关于图的广义距离能量的界

On the Bounds for the Generalized Distance Energy of Graphs

对于简单连通图G,广义距离矩阵Dα(G)是Tr(G)和D(G)的凸组合,即对于0≤α≤1,Dα(G)=αTr(G)+(1-α)D(G).设■是Dα(G)的特征值,则图G的广义距离能量定义为■,其中W(G)是G的Wiener指数.本文首先讨论了当α∈(0,1/2]时,广义距离能量E^(Dα)(G)的一些上下界,研究了当α∈[1/2,1)时的情形,从而扩大了已知界中α的范围.其次,在保留距离能量主要特征情况下,得到广义距离能量E^(Dα)(G)的一些上下界.最后,获得完全k-部图的广义距离能量.

For simple connected graph G,the generalized distance matrix Da(G)is a convex combination of Tr(G)and D(G),and is defined as Dα(G)=αTr(G)+(1-α)D(G),for 0≤α≤1.If■are the eigenvalues of Dα(G).We define the generalized distance energy of G as■,where W(G)is the Wiener index of G.In this paper,we first discuss some upper and lower bounds on the generalized distance energy E^(Dα)(G)of graphs forα∈(0,1/2],and study the case whenα∈[1/2,1),so as to expand the range ofαin known bounds.Secondly,under the condition of preserving the main characteristics of distance energy,we obtain some upper and lower bounds on the generalized distance energy E^(Dα)(G)of graphs.Finally,we get the generalized distance energy of complete k-partite graphs.