Iota energy of weighted digraphs

Document Type : Research Paper

Authors

1 School of Natural Sciences, National university of sciences and Technology Islamabad, Pakistan

2 Department of mathematics, school of Natural Sciences, National University of Sciences and Technology Islamabad, Pakistan

Abstract

The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. The iota energy of a digraph is recently defined as the sum of absolute values of imaginary part of its eigenvalues. In this paper, we extend the concept of iota energy of digraphs to weighted digraphs. We compute the iota energy formulae for the positive and negative weight directed cycles. We also characterize the unicyclic weighted digraphs with cycle weight $ r \in [-1, 1]\backslash \{0\}$ having minimum and maximum iota energy. We obtain well known McClelland upper bound for the iota energy of weighted digraphs. Finally, we find the class of noncospectral equienergetic weighted digraphs.

Keywords

Main Subjects

References

[1] B. D. Achaya, Sp ectral criterion for the cycle balance networks, J. Graph Theor. , 4 (1980) 1{11.
[2] C. Adiga, R. Balakrishnan and W. So, The skew energy of a digraph, Linear Algebra Appl. , 432 (2010) 1825{1835.
[3] R. B. Bapat, D. Kalita and S. Pati, On weighted directed graphs, Linear Algebra Appl. , 436 (2012) 99{111.
[4] M. A. Bhat, Energy of weighted digraphs, Discrete Appl. Math. , 223 (2017) 1{14.
[5] M. A. Bhat and S. Pirzada, On equienergetic signed graphs, Discrete Appl. Math. , 189 (2015) 1{7.
[6] M. A. Bhat and S. Pirzada, Unicyclic signed graphs with minimal energy, Discrete Appl. Math. , 226 (2017) 32{39.
http://dx.doi.org/10.1016/j.dam.2017.03.015 .
[7] D. M. Cvetkovic, M. Do ob and H. Sachs, Spectra of Graphs , Academic Press, New York, 1980.
[8] K. C. Das and R. B. Bapat, A sharp upp er b ound on the sp ectral radius of weighted graphs, Discrete Math. , 308
(2008) 3180{3186.
[9] R. Faro o q, M. Khan and F. Ahmad, Extremal iota energy of bicyclic digraphs, Appl. Math. Comput. , 303 (2017)
24{33.
[10] R. Faro o q, S. Chand and M. Khan, On iota energy of bicyclic signed digraphs, Asian-Europ ean J. Math., http:
//dx.doi.org/10.1142/S1793557119500785 .
[11] R. Faro o q, M. Khan and S. Chand, On iota energy of signed digraphs, Linear Multilinear Algebra , http://dx.doi.
org/10.1080/03081087.2018.1431200 .
[12] F. R. Ganmatcher, The theory of Matrices, I, Translated from the Russian by K. A. Hirsch. Reprint of the 1959
translation, AMS Chelsea Publishing, Providence, RI, 1998.
[13] S. C. Gonga, Y. P. Houb, C. W. Wo o, G. H. Xua and X. L. Shenb, On the integral weighted oriented unicyclic
graphs with minimum skew energy, Linear Algebra Appl. , 439 (2013) 262{272.
[14] K. A. Germina, K. S. Hameed and T. Zaslavsky, On pro ducts and line graphs of signed graphs, their eigenvalues
and energy, Linear Algebra Appl. , 435 (2010) 2432{2450.
[15] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forsch. Graz , 103 (1978) 1{22.
[16] I. Gutman and J. Y. Shao, The energy change of weighted graphs, Linear Algebra Appl. , 435 (2011) 2425{2431.
[17] I. Gutman, B. Furtula, E. Zogic and E. Glogic, Resolvent energy of graphs, MATCH Commun. Math. Comput.
Chem. , 75 (2016) 279{290.
[18] M. Khan, R. Faro o q and A. A. Siddiqui, On the extremal energy of bicyclic digraphs, J. Math. Inequal. , 9 (2015)
799{810.
[19] M. Khan, R. Faro o q and J. Rada, Complex adjacency matrix and energy of digraphs, Linear Multilinear Algebra ,
65 (2017) 2170{2186.
[20] X. Li , Y. Shi and I. Gutman, Graph Energy , Springer-Verlag, New York, 2012.
[21] M. Mateljevic, V. Bozin and I. Gutman, Energy of a p olynomial and the Coulson integral formula, J. Math Chem. ,
48 (2010) 1062{1068.
[22] I. Pe ~na and J. Rada, Energy of digraphs, Linear Multilinear Algebra , 56 (2008) 565{579.
[23] S. Pirzada and M. A. Bhat, Energy of signed digraphs, Discrete Appl. Math. , 169 (2014) 195{205.
Transactions on Combinatorics
Volume 7, Issue 3 - Serial Number 3
September 2018
Pages 55-73

Files

Cited by

History

  • Receive Date: 22 January 2018
  • Revise Date: 21 May 2018
  • Accept Date: 22 May 2018
  • Published Online: 01 September 2018

Share

How to cite

Statistics