An exact formula for the contraction factor of a subdivided Gaussian topological polymer

Abstract We consider the radius of gyration of a Gaussian topological polymer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">G</mml:mi> </mml:mrow> </mml:mrow> </mml:math> formed by subdividing a graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="bold">G</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>′</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> of arbitrary topology (for instance, branched or multicyclic). We give a new exact formula for the expected radius of gyration and contraction factor of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="bold">G</mml:mi> </mml:mrow> </mml:mrow> </mml:math> in terms of the number of subdivisions of each edge of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="bold">G</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>′</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and a new weighted Kirchhoff index for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="bold">G</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>′</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . The formula explains and extends previous results for the contraction factor and Kirchhoff index of subdivided graphs.