20 research outputs found
Largest Laplacian Eigenvalue and Degree Sequences of Trees
We investigate the structure of trees that have greatest maximum eigenvalue
among all trees with a given degree sequence. We show that in such an extremal
tree the degree sequence is non-increasing with respect to an ordering of the
vertices that is obtained by breadth-first search. This structure is uniquely
determined up to isomorphism. We also show that the maximum eigenvalue in such
classes of trees is strictly monotone with respect to majorization.Comment: 9 pages, 2 figure
Sharp Bounds for the Signless Laplacian Spectral Radius in Terms of Clique Number
In this paper, we present a sharp upper and lower bounds for the signless
Laplacian spectral radius of graphs in terms of clique number. Moreover, the
extremal graphs which attain the upper and lower bounds are characterized. In
addition, these results disprove the two conjectures on the signless Laplacian
spectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for the
signless Laplacian index of graphs, Linear Algebra Appl., 432(2010) 3319-3336].Comment: 15 pages 1 figure; linear algebra and its applications 201
Note on Sombor index of connected graphs with given degree sequence
For a simple connected graph , let be the degree of the
vertex of . The general Sombor index of is defined as
where
is the recently invented Sombor index. In this paper, we
show that in the class of connected graphs with a fixed degree sequence (for
which the minimum degree being equal to one), there exists a special extremal
-graph with minimum general Sombor index for (resp. maximum
general Sombor index for either or ). Moreover, for any
given tree, unicyclic, and bicyclic degree sequences with minimum degree 1,
there exists a unique extremal -graph with minimum general Sombor index
for or
Largest eigenvalues of the discrete p-Laplacian of trees with degree sequences
Trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence are characterized. It is shown that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p