Further Results On Resistance Distance And Kirchhoff Index

Considering an electrical network N as a weighted graph G,The node in electrical network N is regarded as the vertex in graph G,the resistance in N is regarded as the edge in graph G,the reciprocal of the conductance is the resistance value of the edge.The effective resistance between the two nodes i,j in electrical network N is called the resistance distance between the two vertices i,j of the graph G,denoted by rij(G).Resistance distance is distance function,the resistance distance is widely used in random walk,electronic engineering,complex networks and chemical graph theory etc.,attracting many domestic and foreign scholars.Kirchhoff index is an important topological index(invariant of a graph),which comes from the molecular structure,Kirchhoff index is a way to digitize,it also can reflect the structural features of the compounds.In addition,degree Kirchhoff indices(including the multiplicative degree Kirchhoff index and the addictive degree Kirchhoff index)are also important.The innovation of this paper is mainly reflected in the following aspects:(1)Constructing a reversible matrix L + ehT by rank 1 perturbation,where e =(1,1,…,1)T,h =(h1,h2,…,hn)T.Use the elements of X=(L+ehT)-1 to get resistance distance rij=xii+xjj-xij-xji.It only needs to solve the inverse matrix of X to get the reisistance distance,corresponding.And can avoid the problem:the inverse of matrix is not sole.Another advantage of this result is that we can choose different column vector h according todifferent practical applications(h satisfies(?)).This paper also gives a matrix Z=L(L+?hT)-1,where ?=(d1,d2,…,dn)T.Similarly,we can use the elements of Z to get resistance distance.(2)In this paper,we use the trace of X to get the formulas of Kirchhoff index and degree Kirchhoff index.(3)In 1982,Godsil and Mckay present several methods to construct cospectral graphs which are not isomorphic.One of the most classical methods is GM switching.Based on GM switching,this paper gives the relationship between the resistance distances,the relationship between degree Kirchhoff indices.{V1,V2} is a division of the vertex set of graph G =(V,E),and G is the switched graph.when i,j?V1(V2),we can get the relations of resistance distance between Gand G,that is rij=rij.When i,j are not in the same set,we have rij=rij-4(S#NC-1)ij.The relationships of Kirchhoff indices and degree Kirchhoff indices between G and G are Kf(G)= Kf(G);Kf*(G)= Kf*(G);Kf+(G)= Kf+(G).The relationships between the resistance distances,Kirchhoff indices and degree Kirchhoff indices,respectively,by the operation GM switching,are given for the first time.