The Smith normal form of product distance matrices

• Autorzy: R. B. Bapat , Sivaramakrishnan Sivasubramanian
• Języki publikacji: EN

Abstrakty

EN

Let G = (V, E) be a connected graph with 2-connected blocks H1, H2, . . . , Hr. Motivated by the exponential distance matrix, Bapat and Sivasubramanian in [4] defined its product distance matrix DG and showed that det DG only depends on det DHi for 1 ≤ i ≤ r and not on the manner in which its blocks are connected. In this work, when distances are symmetric, we generalize this result to the Smith Normal Form of DG and give an explicit formula for the invariant factors of DG.

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2016
• Tom: 4
• Numer: 1
• Strony: 46-55

Daty

wydano

2016-01-01

otrzymano

2015-06-15

zaakceptowano

2015-11-10

online

2015-12-16

Twórcy

autor

R. B. Bapat

• Stat-Math Unit, Indian Statistical Institute, Delhi, 7-SJSS Marg, New Delhi 110 016, India

autor

Sivaramakrishnan Sivasubramanian

• Department of Mathematics, Indian Institute of Technology, Bombay, Mumbai 400 076, India

Bibliografia

• [1] Bapat R. B. Resistance matrix and q-laplacian of a unicyclic graph. In Ramanujan Mathematical Society Lecture Notes Series, 7, Proceedings of ICDM 2006, Ed. R. Balakrishnan and C.E. Veni Madhavan (2008), pp. 63–72.
• [2] Bapat R. B., Lal A. K., Pati S. A q-analogue of the distance matrix of a tree. Linear Algebra and its Applications 416 (2006), 799–814.[WoS]
• [3] Bapat R. B., Raghavan T. E. S. Nonnegative Matrices and Applications. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1997.
• [4] Bapat R. B., Sivasubramanian S. Product Distance Matrix of a Graph and Squared Distance Matrix of a Tree. Applicable Analysis and Discrete Mathematics 7 (2013), 285–301.
• [5] Chebotarev P. A class of graph-geodetic distances generalizing the shortest-path and the resistance distances. Discrete Applied Math 159, Issue 5, (2011), 295–302. [2][WoS]
• [6] Chebotarev P. The graph bottleneck identity. Advances in Applied Mathematics 47, Issue 3, (2011), 403–413.[WoS]
• [7] Dealba L. M. Determinants and Eigenvalues. In Handbook of Linear Algebra, L. Hogben, Ed. Chapman & Hall CRC Press, 2007, ch. 4.
• [8] Developers T. S. Sage Mathematics Software (Version 3.1.1), 2008. .
• [9] Klein D. J., Randić M. Resistance distance. Journal of Mathematical Chemistry 12, Issue 1, (1993), 81–95.[WoS]
• [10] Newman M. Integral Matrices. Academic Press, 1972.
• [11] Shiu W. C. Invariant factors of graphs associated with hyperplane arrangements. Discrete Mathematics 288 (2004), 135–148.

Identyfikatory

DOI

10.1515/spma-2016-0005