On Laplacian resolvent energy of graphs

Document Type : Research Paper


1 Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India

2 Department of Mathematics, University of Kashmir, Srinagar, India


Let $G$ be a simple connected graph of order $n$ and size $m$. The matrix $L(G)=D(G)-A(G)$ is the Laplacian matrix of $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix, respectively. For the graph $G$, let $d_{1}\geq d_{2}\geq \cdots d_{n}$ be the vertex degree sequence and $\mu_{1}\geq \mu_{2}\geq \cdots \geq \mu_{n-1}>\mu_{n}=0$ be the Laplacian eigenvalues. The Laplacian resolvent energy $RL(G)$ of a graph $G$ is defined as $RL(G)=\sum\limits_{i=1}^{n}\frac{1}{n+1-\mu_{i}}$. In this paper, we obtain an upper bound for the Laplacian resolvent energy $RL(G)$ in terms of the order, size and the algebraic connectivity of the graph. Further, we establish relations between the Laplacian resolvent energy $RL(G)$ with each of the Laplacian-energy-Like invariant $LEL$, the Kirchhoff index $Kf$ and the Laplacian energy $LE$ of the graph.


Main Subjects

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Volume 12, Issue 4 - Serial Number 4
December 2023
Pages 217-225
  • Receive Date: 09 April 2022
  • Revise Date: 13 September 2022
  • Accept Date: 18 September 2022
  • Published Online: 01 December 2023