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Strong chromatic index of sparse graphs
A coloring of the edges of a graph is strong if each color class is an
induced matching of . The strong chromatic index of , denoted by
, is the least number of colors in a strong edge coloring
of . In this note we prove that for every -degenerate graph . This confirms the strong
version of conjecture stated recently by Chang and Narayanan [3]. Our approach
allows also to improve the upper bound from [3] for chordless graphs. We get
that for any chordless graph . Both
bounds remain valid for the list version of the strong edge coloring of these
graphs
Index statistical properties of sparse random graphs
Using the replica method, we develop an analytical approach to compute the
characteristic function for the probability that a
large adjacency matrix of sparse random graphs has eigenvalues
below a threshold . The method allows to determine, in principle, all
moments of , from which the typical sample to sample
fluctuations can be fully characterized. For random graph models with localized
eigenvectors, we show that the index variance scales linearly with
for , with a model-dependent prefactor that can be exactly
calculated. Explicit results are discussed for Erd\"os-R\'enyi and regular
random graphs, both exhibiting a prefactor with a non-monotonic behavior as a
function of . These results contrast with rotationally invariant
random matrices, where the index variance scales only as , with an
universal prefactor that is independent of . Numerical diagonalization
results confirm the exactness of our approach and, in addition, strongly
support the Gaussian nature of the index fluctuations.Comment: 10 pages, 5 figure
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