On the maximal <i> A _a </i> -index of graphs with a prescribed number of edges

Abstract For any real number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0,1</m:mn> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> \alpha \in \left[\mathrm{0,1}] , by the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:math> {A}_{\alpha } - matrix of a graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G , we mean the matrix <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>α</m:mi> <m:mi>D</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>α</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>A</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {A}_{\alpha }\left(G)=\alpha D\left(G)+\left(1-\alpha )A\left(G) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> A\left(G) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>D</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> D\left(G) are the adjacency matrix and the diagonal matrix of vertex degrees of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G , respectively. The largest eigenvalue of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {A}_{\alpha }\left(G) is called the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:math> {A}_{\alpha } -index of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G . Chang and Tam ( Graphs of fixed order and size with maximal-index , Linear Algebra Appl. 673 (2023), 69–100) have solved the problem of determining graphs with maximal <jats:inline-graphic xmlns:xlink="http://www.w3.or