The Laplacian Polynomial of Graphs Derived from Regular Graphs and Applications

Abstract

Let $R(G)$ be the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and by joining each new vertex to the end vertices of the corresponding edge. Let $RT(G)$ be the graph obtained from $R(G)$ by adding a new edge corresponding to every vertex of $G$, and by joining the end vertices of each new edge to the corresponding vertex of $G$. In this paper, we determine the Laplacian polynomials of $RT(G)$ of a regular graph $G$. Moreover, we derive formulae and lower bounds of Kirchhoff indices of the graphs. Finally, we also present the formulae for calculating the Kirchhoff indices of some special graphs as applications, which show the correction and efficiency of the proposed results.