On graphs with minimal distance signless Laplacian energy
S. Pirzada, Bilal Ahmad Rather, Rezwan Ul Shaban, Merajuddin
Abstract
Abstract For a simple connected graph G of order n having distance signless Laplacian eigenvalues <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msubsup> <m:mrow> <m:mi>ρ</m:mi> </m:mrow> <m:mn>1</m:mn> <m:mi>Q</m:mi> </m:msubsup> <m:mo>≥</m:mo> <m:msubsup> <m:mrow> <m:mi>ρ</m:mi> </m:mrow> <m:mn>2</m:mn> <m:mi>Q</m:mi> </m:msubsup> <m:mo>≥</m:mo> <m:mo>⋯</m:mo> <m:mo>≥</m:mo> <m:msubsup> <m:mrow> <m:mi>ρ</m:mi> </m:mrow> <m:mi>n</m:mi> <m:mi>Q</m:mi> </m:msubsup> </m:mrow> </m:math> \rho _1^Q \ge \rho _2^Q \ge \cdots \ge \rho _n^Q , the distance signless Laplacian energy DSLE(G) is defined as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mi>D</m:mi> <m:mi>S</m:mi> <m:mi>L</m:mi> <m:mi>E</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msubsup> <m:mo>∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:msubsup> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mrow> <m:msubsup> <m:mrow> <m:mi>ρ</m:mi> </m:mrow> <m:mi>i</m:mi> <m:mi>Q</m:mi> </m:msubsup> <m:mo>-</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mi>W</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mi>n</m:mi> </m:mfrac> </m:mrow> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> 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 ), <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>t</m:mi> <m:mo>≤</m:mo> <m:mrow> <m:mo>⌊</m:mo> <m:mrow> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo>⌋</m:mo> </m:mrow> </m:mrow> </m:math> 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.