35,971 research outputs found
What is the meaning of the graph energy after all?
For a simple graph with eigenvalues of the adjacency matrix
, the energy of the graph
is defined by . Myriads of papers have been
published in the mathematical and chemistry literature about properties of this
graph invariant due to its connection with the energy of (bipartite) conjugated
molecules. However, a structural interpretation of this concept in terms of the
contributions of even and odd walks, and consequently on the contribution of
subgraphs, is not yet known. Here, we find such interpretation and prove that
the (adjacency) energy of any graph (bipartite or not) is a weighted sum of the
traces of even powers of the adjacency matrix. We then use such result to find
bounds for the energy in terms of subgraphs contributing to it. The new bounds
are studied for some specific simple graphs, such as cycles and fullerenes. We
observe that including contributions from subgraphs of sizes not bigger than 6
improves some of the best known bounds for the energy, and more importantly
gives insights about the contributions of specific subgraphs to the energy of
these graphs
Exact Recovery for a Family of Community-Detection Generative Models
Generative models for networks with communities have been studied extensively
for being a fertile ground to establish information-theoretic and computational
thresholds. In this paper we propose a new toy model for planted generative
models called planted Random Energy Model (REM), inspired by Derrida's REM. For
this model we provide the asymptotic behaviour of the probability of error for
the maximum likelihood estimator and hence the exact recovery threshold. As an
application, we further consider the 2 non-equally sized community Weighted
Stochastic Block Model (2-WSBM) on -uniform hypergraphs, that is equivalent
to the P-REM on both sides of the spectrum, for high and low edge cardinality
. We provide upper and lower bounds for the exact recoverability for any
, mapping these problems to the aforementioned P-REM. To the best of our
knowledge these are the first consistency results for the 2-WSBM on graphs and
on hypergraphs with non-equally sized community
Fixed-Dimensional Energy Games are in Pseudo-Polynomial Time
We generalise the hyperplane separation technique (Chatterjee and Velner,
2013) from multi-dimensional mean-payoff to energy games, and achieve an
algorithm for solving the latter whose running time is exponential only in the
dimension, but not in the number of vertices of the game graph. This answers an
open question whether energy games with arbitrary initial credit can be solved
in pseudo-polynomial time for fixed dimensions 3 or larger (Chaloupka, 2013).
It also improves the complexity of solving multi-dimensional energy games with
given initial credit from non-elementary (Br\'azdil, Jan\v{c}ar, and
Ku\v{c}era, 2010) to 2EXPTIME, thus establishing their 2EXPTIME-completeness.Comment: Corrected proof of Lemma 6.2 (thanks to Dmitry Chistikov for spotting
an error in the previous proof
Hardy-Muckenhoupt Bounds for Laplacian Eigenvalues
We present two graph quantities Psi(G,S) and Psi_2(G) which give constant factor estimates to the Dirichlet and Neumann eigenvalues, lambda(G,S) and lambda_2(G), respectively. Our techniques make use of a discrete Hardy-type inequality due to Muckenhoupt
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