Study On The Extreme Value Of Product Form Kirchhoff Index

In this paper,we study the extremal structure of Kirchhoff index with the form of the product by use the method of graph transformation and graph symmetry,characterizes the extremal structure of the cacti graph with the multiplicative eccentricity resistance-distance and extremal structure graph of the multiplicative eccentricity resistance-distance and resistance-distance spectral radius for graphs of diameter 2.First,we introduce the problems and their background,establish the definition of used symbols and then display the main research results and the main structure.In the first part of this paper,the extremal structure of the minimum,second minimum and maximum multiplicative eccentricity resistance-distance of cacti graph is mainly studied.We use graph transformation monotonically decreasing and increasing,the first graph transformation of cacti graph generally by moving or bordered then minus edge,calculate,and reasoning and proof,some conclusions are obtained.By continuous transformation,the extreme value is obtained,and the corresponding extremal structure graph is characterized.In the second part of this paper,the extremal structure of multiplicative eccentricity resistance-distance and resistance-distance spectral radius for graphs of diameter 2 is mainly studied.Firstly,we introduce some relevant lemma.Through using the transform of graphs,we determine the extremal structure graphs of some possible multiplicative eccentricity resistance-distance and resistance-distance spectral radius.Secondly,according to the method of slove the multiplicative eccentricity resistance-distance and resistance-distance spectral radius,we get the possible extremal structure graph correspond to the multiplicative eccentricity resistance-distance and resistance-distance spectral radius equations,determine their extremal structure,and prove the conclusion.Finally,the extreme value of the multiplicative eccentricity resistance-distance is obtained by continuous transformation and the corresponding extremal structure graph is characterized.Through differential comparison,which characterize the extremal structure graph of corresponding resistance-distance spectral radius of maximum,minimum,second maximum and second minimum,and so on.