On the $A_{\alpha}$ and $RD_{\alpha}$ matrices over certain groups

Abstract

The power graph $G = P(\Omega)$ of a finite group $\Omega$ is a graph with the vertex set $\Omega$ and two vertices $u, v \in \Omega$ form an edge if and only if one is an integral power of the other. Let $D(G)$, $A(G)$, $RT(G)$, and $RD(G)$ denote the degree diagonal matrix, adjacency matrix, the diagonal matrix of the vertex reciprocal transmission, and Harary matrix of the power graph $G$ respectively. Then the $A_{\alpha}$ and $RD_{\alpha}$ matrices of $G$ are defined as $A_{\alpha}(G) = \alpha D(G) + (1-\alpha)A(G)$ and $RD_{\alpha}(G) = \alpha RT(G) + (1-\alpha)RD(G)$. In this article, we determine the eigenvalues of $A_{\alpha}$ and $RD_{\alpha}$ matrices of the power graph of group $ \mathcal{G} = \langle s,r \, : r^{2^kp} = s^2 = e,~ srs^{-1} = r^{2^{k-1}p-1}\rangle$. In addition, we calculate its distant and detotar distance degree sequences, metric dimension, and strong metric dimension.