On Normalized Signless Laplacian Resolvent Energy

Ş. B. Bozkurt Altindağ ; I. Milovanović ; E. Milovanović ; M. Matejić

Kragujevac Journal of Mathematics, Tome 48 (2024) no. 5, p. 673

Résumé

Let $G$ be a simple connected graph with $n$ vertices. Denote by $\mathcal{L}^{+}\left( G\right) =D\left( G\right) ^{-1/2}Q\left( G\right) D\left( G\right) ^{-1/2}$ the normalized signless Laplacian matrix of graph $G$, where $Q\left( G\right) $ and $D\left( G\right) $ are the signless Laplacian and diagonal degree matrices of $G,$ respectively. The eigenvalues of matrix $\mathcal{L}^{+}(G)$, $2=\gamma _{1}^{+}\geq \gamma _{2}^{+}\geq \cdots \geq \gamma _{n}^{+}\geq 0$, are normalized signless Laplacian eigenvalues of $G$. In this paper, we introduce the normalized signless Laplacian resolvent energy of $G$ as $ERNS\left( G\right) =\sum_{i=1}^{n}\frac{1}{3-\gamma _{i}^{+}}$. We also obtain some lower and upper bounds for $ERNS\left( G\right) $ as well as its relationships with other energies and signless Kemeny's constant.

Détail

Classification : 05C50 05C90
Keywords: normalized signless Laplacian eigenvalues, normalized signless Laplacian resolvent energy, bounds.