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SUMMARY:Distance Matrices of Trees: Invariants\, Old and New
DTSTART;VALUE=DATE-TIME:20191022T090000Z
DTEND;VALUE=DATE-TIME:20191022T100000Z
DTSTAMP;VALUE=DATE-TIME:20200806T093516Z
UID:indico-event-7269@cern.ch
DESCRIPTION:Abstract: In 1971\, Graham and Pollak showed that if DT is the
distance matrix of a tree T on n nodes\, then det(DT) depends only on n\,
not T. This independence from the tree structure has been verified for ma
ny different variants of weighted bi-directed trees. In my talk:\n\n1. I w
ill present a general setting which strictly subsumes every known variant\
, and where we show that det(DT) - as well as another graph invariant\, th
e cofactor-sum - depends only on the edge-data\, not the tree-structure.\n
\n2. More generally - even in the original unweighted setting - we strengt
hen the state-of-the-art\, by computing the minors of DT where one removes
rows and columns indexed by equal-sized sets of pendant nodes. (In fact w
e go beyond pendant nodes.)\n\n3. We explain why our result is the "most g
eneral possible"\, in that allowing greater freedom in the parameters lead
s to dependence on the tree-structure.\n\n4. Our results hold over an arbi
trary unital commutative ring. This uses Zariski density\, which seems to
be new in the field\, yet is richly rewarding.\n\nWe then discuss related
results for arbitrary strongly connected graphs\, including a third\, nove
l invariant. If time permits\, a formula for Dâˆ’1T will be presented for
trees T\, whose special case answers an open problem of Bapat-Lal-Pati (Li
near Alg. Appl. 2006)\, and which extends to our general setting a result
of Graham-Lovasz (Advances in Math. 1978). (Joint with Projesh Nath Choudh
ury).\n\nhttps://indico.tifr.res.in/indico/conferenceDisplay.py?confId=726
9
LOCATION: A-201 (STCS Seminar Room)
URL:https://indico.tifr.res.in/indico/conferenceDisplay.py?confId=7269
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