153,801 research outputs found
A New Quasi-local Mass and Positivity
We use an idea of Wang and Yau to give a new definition of quasi-local mass
for a topological sphere in an initial date set. The new definition modifies
Brown-York's definition by using certain spinor norm as lapse function. And it
requires mean curvature of the topological sphere satisfies apparent horizon
conditions, the topological sphere can be isometrically into Euclidean 3-space
and mean curvature of the initial date set does not change sign. The positivity
holds if we further assume the image of the topological sphere in Euclidean
3-space has nonnegative mean curvature.Comment: 13 pages, final version, the limiting case in Theorem 1 corrected,
remark 1 and new references added, Acta Mathematica Sinica (English Series),
to appea
Laplacian coefficients of unicyclic graphs with the number of leaves and girth
Let be a graph of order and let be the characteristic polynomial of its
Laplacian matrix. Motivated by Ili\'{c} and Ili\'{c}'s conjecture [A. Ili\'{c},
M. Ili\'{c}, Laplacian coefficients of trees with given number of leaves or
vertices of degree two, Linear Algebra and its Applications
431(2009)2195-2202.] on all extremal graphs which minimize all the Laplacian
coefficients in the set of all -vertex unicyclic graphs
with the number of leaves , we investigate properties of the minimal
elements in the partial set of the Laplacian
coefficients, where denote the set of -vertex
unicyclic graphs with the number of leaves and girth . These results are
used to disprove their conjecture. Moreover, the graphs with minimum
Laplacian-like energy in are also studied.Comment: 19 page, 4figure
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