Extremal skew energy of digraphs with no even cycles

Document Type: Research Paper

Authors

1 Department of Applied Mathematics, Northwestern Polytechnical University

2 Center for Combinatorics and LPMC-TJKLC, Nankai University

3 Center for Combinatorics, Nankai University

Abstract

‎Let $D$ be a digraph with skew-adjacency matrix $S(D)$‎. ‎Then the‎ ‎skew energy of $D$ is defined as the sum of the norms of all‎ ‎eigenvalues of $S(D)$‎. ‎Denote by $\mathcal{O}_n$ the class of‎ ‎digraphs of order $n$ with no even cycles‎, ‎and by‎ ‎$\mathcal{O}_{n,m}$ the class of digraphs in $\mathcal{O}_n$ with‎ ‎$m$ arcs‎. ‎In this paper‎, ‎we first give the minimal skew energy‎ ‎digraphs in $\mathcal{O}_n$ and $\mathcal{O}_{n,m}$ with $n-1\leq‎ ‎m\leq \frac{3}{2}(n-1)$‎. ‎Then we determine the maximal skew energy‎ ‎digraphs in $\mathcal{O}_{n,n}$ and $\mathcal{O}_{n,n+1}$‎, ‎and in‎ ‎the latter case we assume that $n$ is even‎.

Keywords

Main Subjects

References

C. Adiga, R. Balakrishnan and W. So (2010). The skew energy of digraph. Linear Algebra Appl.. 432, 1825-1835
X. Chen, X. Li and H. Lian (2013). 4-Regular oriented graphs with optimum skew energy. Linear Algebra Appl.. 439, 2948-2960
S. Gong and G. Xu (2012). The characteristic polynomial and the matchings polynomial of a weighted oriented graph. Linear Algebra Appl.. 436, 3597-3607
S. Gong and G. Xu (2012). 3-Regular digraphs with optimum skew energy. Linear Algebra Appl.. 436, 465-471
S. Gong, W. Zhong and G. Xu (2014). 4-Regular oriented graphs with optimum skew energies. European J. Combin.. 36, 77-85
I. Gutman (1978). The energy of a graph. Ber. Math.-Statist. Sekt. Forsch. Graz. (103), 1-22
I. Gutman, X. Li and J. Zhang (2009). Graph Energy. in: M. Dehmer, F. Emmert-Streib (Eds.), Analysis of Complex Network: From Biology to Linguistics, Wiley-VCH Verlag, Weinheim. , 145-174
I. Gutman and O.E. Polansky (1986). Mathematical Concepts in Organic Chemistry. Springer-Verlag, berlin.
Y. Hou and T. Lei (2011). Characteristic polynomials of skew-adjacency matrices of oriented graphs. Electron. J. Combin.. 18, 156-167
Y. Hou, X. Shen and C. Zhang Oriented unicyclic graphs with extremal skew energy. http://arxiv.org/pdf/1108.6229v1.pdf.
B. Lass (2004). Matching polynomials and duality. Combinatorica. 24, 427-440
J. Li, X. Li and Y. Shi (2011). On the maximal energy tree with two maximum degree vertices. Linear Algebra Appl.. 435, 2272-2284
X. Li and H. Lian A survey on the skew energy of oriented graphs. http://arxiv.org/pdf/1304.5707v4.pdf.
X. Li, Y. Shi and I. Gutman (2012). Graph Energy. Springer, New York.
X. Shen, Y. Hou and C. Zhang (2012). Bicyclic digraphs with extremal skew energy. Electron. J. Linear Algebra. 23, 340-355
R. Thomas (2006). A survey of Pfaffian orientations of graph. Proc. Int. Congress of Mathematicians, ICM, Madrid, Spain. , 963-984
V. A. Zorich (2002). Mathematical Analysis. MCCME, Moscow.
Transactions on Combinatorics

Volume 3, Issue 1
March 2014
Pages 37-49

Files

Share

How to cite

Statistics