On the eccentric atom-bond sum-connectivity index of trees and unicyclic graphs with given diameter

The eccentric atom-bond sum-connectivity index ( ABSC e ) of a graph G is defined as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>A</m:mi> <m:mi>B</m:mi> <m:mi>S</m:mi> <m:msub> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mi>e</m:mi> </m:mrow> </m:msub> <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:msub> <m:mrow> <m:mo form="prefix" movablelimits="false">∑</m:mo> </m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:mi>E</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:mrow> </m:msub> <m:msqrt> <m:mrow> <m:mfrac> <m:mrow> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>v</m:mi> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mi>v</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfrac> </m:mrow> </m:msqrt> </m:math> ABSCₑ(G)=∑ uv∈ E(G)√eᵤ+eᵥ-2eᵤ+eᵥ , where e u and e v represent the eccentricities of u and v , respectively. In this paper, we investigate the eccentric atom-bond sum-connectivity index of trees and unicyclic graphs on n vertices with diameter d . We also characterize the extremal graphs.