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定向图的斜 Randi\'{c} 能量

郭立峰1  王力工1,*   

  1. 1. 西北工业大学理学院应用数学系, 西安  710072
  • 收稿日期:2016-03-11 出版日期:2016-12-15 发布日期:2016-12-15
  • 通讯作者: 王力工 lgwangmath@163.com
  • 基金资助:

    国家自然科学基金(No. 11171273)

Skew Randi\'{c} energy of an oriented graph

GUO Lifeng1 WANG Ligong1,*   

  1. 1. Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi'an 710072, China
  • Received:2016-03-11 Online:2016-12-15 Published:2016-12-15

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摘要：

图G是一个简单无向图, G^\sigma 是图 G 在定向 \sigma 下的定向图, G 被称作 G^\sigma 的基础图. 定向图G^\sigma 的斜 Randi\'{c} 矩阵是实对称n\times n矩阵
R_{s}(G^\sigma)=[(r_s)_{ij}]. 如果(v_{i},v_{j})是G^\sigma 的弧, 那么(r_s)_{ij}=(d_id_j)^{-\frac{1}{2}} 且(r_s)_{ji}=-(d_id_j)^{-\frac{1}{2}}, 否则(r_s)_{ij}=(r_s)_{ji}=0. 定向图G^\sigma 的斜Randi\'{c}~能量RE_s(G^\sigma)是指R_{s}(G^\sigma) 的所有特征值的绝对值的和. 首先刻画了定向图G^\sigma 的斜Randi\'{c}矩阵R_{s}(G^\sigma)的特征多项式的系数. 然后给出了定向图G^\sigma 的斜Randi\'{c}能量RE_s(G^\sigma) 的积分表达式. 之后给出了RE_s(G^\sigma) 的上界. 最后计算了定向圈的斜~Randi\'{c}~能量RE_s(G^\sigma).

 

关键词: 斜 Randi\'{c} 矩阵, 斜 Randi\'{c} 能量

Abstract:

Let G be a simple undirected graph and G^\sigma the corresponding oriented graph of G with the orientation \sigma. G is said to be the underlying graph of G^\sigma. The skew Randi\'{c} matrix of an oriented graph G^\sigma is the real symmetric matrix R_{s}(G^\sigma)=[(r_s)_{ij}], where
(r_s)_{ij}=(d_id_j)^{-\frac{1}{2}} and (r_s)_{ji}=-(d_id_j)^{-\frac{1}{2}} if (v_i, v_j) is an arc of \sigma, otherwise (r_s)_{ij}=(r_s)_{ji}=0. The skew Randi\'{c} energy RE_s(G^\sigma) of G^\sigma is the sum of absolute values of the eigenvalues of R_{s}(G^\sigma). In this paper, we firstly
characterize the coefficients of the characteristic polynomial of R_{s}(G^\sigma). Secondly we give an integral representation for the skew Randi\'{c} energy of G^\sigma. Thirdly we show a new upper bound of RE_s(G^\sigma). Finally we compute RE_s(G^\sigma) of oriented cycles.

Key words: skew Randi\'{c} matrix, skew Randi\'{c} energy