466 research outputs found
The extremal problems on the inertia of weighted bicyclic graphs
Let be a weighted graph. The number of the positive, negative and zero
eigenvalues in the spectrum of are called positive inertia index,
negative inertia index and nullity of , and denoted by ,
, , respectively. In this paper, sharp lower bound on
the positive (resp. negative) inertia index of weighted bicyclic graphs of
order with pendant vertices is obtained. Moreover, all the weighted
bicyclic graphs of order with at most two positive, two negative and at
least zero eigenvalues are identified, respectively.Comment: 12 pages, 5 figures, 2 tables. arXiv admin note: text overlap with
arXiv:1307.0059 by other author
Remainder terms in the fractional Sobolev inequality
We show that the fractional Sobolev inequality for the embedding , can be sharpened by
adding a remainder term proportional to the distance to the set of optimizers.
As a corollary, we derive the existence of a remainder term in the weak
-norm for functions supported in a domain of finite measure.
Our results generalize earlier work for the non-fractional case where is an
even integer.Comment: 13 page
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