Solution to a conjecture on the maximal energy of bipartite bicyclic graphs
Solution to a conjecture on the maximal energy of bipartite bicyclic graphs
复制标题
DOI：
10.1016/j.laa.2011.02.001
复制
发表时间：
2010-12
影响因子：
1.1
通讯作者：
Bofeng Huo;Shengjin Ji;Xueliang Li;Yongtang Shi
中科院分区：
文献类型：
--
作者：
Bofeng Huo;Shengjin Ji;Xueliang Li;Yongtang Shi
一个简单图G的能量，记为E（G），定义为它的邻接矩阵的所有特征值的绝对值之和。设Cn表示n阶圈，Pn 6，6表示两个圈C6通过一条有两片叶子的路Pn-12连接而成的图。设Bn表示除图Ra，B以外的所有二部双圈图类，其中图Ra，B是由两个圈Ca和C B b（a，b≥10）通过一条边连接而得到的，a，b B ≥ 2（mod 4）.在[I. Gutman，D. Vidović，Quest for molecular graphs with maximum energy：a computer experiment，J. Chem. Inf. Sci. 41（2001）1002-1005]，Gutman和Vidović证明了具有最大能量的双圈图是Pn 6，6，其中n=14且n≥16。在[X. Li，J. Zhang，On bicycle graphs with maximum energy，Linear Algebra Appl.427（2007）87-98]，Li和Zhang证明了该猜想对于Bn类中的图是成立的。然而，他们不能确定两个图Ra，带Pn 6，6中的哪一个具有最大的能量值。在[B. Furtula，S. Radenković岛Gutman，Bicyclic molecular graphs with the greatest energy，J. Serbian。Chem.Soc.73（4）（2008）431-433]，报道了高达a+B=50的数值计算，支持该猜想。因此，这一猜想还需要数学证明。本文证明了Pn 6，6的能量大于Ra，B的能量，从而证明了二部双圈图的猜想.对于非二部双圈图，该猜想仍然是开放的。
The energy of a simple graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Cndenote the cycle of order n and Pn6,6the graph obtained from joining two cycles C6by a path Pn-12with its two leaves. Let Bndenote the class of all bipartite bicyclic graphs but not the graph Ra,b, which is obtained from joining two cycles Caand Cb(a,b≥10 and a≡b≡2(mod4)) by an edge. In [I. Gutman, D. Vidović, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41(2001) 1002–1005], Gutman and Vidović conjectured that the bicyclic graph with maximal energy is Pn6,6, for n=14 and n≥16. In [X. Li, J. Zhang, On bicyclic graphs with maximal energy, Linear Algebra Appl. 427(2007) 87–98], Li and Zhang showed that the conjecture is true for graphs in the class Bn. However, they could not determine which of the two graphs Ra,band Pn6,6has the maximal value of energy. In [B. Furtula, S. Radenković, I. Gutman, Bicyclic molecular graphs with the greatest energy, J. Serb. Chem. Soc. 73(4)(2008) 431–433], numerical computations up to a+b=50 were reported, supporting the conjecture. So, it is still necessary to have a mathematical proof to this conjecture. This paper is to show that the energy of Pn6,6is larger than that of Ra,b, which proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, the conjecture is still open.