Mixed Energy of a Mixed Hourglass Graph

Authors

  • Olayiwola Babarinsa Department of Mathematical Sciences, Federal University Lokoja, 1154 Kogi State
  • Hailiza Kamarulhaili School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Pulau Pinang

DOI:

https://doi.org/10.26713/cma.v10i1.1143

Keywords:

Hourglass matrix, Adjacency matrix, Mixed graph, Mixed energy

Abstract

In this paper we discuss a complete mixed graph called mixed hourglass graph. The direct representation of hourglass matrix in graph gives a weighted mixed hourglass graph. Then, we obtain a mixed hourglass graph from the weighted mixed hourglass graph by assigning its edge-labelled a numerical value of weight 1. Next, we derive the determinant, spectrum and mixed energy of the graph to conclude that the energy of a mixed hourglass graph coincides with twice the number of edges in the graph and the sum of the square of its eigenvalues.

Downloads

Download data is not yet available.

References

C. Adiga, B.R. Rakshith and W. So, On the mixed adjacency matrix of a mixed graph, Linear Algebra Appl. 495 (2016), 223 – 241, DOI: 10.1016/j.laa.2016.01.033.

S. Alikhani, J.I. Brown and S. Jahari, On the domination polynomials of friendship graphs, Filomat 30 (2016), 169 – 178.

O. Babarinsa and H. Kamarulhaili, On determinant of Laplacian matrix and signless Laplacian matrix of a simple graph, in Theoretical Computer Science and Discrete Mathematics, S. Arumugam (ed.), Lecture Notes in Computer Science, 2017, Springer, DOI: 10.1007/978-3-319-64419-628.

O. Babarinsa and H. Kamarulhaili, Quadrant interlocking factorization of hourglass matrix, in AIP Conference Proceedings, D. Mohamad (ed.), AIP Publishing, 2018, DOI: 10.1063/1.5041653.

M. Beck, D. Blado, J. Crawford, T. Jean-Louis and M. Young, On weak chromatic polynomials of mixed graphs, Graphs Combin. 31(1) (2015), 91 – 98, DOI: 10.1007/s00373-013-1381-1.

B. Bylina, The block wz factorization, J. Comput. Appl. Math. 331 (2018), 119 – 132, DOI: 10.1016/j.cam.2017.10.004.

K. Das, S. Mojallal and I. Gutman, Improving mcclellands lower bound for energy, MATCH Commun. Math. Comput. Chem. 70(2) (2013), 663 – 668.

C. Demeure, Bowtie factors of toeplitz matrices by means of split algorithms, IEEE Trans. Acoust., Speech, Signal Process 37(10) (1989), 1601 – 1603, DOI: 10.1109/29.35401.

D. Evans and M. Hatzopoulos, A parallel linear system solver, Int. J. Comput. Math. 7(3) (1979), 227 – 238, DOI: 10.1080/00207167908803174.

K. Guo and B. Mohar, Hermitian adjacency matrix of digraphs and mixed graphs, J. Graph Theory. 85(1) (2015), 217 – 248, DOI: 10.1002/jgt.22057.

I. Gutman, Hyperenergetic and hypoenergetic graphs: Selected Topics on Applications of Graph Spectra, Math. Inst., Belgrade 14(22) (2011), 113 – 135.

I. Gutman and F. Boris, Survey of graph energies, Math. Interdisc. Res. 2 (2017), 85 – 129.

D. Kalita, Determinant of the Laplacian matrix of a weighted directed graph, in Combinatorial Matrix Theory and Generalized Inverses of Matrices, Springer (2013).

J. Liu and X. Li, Hermitian-adjacency matrices and hermitian energies of mixed graphs, Linear Algebra Appl. 466 (2015), 182 – 207, DOI: 10.1016/j.laa.2014.10.028.

M. Liu, Y. Zhu, H. Shan and K.C. Das, The spectral characterization of butterfly-like graphs, Linear Algebra Appl. 513 (2017), 55 – 68, DOI: 10.1016/j.laa.2016.10.003.

R. Ponraj, S.S. Narayanan and A. Ramasamy, Total mean cordiality of umbrella, butterfly and dumbbell graphs, Jordan J. Math. and Stat. 8(1) (2015), 59 – 77.

K. Rosen and K. Krithivasan, Discrete mathematics and its applications, McGraw-Hill Education, Singapore (2015).

G. Yu and H. Qu, Hermitian laplacian matrix and positive of mixed graphs, Appl. Math. Comput. 269 (2015), 70 – 76, DOI: 10.1016/j.amc.2015.07.045.

J. Zhang and H. Kan, On the minimal energy of graphs, Linear Algebra Appl. 453 (2014), 141 – 153, DOI: 10.1016/j.laa.2014.04.009.

S. Zimmerman, Huckel Energy of a Graph: Its Evolution from Quantum Chemistry to Mathematics, Ph.D. Thesis, University of Central Florida Masters (2011).

Downloads

Published

31-03-2019
CITATION

How to Cite

Babarinsa, O., & Kamarulhaili, H. (2019). Mixed Energy of a Mixed Hourglass Graph. Communications in Mathematics and Applications, 10(1), 45–53. https://doi.org/10.26713/cma.v10i1.1143

Issue

Vol. 10 No. 1 (2019)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.