# On The Determinant of q-Distance Matrix of a Graph

Hong-Hai Li; Li Su; Jing Zhang

Discussiones Mathematicae Graph Theory (2014)

- Volume: 34, Issue: 1, page 103-111
- ISSN: 2083-5892

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topHong-Hai Li, Li Su, and Jing Zhang. "On The Determinant of q-Distance Matrix of a Graph." Discussiones Mathematicae Graph Theory 34.1 (2014): 103-111. <http://eudml.org/doc/267842>.

@article{Hong2014,

abstract = {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.},

author = {Hong-Hai Li, Li Su, Jing Zhang},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {q-distance matrix; determinant; weighted graph; directed graph; -distance matrix},

language = {eng},

number = {1},

pages = {103-111},

title = {On The Determinant of q-Distance Matrix of a Graph},

url = {http://eudml.org/doc/267842},

volume = {34},

year = {2014},

}

TY - JOUR

AU - Hong-Hai Li

AU - Li Su

AU - Jing Zhang

TI - On The Determinant of q-Distance Matrix of a Graph

JO - Discussiones Mathematicae Graph Theory

PY - 2014

VL - 34

IS - 1

SP - 103

EP - 111

AB - 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.

LA - eng

KW - q-distance matrix; determinant; weighted graph; directed graph; -distance matrix

UR - http://eudml.org/doc/267842

ER -

## References

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- [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] Zbl1092.05041
- [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] Zbl1175.05031
- [4] R.L. Graham and H.O. Pollak, On the addressing problem for loop switching, Bell. System Tech. J. 50 (1971) 2495-2519. Zbl0228.94020
- [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] Zbl0363.05034
- [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] Zbl1073.05550
- [7] S. Sivasubramanian, A q-analogue of Graham, Hoffman and Hosoya’s result , Electron. J. Combin. 17 (2010) #21. Zbl1188.05027
- [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]

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