Mixed Energy of a Mixed Hourglass Graph

Olayiwola Babarinsa, Hailiza Kamarulhaili

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.

Keywords


Hourglass matrix; Adjacency matrix; Mixed graph; Mixed energy

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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).




DOI: http://dx.doi.org/10.26713%2Fcma.v10i1.1143

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