On Relation Between Kirchhoff Index, Laplacian-Energy-Like Invariant and Laplacian Energy of Graphs

Let $$G=(V,E)$$G=(V,E) be a simple graph of order n with m edges and Laplacian eigenvalues $$\mu _1\ge \mu _2\ge \cdots \ge \mu _{n-1}\ge \mu _n=0$$μ1≥μ2≥⋯≥μn-1≥μn=0. The Kirchhoff index and the Laplacian-energy-like invariant of G are defined as $$\begin{aligned} \mathrm{Kf}(G)=n\sum _{k=1}^{n-1}\frac{1}{\mu _k}~~ \text{ and } ~~\mathrm{LEL}(G)=\sum _{k=1}^{n-1}\sqrt{\mu _k}, \end{aligned}$$Kf(G)=n∑k=1n-11μkandLEL(G)=∑k=1n-1μk,respectively. The Laplacian energy of the graph G is defined as $$\begin{aligned} \mathrm{LE}(G)=\sum ^n_{i=1}\Big |\mu _i-\frac{2m}{n}\Big |. \end{aligned}$$LE(G)=∑i=1n|μi-2mn|.In this paper, we present an upper bound on Kf of graphs. Also, we obtain some relations between Kf, LEL and first Zagreb index of G. Finally, we give a relation between LEL and LE of G.