2,808 research outputs found
Weighted random--geometric and random--rectangular graphs: Spectral and eigenfunction properties of the adjacency matrix
Within a random-matrix-theory approach, we use the nearest-neighbor energy
level spacing distribution and the entropic eigenfunction localization
length to study spectral and eigenfunction properties (of adjacency
matrices) of weighted random--geometric and random--rectangular graphs. A
random--geometric graph (RGG) considers a set of vertices uniformly and
independently distributed on the unit square, while for a random--rectangular
graph (RRG) the embedding geometry is a rectangle. The RRG model depends on
three parameters: The rectangle side lengths and , the connection
radius , and the number of vertices . We then study in detail the case
which corresponds to weighted RGGs and explore weighted RRGs
characterized by , i.e.~two-dimensional geometries, but also approach
the limit of quasi-one-dimensional wires when . In general we look for
the scaling properties of and as a function of , and .
We find that the ratio , with , fixes the
properties of both RGGs and RRGs. Moreover, when we show that
spectral and eigenfunction properties of weighted RRGs are universal for the
fixed ratio , with .Comment: 8 pages, 6 figure
Spectral statistics of random geometric graphs
We use random matrix theory to study the spectrum of random geometric graphs,
a fundamental model of spatial networks. Considering ensembles of random
geometric graphs we look at short range correlations in the level spacings of
the spectrum via the nearest neighbour and next nearest neighbour spacing
distribution and long range correlations via the spectral rigidity Delta_3
statistic. These correlations in the level spacings give information about
localisation of eigenvectors, level of community structure and the level of
randomness within the networks. We find a parameter dependent transition
between Poisson and Gaussian orthogonal ensemble statistics. That is the
spectral statistics of spatial random geometric graphs fits the universality of
random matrix theory found in other models such as Erdos-Renyi, Barabasi-Albert
and Watts-Strogatz random graph.Comment: 19 pages, 6 figures. Substantially updated from previous versio
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
On the sum of k largest singular values of graphs and matrices
In the recent years, the trace norm of graphs has been extensively studied
under the name of graph energy. The trace norm is just one of the Ky Fan
k-norms, given by the sum of the k largest singular values, which are studied
more generally in the present paper. Several relations to chromatic number,
spectral radius, spread, and to other fundamental parameters are outlined. Some
results are extended to more general matrices.Comment: Some corrections applied in v
Beyond graph energy: norms of graphs and matrices
In 1978 Gutman introduced the energy of a graph as the sum of the absolute
values of graph eigenvalues, and ever since then graph energy has been
intensively studied.
Since graph energy is the trace norm of the adjacency matrix, matrix norms
provide a natural background for its study. Thus, this paper surveys research
on matrix norms that aims to expand and advance the study of graph energy.
The focus is exclusively on the Ky Fan and the Schatten norms, both
generalizing and enriching the trace norm. As it turns out, the study of
extremal properties of these norms leads to numerous analytic problems with
deep roots in combinatorics.
The survey brings to the fore the exceptional role of Hadamard matrices,
conference matrices, and conference graphs in matrix norms. In addition, a vast
new matrix class is studied, a relaxation of symmetric Hadamard matrices.
The survey presents solutions to just a fraction of a larger body of similar
problems bonding analysis to combinatorics. Thus, open problems and questions
are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia
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