BEGIN:VCALENDAR
PRODID:-//eluceo/ical//2.0/EN
VERSION:2.0
CALSCALE:GREGORIAN
BEGIN:VEVENT
UID:www.tcs.tifr.res.in/event/1008
DTSTAMP:20230914T125946Z
SUMMARY:Distance Matrices of Trees: Invariants\, Old and New
DESCRIPTION:Speaker: Apoorva Khare (Indian Institute of Science \nand Anal
ysis & Probability Research Group\n(Bangalore\, India))\n\nAbstract: \nAbs
tract: In 1971\, Graham and Pollak showed that if $D_T$ is the distance m
atrix of a tree $T$ on $n$ nodes\, then $\\det(D_T)$ depends only on $n$\,
not $T$. This independence from the tree structure has been verified for
many different variants of weighted bi-directed trees. In my talk:\n1. I w
ill present a general setting which strictly subsumes every known variant\
, and where we show that $\\det(D_T)$ - as well as another graph invariant
\, the cofactor-sum - depends only on the edge-data\, not the tree-structu
re.\n2. More generally - even in the original unweighted setting - we stre
ngthen the state-of-the-art\, by computing the minors of $D_T$ where one r
emoves rows and columns indexed by equal-sized sets of pendant nodes. (In
fact we go beyond pendant nodes.)\n3. We explain why our result is the "mo
st general possible"\, in that allowing greater freedom in the parameters
leads to dependence on the tree-structure.\n4. Our results hold over an ar
bitrary unital commutative ring. This uses Zariski density\, which seems t
o be new in the field\, yet is richly rewarding.\nWe then discuss related
results for arbitrary strongly connected graphs\, including a third\, nove
l invariant. If time permits\, a formula for $D_T^{-1}$ will be presented
for trees $T$\, whose special case answers an open problem of Bapat-Lal-Pa
ti (Linear Alg. Appl. 2006)\, and which extends to our general setting a r
esult of Graham-Lovasz (Advances in Math. 1978). (Joint with Projesh Nath
Choudhury.)\n
URL:https://www.tcs.tifr.res.in/web/events/1008
DTSTART;TZID=Asia/Kolkata:20191022T143000
DTEND;TZID=Asia/Kolkata:20191022T153000
LOCATION:A-201 (STCS Seminar Room)
END:VEVENT
END:VCALENDAR