More on the bounds for the skew Laplacian energy of weighted digraphs

Document Type : Original paper


1 Department of Mathematical Sciences IUST Awantipora Pulwama Jammu and Kashmir India

2 Institute of Technology University of Kashmir

3 Department of Mathematics, Hazratbal


Let $\mathscr{D}$ be a simple connected digraph with $n$ vertices and $m$ arcs and let $W(\mathscr{D})=\mathscr{D},w)$ be the weighted digraph corresponding to $\mathscr{D}$, where the weights are taken from the set of non-zero real numbers. Let $nu_1,nu_2, \dots,nu_n$ be the eigenvalues of the skew Laplacian weighted matrix $\widetilde{SL}W(\mathscr{D})$ of the weighted digraph $W(\mathscr{D})$. In this paper, we discuss the skew Laplacian energy $\widetilde{SLE}W(\mathscr{D})$ of weighted digraphs and obtain the skew Laplacian energy of the weighted star $W(\mathscr{K}_{1, n})$ for some fixed orientation to the weighted arcs. We obtain lower and upper bounds for $\widetilde{SLE}W(\mathscr{D})$ and show the existence of weighted digraphs attaining these bounds. 


Main Subjects

[1] B.D. Acharya, Spectral criterion for cycle balance in networks, J. Graph Theory 4 (1980), no. 1, 1–11.
[2] A. Anuradha and R. Balakrishnan, Skew spectrum of the Cartesian product of an oriented graph with an oriented hypercube, Combinatorial Matrix Theory and Generalized Inverses of Matrices (R.B. Bapat, S.J. Kirkland, K.M. Prasad, and
S. Puntanen, eds.), Springer, New Delhi, 2013, pp. 1–12.
[3] R.B. Bapat, D. Kalita, and S. Pati, On weighted directed graphs, Linear Algebra Appl. 436 (2012), no. 1, 99–111.
[4] M.A. Bhat, Energy of weighted digraphs, Discrete Appl. Math. 223 (2017), 1–14.
[5] Q. Cai, X. Li, and J. Song, New skew Laplacian energy of simple digraphs, Trans. Comb. 2 (2013), no. 1, 27–37.
[6] B.A. Chat, H.A. Ganie, and S. Pirzada, Bounds for the skew Laplacian spectral radius of oriented graphs, Carpathian J. Math. 35 (2019), no. 1, 31–40.
[7] B.A. Chat, H.A. Ganie, and S. Pirzada, Bounds for the skew Laplacian energy of weighted digraphs, Afrika Matematika 32 (2021), no. 5, 745–756.
[8] X. Chen, X. Li, and H. Lian, The skew energy of random oriented graphs, Linear Algebra Appl. 438 (2013), no. 11, 4547–4556.
[9] D.M. Cvetkovic, M. Doob, and H. Sachs, Spectra of Graphs, Academic Press, New York, 1980.
[10] K.C. Das and R.B. Bapat, A sharp upper bound on the spectral radius of weighted graphs, Discrete Math. 308 (2008), no. 15, 3180–3186.
[11] H.A. Ganie and B.A. Chat, Bounds for the energy of weighted graphs, Discrete Appl. Math. 268 (2019), 91–101.
[12] H.A. Ganie, B.A. Chat, and S. Pirzada, Signless Laplacian energy of a graph and energy of a line graph, Linear Algebra Appl. 544 (2018), 306–324.
[13] H.A. Ganie, B.A. Chat, and S. Pirzada, On skew Laplacian spectra and skew Laplacian energy of digraphs, Kragujevac J. Math. 43 (2019), no. 1, 87–98.
[14] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forsch. Graz. 103 (1978), 1–22.
[15] I. Gutman and J.-Y. Shao, The energy change of weighted graphs, Linear Algebra Appl. 435 (2011), no. 10, 2425–2431.
[16] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
[17] X. Li, Y. Shi, and I. Gutman, Graph Energy, Springer, New York, 2012.
[18] I. Pena and J. Rada, Energy of digraphs, Linear Multilinear Algebra 56 (2008), no. 5, 565–579.
[19] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient Blackswan, Hyderabad, 2012.
[20] B. Shader and W. So, Skew spectra of oriented graphs, Electron. J. Combin. 16 (2009), no. 1, ID: N32.
[21] Y. Wang and B. Zhou, A note on skew spectrum of graphs, Ars Combin. 110 (2013), 481–485.
[22] G.-H. Xu and S.-C. Gong, On oriented graphs whose skew spectral radii do not exceed 2, Linear Algebra Appl. 439 (2013), no. 10, 2878–2887.