×

Matrices associated with \(D\)-distance magic graphs and their properties. (English. Russian original) Zbl 1417.15043

Cybern. Syst. Anal. 55, No. 3, 441-448 (2019); translation from Kibern. Sist. Anal. 2019, No. 3, 112-120 (2019).
Summary: Matrices associated with \(D\)-distance magic graphs are considered in the paper. Results regarding the spectral properties of these matrices have been obtained. It has been proved that if two graphs \(G\) and \(H\) of the same order have similar distance matrices \( {A}_{D_1} \) and \( {A}_{D_2} \), respectively, then graph \(G\) is \(D_1\)-distance magic if and only if \(H\) is a \(D_2\)-distance magic graph. Graphs \(G\) and \(H\) are called magic distance-similar and their distance magic constants have been proved to coincide.

MSC:

15A99 Basic linear algebra
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. A. Gallian, “A dynamic survey of graph labeling,” The Electronic J. of Combinatorics, DS6: Dec 22 (2017). · Zbl 0953.05067
[2] M. Miller, C. Rodger, and R. Simanjuntak, “Distance magic labelings of graphs,” Australasian J. of Combinatorics, Vol. 28, 305-315 (2003). · Zbl 1031.05117
[3] V. Vilfred, “Sigma labelled graphs and circulant graphs,” Ph.D. Thesis, University of Kerala, India, March (1994).
[4] S. Beena, “On Σ and Σ′ labelled graphs,” Discrete Mathematics, Vol. 309, 1783-1787 (2009). · Zbl 1188.05108 · doi:10.1016/j.disc.2008.02.038
[5] S. Arumugam, D. Froncek, and N. Kamatchi, “Distance magic graphs — a survey,” J. of the Indonesian Mathematical Society, Special Edition, 11-26 (2011). · Zbl 1288.05216
[6] A. O’Neal and P. Slater, “An introduction to distance <Emphasis Type=”Italic“>D magic graphs,” J. of the Indonesian Mathematical Society, Special Edition, 89-107 (2011). · Zbl 1288.05227
[7] M. Anholcer, S. Cichacz, and I. Peterin, “Spectra of graphs and closed distance magic labelings,” Discrete Mathematics, Vol. 339, 1915-1923 (2016). · Zbl 1334.05134 · doi:10.1016/j.disc.2015.12.025
[8] S. Arumugam and N. Kamatchi, “On the uniqueness of <Emphasis Type=”Italic“>D-vertex magic constant,” Discussiones Mathematicae Graph Theory, Vol. 34, 279-286 (2014). · Zbl 1290.05126 · doi:10.7151/dmgt.1728
[9] G. A. Donets, “Solution of the safe problem on (0, 1)-matrices,” Cybern. Syst. Analysis, Vol. 38, No. 1, 83-88 (2002). · Zbl 1054.15004 · doi:10.1023/A:1015596100010
[10] D. M. Cvetkovic, M. Doob, and H. Zachs, Spectra of Graphs: Theory and Application, Pure and Appl. Mathematics, Acad. Press, July (1984).
[11] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press (2012).
[12] N. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, New York (1993). · Zbl 0797.05032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.