On graphs with minimal distance signless Laplacian energy

S. Pirzada; Bilal A. Rather; Rezwan Ul Shaban; Merajuddin

doi:10.2478/ausm-2021-0028

Abstract

For a simple connected graph G of order n having distance signless Laplacian eigenvalues $ρ_{1}^{Q} \geq ρ_{2}^{Q} \geq \dots \geq ρ_{n}^{Q}$ \rho _1^Q \ge \rho _2^Q \ge \cdots \ge \rho _n^Q , the distance signless Laplacian energy DSLE(G) is defined as $D S L E (G) = \sum_{i = 1}^{n} | ρ_{i}^{Q} - \frac{2 W (G)}{n} |$ DSLE\left( G \right) = \sum\nolimits_{i = 1}^n {\left| {\rho _i^Q - {{2W\left( G \right)} \over n}} \right|} where W(G) is the Weiner index of G. We show that the complete split graph has the minimum distance signless Laplacian energy among all connected graphs with given independence number. Further, we prove that the graph K_k ∨ ( K_t∪ K_n−k−t), $1 \leq t \leq ⌊ \frac{n - k}{2} ⌋$ 1 \le t \le \left\lfloor {{{n - k} \over 2}} \right\rfloor has the minimum distance signless Laplacian energy among all connected graphs with vertex connectivity k.

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On graphs with minimal distance signless Laplacian energy

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