Some new bounds on resolvent energy of a graph

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G be a simple graph of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n with eigenvalues <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>≥</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo form="prefix">≥</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>≥</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>.</m:mo> </m:math> {\lambda }_{1}\ge {\lambda }_{2}\ge \ldots \ge {\lambda }_{n}. The resolvent energy of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G is a spectrum-based graph invariant defined as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">ER</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msubsup> <m:mrow> <m:mrow> <m:mo>∑</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>.</m:mo> </m:math> {\rm{ER}}(G)={\sum }_{i=1}^{n}{(n-{\lambda }_{i})}^{-1}. In this work, we propose some new bounds for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">ER</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {\rm{ER}}(G) . As a direct consequence of these bounds, we present some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (n,m) -type results for triangle-free graphs.