不含偶圈双圈图的极小斜能量

运筹学学报 ›› 2014, Vol. 18 ›› Issue (4): 85-95.

不含偶圈双圈图的极小斜能量

肖毛¹, 王文环^1,*

1. 上海大学理学院数学系, 上海 200444

收稿日期:2014-03-05 出版日期:2014-12-15 发布日期:2014-12-15
通讯作者: 王文环 E-mail:whwang@shu.edu.cn
基金资助:
国家自然科学基金(No. 11001166), 上海市重点学科建设基金(No. S30104)

Minimal skew energies of oriented bicyclic graphs without even cycles

XIAO Mao¹, WANG Wenhuan^1,*

1. Department of Mathematics, College of Sciences, Shanghai University, Shanghai 200444, China

Received:2014-03-05 Online:2014-12-15 Published:2014-12-15

摘要/Abstract

摘要： 令G为简单连通图. 给图G的每条边赋予一个方向, 得到的有向图, 记为G^\sigma. 有向图G^\sigma的斜能量E_{s}(G^{\sigma})定义为G^\sigma的斜邻接矩阵特征值的绝对值之和. 令\mathcal{B}^\circ_{n}表示顶点个数为n不含偶圈的双圈图的集合. 考虑了\mathcal{B}^\circ_{n}中图依斜能量从小到大的排序问题. 利用有向图斜能量的积分公式和实分析的方法, 当n \geq 156和155 \geq n\geq 12时, 分别得到了\mathcal{B}^\circ_{n}中具有最小、次二小和次三小斜能量的双圈图.

关键词: 有向图, 双圈图, 斜能量

Abstract: Let G be a simple connected graph. By assigning an orientation to each edge of G, we obtained an oriented graph G^{\sigma}. The skew energy E_{s}(G^{\sigma}) of an oriented graph G^\sigma is defined as the sum of the absolute eigenvalues of the skew adjacency matrix for G^\sigma. Let \mathcal{B}^\circ_{n} be the set of bicyclic graphs without even cycles having n vertices. The ordering of graphs in \mathcal{B}^\circ_{n} in terms of their minimal skew energies was considered. By employing the integral formula of skew energy and knowledge of real analysis, we deduced the first threegraphs with minimal skew energies in \mathcal{B}^\circ_{n} for n\geq 156 and 155 \geq n\geq 12, respectively.

Key words: oriented graphs, bicyclic graphs, skew energy

中图分类号:

肖毛, 王文环. 不含偶圈双圈图的极小斜能量[J]. 运筹学学报, 2014, 18(4): 85-95.

XIAO Mao, WANG Wenhuan. Minimal skew energies of oriented bicyclic graphs without even cycles[J]. Operations Research Transactions, 2014, 18(4): 85-95.

参考文献

Gutman I, Polansky O E. Mathematical Concepts in Organic Chemistry [M]. Berlin: Springer-Verlag, 1986.

Li X L, Shi Y T, Gutman I. Graph Energy [M]. New York: Springer, 2012.

Adiga C, Balakrishnan R, So W. The skew energy of a digraph [J]. Linear Algebra and Its Applications, 2010, 432(7): 1825-1835.
Adiga C, Balakrishnan R, So W. The skew energy of a digraph [J]. Linear Algebra and Its Applications, 2010, 432(7): 1825-1835.

Hou Y P, Sun X N, Zhang C Y. Oriented unicyclic graphs with extremal skew energy [EB/OL]. [2014-10-28]. http://www.researchgate.net/publication/51934549.

Zhu J M. Oriented unicyclic graphs with the first largest skew energies [J]. Linear Algebra and Its Applications, 2012, 437(10): 2630-2649.

Shen X L, Hou Y P, Zhang C Y. Bicyclic digraphs with extremal skew energy [J]. Electronic Journal of Linear Algebra, 2012, 23: 340-355.

Gong S C, Li X L, Xu G H. On oriented graphs with minimal skew energy [J]. Electronic Journal of Linear Algebra, 2014, 47: 692-704

Gong S C, Xu G H. 3-regular digraphs with optimum skew energy [J]. Linear Algebra and Its Applications, 2012, 436: 465-471.

Chen X L, Li X L, Lian H S. 4-Regular oriented graphs with optimum skew energy [J]. Linear Algebra and Its Applications, 2013, 439(10): 2948-2960.

Tian G X. On the skew energy of orientations of hypercubes [J]. Linear Algebra and Its Applications, 2011, 435(9): 2140-2149.

Anuradha A, Balakrishnan R, So W. Skew spectra of graphs without even cycles [J]. Linear Algebra and Its Applications, 2014, 444(1): 67-80.

Hou Y P, Lei T G. Characteristic polynomials of skew-adjacency matrices of oriented graphs [J]. The Electronic Journal of Combinatorics, 2011, 18(1): 156-167.

Gutman I. Acyclic systems with extremal Huckel \pi-Electron energy [J]. Theoretica Chimica Acta, 1977, 45: 79-87.

Yan W G, Ye L Z. On the minimal energy of trees with a given diameter [J]. Applied Mathematics Letters, 2005, 18(9): 1046-1052.

Zhou B, Li F. On minimal energies of trees of a prescribed diameter [J]. Journal of Mathematical Chemistry, 2006, 39(3/4): 465-473.

Huo B F, Li X L, Shi Y T, et al. Determining the conjugated trees with the third through the sixth minimal energies [J]. Match-Communications in Mathematical and in Computer Chemistry, 2010, 65: 521-532.

Wang W H, Xu W W. Graphs with the maximal Estrada indices [J]. Linear Algebra and Its Applications, 2014, 446: 314-328.

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