Revan and hyper-Revan indices of Octahedral and icosahedral networks

Let G be a connected graph with vertex set V ( G ) and edge set E ( G ). Recently, the Revan vertex degree concept is defined in Chemical Graph Theory. The first and second Revan indices of G are defined as R 1 ( G ) = <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <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:mrow> </m:munder> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> array ∑uv∈ E array [ r G ( u ) + r G ( v )] and R 2 ( G ) = <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <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:mrow> </m:munder> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> array ∑uv∈ E array [ r G ( u ) r G ( v )], where uv means that the vertex u and edge v are adjacent in G . The first and second hyper-Revan indices of G are defined as HR 1 ( G ) = <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <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:mrow> </m:munder> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> array ∑uv∈ E array [ r G ( u ) + r G ( v )] 2 and HR 2 ( G ) = <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable> <m:mtr> <m:mtd> <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:mrow> </m:munder> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> array ∑uv∈ E array [ r G ( u ) r G ( v )] 2 . In this paper, we compute the first and second kind of Revan and hyper-Revan indices for the octahedral and icosahedral networks.