The tree number of power graphs associated with specific groups and applications

A. Moghaddamfar

The power graph \(\mathcal{P}(G)\) of a group \(G\) is an undirected graph whose vertex set is \(G\) and two vertices \(x, y\in G\) are adjacent if and only if \(\langle x\rangle\subseteq \langle y\rangle\) or \(\langle y\rangle\subseteq \langle x\rangle\) (which is equivalent to saying that \(x\neq y\) and \(x^m=y\) or \(y^m=x\) for some non-negative integer \(m\)). Clearly, the power graph \(\mathcal{P}(G)\) of any group \(G\) is always connected. The number of spanning trees of the power graph \(\mathcal{P}(G)\) of a group \(G\), which is denoted by \(\kappa(G)\) and called the tree-number of \(G\), will be investigated for certain finite groups \(G\) in this talk. Indeed, the explicit formula for the tree-number of a cyclic group or a generalized quaternion group is obtained. We have also determined, up to isomorphism, the structure of any finite group \(G\) for which \(\kappa(G) < 125\).

References

1. I. Chakrabarty, S. Ghosh, M. K. Sen, Undirected power graphs of semigroups, Semigroup Forum, 78, N3 (2009), 410–426.
2. A. R. Moghaddamfar, S. Rahbariyan, W. J. Shi, Certain properties of the power graph associated with a finite group. (submitted)
3. A. R. Moghaddamfar, S. Rahbariyan, S. Navid Salehy, S. Nima Salehy, The number of spanning trees of power graphs associated with specific groups and some applications. (submitted)

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