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Friday, August 19, 2016 - 15:30 to 16:30
SMS, Seminar Room
Sriparna Chattopadhyay
IIT, Kharagpur
Some spectral results on power graphs of finite cyclic and dihedral groups
Abstract: There are a number of graphs associated with semigroups and groups. The Cayley graph is the oldest and most popular among them whereas the concept of power graph is a recent development. The power graph $G(G)$ of a finite group $G$ is the graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if and only if one is an integral power of the other. So far mostly graph theoretic properties of power graphs has been studied by the researchers. Here I discuss some of our recent results on spectrum and energy of power graphs of the additive cyclic group $\mathbb{Z}_n$ and the dihedral group $D_n$. We give formulae for adjacency (resp. Laplacian) characteristic polynomial of the power graph of $\mathbb{Z}_n$ in terms of a suitable minor of adjacency matrix (resp. Laplacian matrix) of $G(\mathbb{Z}_n)$ formed by non-identity and non-generator elements. We also find the adjacency and Laplacian characteristic polynomial of $G(D_{2n})$ in terms of that of $G(\mathbb{Z}_n)$. Further we give bounds for spectral radius and algebraic connectivity of $G(\mathbb{Z}_n)$ and find that the algebraic connectivity of $G(D_{2n})$ is 1. Finally we have solved partially an open problem of Abawajy et al. given in their survey paper which asked for link between power graphs and Cayley graphs. We have found some relations between power graphs and Cayley graphs and applying these connections we study the eigenvalues and energy of power graphs and the related Cayley graphs. All are cordially invited.

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