ defined on $G$ is a bipartite graph whose vertex set is partitioned into two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. In this paper, we deduce expressions for first and second Zagreb indices of $\mathcal{B}(G)$ and obtain a condition such that $\mathcal{B}(G)$ satisfy Hansen-Vuki{č}evi{ć} conjecture [Hansen, P. and Vuki{č}evi{ć}, D. Comparing the Zagreb indices, {\em Croatica Chemica Acta}, \textbf{80}(2), 165-168, 2007]. It is shown that $\mathcal{B}(G)$ satisfies Hansen-Vuki{č}evi{ć} conjecture if $G$ is a cyclic group of order $2p, 2p^2, 4p$, $4p^2$ and $p^n$; dihedral group of order $2p$ and $2p^2$; and dicyclic group of order $4p$ and $4p^2$ for any prime $p$. While computing Zagreb indices of $\mathcal{B}(G)$ we have computed $°_{\mathcal{B}(G)}(H)$ for all $H \in L(G)$ for the above mentioned groups. Using these information we also compute Randic Connectivity index, Atom-Bond Connectivity index, Geometric-Arithmetic index, Harmonic index and Sum-Connectivity index of $\mathcal{B}(G)$. |
| title | Zagreb indices of subgroup generating bipartite graph |
| topic | Group Theory |
| url | https://arxiv.org/abs/2501.06124 |