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

Document Type : Original paper

Authors

1 Faculty of Electronic Engineering, Nis, Serbia

2 State University of Novi Pazar, Novi Pazar, Serbia

3 Faculty of Electronic Engineering, Nis, Srbia

Abstract

Let $G=(V,E)$, $V={1,2,ldots,n}$, $E={e_1,e_2,ldots,e_m}$,
be a simple connected graph,
with sequence of vertex degrees
$Delta =d_1geq d_2geqcdotsgeq d_n=delta >0$ and Laplacian eigenvalues
$mu_1geq mu_2geqcdotsgeqmu_{n-1}>mu_n=0$. Denote by $Kf(G)=nsum_{i=1}^{n-1} frac{1}{mu_i}$ and $t=t(G)=frac 1n prod_{i=1}^{n-1} mu_i$ the Kirchhoff index and number of spanning trees of $G$, respectively. In this paper we determine several lower bounds for $Kf(G)$ depending on $t(G)$ and some of the graph parameters $n$, $m$, or $Delta$.

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