On The Determinant of q-Distance Matrix of a Graph

• Autorzy: Hong-Hai Li , Li Su , Jing Zhang
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2014
• Tom: 34
• Numer: 1
• Strony: 103-111

Daty

wydano

2014-02-01

online

2014-02-14

Twórcy

autor

Hong-Hai Li

• College of Mathematics and Information Science Jiangxi Normal University Nanchang, 330022, P.R. China

autor

Li Su

• College of Mathematics and Information Science Jiangxi Normal University Nanchang, 330022, P.R. China

autor

Jing Zhang

• College of Mathematics and Information Science Jiangxi Normal University Nanchang, 330022, P.R. China

Bibliografia

• [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]
• [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]
• [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]
• [4] R.L. Graham and H.O. Pollak, On the addressing problem for loop switching, Bell. System Tech. J. 50 (1971) 2495-2519.
• [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]
• [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]
• [7] S. Sivasubramanian, A q-analogue of Graham, Hoffman and Hosoya’s result , Electron. J. Combin. 17 (2010) #21.
• [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]

Identyfikatory

DOI

10.7151/dmgt.1720