The Smith normal form of product distance matrices

R. B. Bapat; Sivaramakrishnan Sivasubramanian

Special Matrices (2016)

  • Volume: 4, Issue: 1, page 46-55
  • ISSN: 2300-7451

Abstract

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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.

How to cite

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R. B. Bapat, and Sivaramakrishnan Sivasubramanian. "The Smith normal form of product distance matrices." Special Matrices 4.1 (2016): 46-55. <http://eudml.org/doc/276949>.

@article{R2016,
abstract = {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.},
author = {R. B. Bapat, Sivaramakrishnan Sivasubramanian},
journal = {Special Matrices},
language = {eng},
number = {1},
pages = {46-55},
title = {The Smith normal form of product distance matrices},
url = {http://eudml.org/doc/276949},
volume = {4},
year = {2016},
}

TY - JOUR
AU - R. B. Bapat
AU - Sivaramakrishnan Sivasubramanian
TI - The Smith normal form of product distance matrices
JO - Special Matrices
PY - 2016
VL - 4
IS - 1
SP - 46
EP - 55
AB - 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.
LA - eng
UR - http://eudml.org/doc/276949
ER -

References

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  1. [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. Zbl1194.05030
  2. [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] Zbl1092.05041
  3. [3] Bapat R. B., Raghavan T. E. S. Nonnegative Matrices and Applications. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1997. Zbl0879.15015
  4. [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. Zbl1299.05077
  5. [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] Zbl1209.05068
  6. [6] Chebotarev P. The graph bottleneck identity. Advances in Applied Mathematics 47, Issue 3, (2011), 403–413.[WoS] Zbl1232.05064
  7. [7] Dealba L. M. Determinants and Eigenvalues. In Handbook of Linear Algebra, L. Hogben, Ed. Chapman & Hall CRC Press, 2007, ch. 4. 
  8. [8] Developers T. S. Sage Mathematics Software (Version 3.1.1), 2008. . 
  9. [9] Klein D. J., Randić M. Resistance distance. Journal of Mathematical Chemistry 12, Issue 1, (1993), 81–95.[WoS] 
  10. [10] Newman M. Integral Matrices. Academic Press, 1972. Zbl0254.15009
  11. [11] Shiu W. C. Invariant factors of graphs associated with hyperplane arrangements. Discrete Mathematics 288 (2004), 135–148. Zbl1056.05120

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