Computing multiple ABC index and multiple GA index of some grid graphs

Topological indices are the atomic descriptors that portray the structures of chemical compounds and they help us to anticipate certain physico-compound properties like boiling point, enthalpy of vaporization and steadiness. The atom bond connectivity ( ABC ) index and geometric arithmetic ( GA ) index are topological indices which are defined as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mi>B</m:mi> <m:mi>C</m:mi> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> <m:mo>=</m:mo> <m:mstyle> <m:munder> <m:mo>∑</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:mi>E</m:mi> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:munder> <m:mrow> <m:msqrt> <m:mrow> <m:mfrac> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>u</m:mi> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi>d</m:mi> <m:mi>v</m:mi> </m:msub> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>u</m:mi> </m:msub> <m:msub> <m:mi>d</m:mi> <m:mi>v</m:mi> </m:msub> </m:mrow> </m:mfrac> </m:mrow> </m:msqrt> </m:mrow> </m:mstyle> </m:mrow> </m:math> ABC(G)=∑uv∈ E(G)√dᵤ+dᵥ-2/dᵤdᵥ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mi>A</m:mi> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> <m:mo>=</m:mo> <m:mstyle> <m:munder> <m:mo>∑</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:mi>E</m:mi> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:munder> <m:mrow> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:msqrt> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>u</m:mi> </m:msub> <m:msub> <m:mi>d</m:mi> <m:mi>v</m:mi> </m:msub> </m:mrow> </m:msqrt> </m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>u</m:mi> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi>d</m:mi> <m:mi>v</m:mi> </m:msub> </m:mrow> </m:mfrac> </m:mrow> </m:mstyle> </m:mrow> </m:math> GA(G)=∑uv∈ E(G)2√dᵤdᵥdᵤ+dᵥ , respectively, where d u is the degree of the vertex u . The aim of this paper is to introduced the new versions of ABC index and GA index namely multiple atom bond connectivity ( ABC ) index and multiple geometric arithmetic ( GA ) index. As an application, we have computed these newly defined indices for the octagonal grid <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>O</m:mi> <m:mi>p</m:mi> <m:mi>q</m:mi> </m:msubsup> </m:mrow> </m:math> Oₚq , the hexagonal grid H ( p , q ) and the square grid G p, q . Also, we compared these results obtained with the ones by other indices like the ABC 4 index and the GA 5 index.