On the Relationship between the Kirchhoff and the Narumi-Katayama Indices

Abstract

Let $G$ be a simple connected graph with $n$ vertices and $m$
edges, sequence of vertex degrees $\Delta=d_1\geq d_2\geq\cdots\geq d_n=\delta>0$ and diagonal matrix $D={\rm{diag}}(d_1,d_2,\ldots,d_n)$ of its vertex degrees. Denote by $Kf(G)=n\sum_{i=1}^{n-1} \frac{1}{\mu_i}$, where $\mu_i$ are the Lapacian eigenvalues of graph $G$, the Kirchhoff index of $G$, and by $NK=\prod_{i=1}^n d_i$ the Narumi-Katayama index. In this paper we prove some inequalities that set up relationship between the Kirchhoff and Narumi-Katayama index.