On relation between the Kirchhoff index and number of spanning trees of graph

Abstract

Let $G$ be a simple connected graph with degree sequence $(d_1,d_2,\ldots, d_n)$ where $\Delta =d_1\geq d_2\geq\cdots\geq d_n=\delta >0$ and let $\mu_1\geq \mu_2\geq\cdots\geq\mu_{n-1}>\mu_n=0$ be the Laplacian eigenvalues of $G$. Let $Kf(G)=n\sum_{i=1}^{n-1} \frac{1}{\mu_i}$ and $\tau(G)=\frac 1n \prod_{i=1}^{n-1} \mu_i$ denote the Kirchhoff index and the number of spanning trees of $G$, respectively. In this paper we establish several lower bounds for $Kf(G)$ in terms of $\tau(G)$, the order, the size and maximum degree of $G$.

Keywords

• topological indices
• kirchhoff index
• spanning tree