7,406 research outputs found
The Laplacian energy of random graphs
Gutman {\it et al.} introduced the concepts of energy \En(G) and Laplacian
energy \EnL(G) for a simple graph , and furthermore, they proposed a
conjecture that for every graph , \En(G) is not more than \EnL(G).
Unfortunately, the conjecture turns out to be incorrect since Liu {\it et al.}
and Stevanovi\'c {\it et al.} constructed counterexamples. However, So {\it et
al.} verified the conjecture for bipartite graphs. In the present paper, we
obtain, for a random graph, the lower and upper bounds of the Laplacian energy,
and show that the conjecture is true for almost all graphs.Comment: 14 page
Asymptotic Laplacian-Energy-Like Invariant of Lattices
Let denote the Laplacian eigenvalues of
with vertices. The Laplacian-energy-like invariant, denoted by , is a novel topological index. In this paper, we
show that the Laplacian-energy-like per vertex of various lattices is
independent of the toroidal, cylindrical, and free boundary conditions.
Simultaneously, the explicit asymptotic values of the Laplacian-energy-like in
these lattices are obtained. Moreover, our approach implies that in general the
Laplacian-energy-like per vertex of other lattices is independent of the
boundary conditions.Comment: 6 pages, 2 figure
Essential spectrum and Weyl asymptotics for discrete Laplacians
In this paper, we investigate spectral properties of discrete Laplacians. Our
study is based on the Hardy inequality and the use of super-harmonic functions.
We recover and improve lower bounds for the bottom of the spectrum and of the
essential spectrum. In some situation, we obtain Weyl asymptotics for the
eigenvalues. We also provide a probabilistic representation of super-harmonic
functions. Using coupling arguments, we set comparison results for the bottom
of the spectrum, the bottom of the essential spectrum and the stochastic
completeness of different discrete Laplacians. The class of weakly spherically
symmetric graphs is also studied in full detail
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