On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

Abstract

Let $G$ be a simple connected graph of order $n$ and size $m$, vertex degree sequence $d_1\geq d_2\geq\cdots\geq
d_n>0$, and let $\mu_1\geq \mu_2\geq\cdots\geq\mu_{n-1}>\mu_n=0$
be the eigenvalues of its Laplacian matrix. Laplacian energy $LE$, Laplacian-energy-like invariant $LEL$ and Kirchhoff index $Kf$, are graph invariants defined in terms of Laplacian eigenvalues. These are, respectively,
defined as $LE(G)=\sum_{i=1}^{n}\left|\mu_i-\frac{2m}{n}\right|$,
$LEL(G)=\sum_{i=1}^{n-1}\sqrt{\mu_{i}}$ and
$Kf(G)=n\sum_{i=1}^{n-1} \frac{1}{\mu_i}$. A vertex--degree--based topological index referred to as degree deviation is defined as $S(G)=\sum_{i=1}^{n}\left|d_i-\frac{2m}{n}\right|$. Relations between $Kf$ and $LE$, $Kf$ and $LEL$, as well as $Kf$ and $S$ are obtained.