Generating Graphs of Finite Dihedral Groups

For a group $G$, the generating graph $\mathrm{\Gamma }(G)$ is defined as the graph with the vertex set $G$, and any two distinct vertices of $\mathrm{\Gamma }(G)$ are adjacent if they generate $G$. In this paper, we study the generating graph of $D_n,$ where $D_n$ is a Dihedral group of order $2n$. We explore various graph theoretic properties, and determine complete spectrum of the adjacency and the Laplacian matrix of $\mathrm{\Gamma }(D_n)$. Moreover, we compute some distance and degree based topological indices of $\mathrm{\Gamma }(D_n)$.