106 research outputs found
Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energy
The Laplacian energy of a graph is the sum of the distances of the
eigenvalues of the Laplacian matrix of the graph to the graph's average degree.
The maximum Laplacian energy over all graphs on nodes and edges is
conjectured to be attained for threshold graphs. We prove the conjecture to
hold for graphs with the property that for each there is a threshold graph
on the same number of nodes and edges whose sum of the largest Laplacian
eigenvalues exceeds that of the largest Laplacian eigenvalues of the graph.
We call such graphs spectrally threshold dominated. These graphs include split
graphs and cographs and spectral threshold dominance is preserved by disjoint
unions and taking complements. We conjecture that all graphs are spectrally
threshold dominated. This conjecture turns out to be equivalent to Brouwer's
conjecture concerning a bound on the sum of the largest Laplacian
eigenvalues
Unicyclic Graphs with equal Laplacian Energy
We introduce a new operation on a class of graphs with the property that the
Laplacian eigenvalues of the input and output graphs are related. Based on this
operation, we obtain a family of order (square root of n) noncospectral
unicyclic graphs on n vertices with the same Laplacian energy.Comment: 11 pages, 11 figures, slightly modified version of Theorem 1 when
compared with original pape
Laplacian Distribution and Domination
Let denote the number of Laplacian eigenvalues of a graph in an
interval , and let denote its domination number. We extend the
recent result , and show that isolate-free graphs also
satisfy . In pursuit of better understanding Laplacian
eigenvalue distribution, we find applications for these inequalities. We relate
these spectral parameters with the approximability of , showing that
. However, for -cyclic graphs, . For trees ,
A note on Gao’s algorithm for polynomial factorization
AbstractShuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao’s construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f
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