图的最大度能量及删边操作下最大度能量的变化

The Maximum Degree Energy of Graphs and the Change of Maximum Degree Energy Under Edge Deletion

设G是一个n阶无向图,其顶点集为V={v1,v2,…,vn},边集为E(G),用d_(vi)表示顶点vi的度.图G的最大度矩阵是一个n阶方阵MD(G)=(mij),其中当vi,vj邻接时,mij=max{d_(vi),d_(vj)},否则为0.图G的最大度能量EMD(G)等于MD(G)的所有特征值的绝对值之和.给出了连通二部图G的最大度能量E_(MD)(G)的上下界,并讨论了对于一个连通图G,当删除满足条件d_(u)≥2,dv≥2且N(u)∩N(v)=■的边e=uv∈E(G)时,最大度能量的变化.

Let G be a graph of order n whit vertex set V(G)={v1,v2,…,vn}and edge E(G),and d(vi)be the degree of vertex vi.The maximum degree matrix of G is the n×nmatrix MD(G)=(m_(ij)),where mij=max{d_(vj),d_(vj)}if vi and vj is adjacent,and mij=0 otherwise.The maximum degree energy EMD(G)of graph G is equal to the sum of absolute values of all eigenvalues of MD(G).In this paper,the upper and lower bounds on the maximum degree energy of a connected bipartite graph are established.It is also discussed that for a connected graph G,if the deleted edge e=uv∈E(G)satisfies the condition du≥2,dv≥2 and N(u)∩N(v)=■,the maximum degree energy change.