In this note, we show how the determinant of the q-distance matrix Dq(T) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85-88]. Further, by means of the result, we determine the determinant of the q-distance matrix of the graph obtained from a connected weighted graph G by adding the weighted branches to G, and so generalize in part the results obtained by Bapat et al. [R.B. Bapat, S. Kirkland and M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl. 401 (2005) 193- 209]. In particular, as a consequence, determinantal formulae of q-distance matrices for unicyclic graphs and one class of bicyclic graphs are presented.
@article{bwmeta1.element.doi-10_7151_dmgt_1720,
author = {Hong-Hai Li and Li Su and Jing Zhang},
title = {On The Determinant of q-Distance Matrix of a Graph},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {103-111},
zbl = {1292.05174},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1720}
}
Hong-Hai Li; Li Su; Jing Zhang. On The Determinant of q-Distance Matrix of a Graph. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 103-111. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1720/
[1] R.B. Bapat, S. Kirkland and M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl. 401 (2005) 193-209. doi:10.1016/j.laa.2004.05.011[Crossref] | Zbl 1064.05097
[2] R.B. Bapat, A.K. Lal and S. Pati, A q-analogue of the distance matrix of a tree, Linear Algebra Appl. 416 (2006) 799-814. doi:10.1016/j.laa.2005.12.023[Crossref] | Zbl 1092.05041
[3] R.B. Bapat and Pritha Rekhi, Inverses of q-distance matrices of a tree, Linear Algebra Appl. 431 (2009) 1932-1939. doi:10.1016/j.laa.2009.06.032[WoS][Crossref] | Zbl 1175.05031
[4] R.L. Graham and H.O. Pollak, On the addressing problem for loop switching, Bell. System Tech. J. 50 (1971) 2495-2519. | Zbl 0228.94020
[5] R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85-88. doi:10.1002/jgt.3190010116[Crossref] | Zbl 0363.05034
[6] S.G. Guo, The spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices, Linear Algebra Appl. 408 (2005) 78-85. doi:10.1016/j.laa.2005.05.022[Crossref][WoS] | Zbl 1073.05550
[7] S. Sivasubramanian, A q-analogue of Graham, Hoffman and Hosoya’s result , Electron. J. Combin. 17 (2010) #21. | Zbl 1188.05027
[8] P. Lancaster, Theory of Matrices (Academic Press, NY, 1969).
[9] W. Yan, Y.-N. Yeh, The determinants of q-distance matrices of trees and two quantities relating to permutations, Adv. in Appl. Math. 39 (2007) 311-321. doi:10.1016/j.aam.2006.04.002[Crossref][WoS]