On the spectral radius and energy of the degree distance matrix of a connected 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 connected graph on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n vertices. The degree of a vertex <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:mi>V</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> v\in V\left(G) , denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>v</m:mi> </m:mrow> </m:msub> </m:math> {d}_{v} , is the number of edges incident with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>v</m:mi> </m:math> v and the distance between any two vertices <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:mi>V</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> u,v\in V\left(G) , denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mi>v</m:mi> </m:mrow> </m:msub> </m:math> {d}_{uv} , is defined as the length of the shortest path from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> </m:math> u to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>v</m:mi> </m:math> v . The distance matrix of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G , denoted by, <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) , is defined as <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:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mi>v</m:mi> </m:mrow> </m:msub> </m:math> D\left(G)={d}_{uv} . We now define and investigate the degree distance matrix of a connected graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G , defined as <jats:inline-graphic xm