具有乘积度-基尔霍夫指标极值的双圈图（英文）

Bicyclic Graphs with Extremal Multiplicative Degree-Kirchhoff Index

对于一个图G,乘积度-基尔霍夫指标定义为R*(G)=∑{x,y}■V(G)dG(x)dG(y)rG(x,y).基于前人的一些研究成果,用类似于和的度-基尔霍夫指标应用在双圈图中的方法,把乘积度-基尔霍夫指标运用到双圈图中.首先给出了关于R*(G)的一些图变换,然后根据这些图变换,确定了恰好有两个圈的n阶双圈图的最小和最大的乘积度-基尔霍夫指标的值及其对应的极值图.度-基尔霍夫指标广泛应用于电流网络、化学、马尔可夫链和欧氏距离等各个方面.

For a graph G, the multiplicative degree-Kirchhoff index is defined as R＊（G）=∑{x,y}V（G）dG（x）dG（y）rG（x,y）.Based on the previous research results,similar to the additive degree-Kirchhoff index in the bicyclic graphs,the method of the same problem on the multiplicative degree-Kirchhoff index to the bicyclic graphs are discussed.First,some transformations on R＊（G） are given,and then according to these transformations,the values of the minimum and maximum multiplicative degree-Kirchhoff index of bicyclic graphs with n-vertex having precisely two cycles are obtained and the corresponding extremal graphs are characterized.The degree-Kirchhoff index is widely used in the fields of electrical network,chemistry,Markov chain and Euclidean distance,and so on.