54 research outputs found
Edge reconstruction of the Ihara zeta function
We show that if a graph has average degree , then the
Ihara zeta function of is edge-reconstructible. We prove some general
spectral properties of the edge adjacency operator : it is symmetric for an
indefinite form and has a "large" semi-simple part (but it can fail to be
semi-simple in general). We prove that this implies that if , one can
reconstruct the number of non-backtracking (closed or not) walks through a
given edge, the Perron-Frobenius eigenvector of (modulo a natural
symmetry), as well as the closed walks that pass through a given edge in both
directions at least once.
The appendix by Daniel MacDonald established the analogue for multigraphs of
some basic results in reconstruction theory of simple graphs that are used in
the main text.Comment: 19 pages, 2 pictures, in version 2 some minor changes and now
including an appendix by Daniel McDonal
Open Systems Viewed Through Their Conservative Extensions
A typical linear open system is often defined as a component of a larger
conservative one. For instance, a dielectric medium, defined by its frequency
dependent electric permittivity and magnetic permeability is a part of a
conservative system which includes the matter with all its atomic complexity. A
finite slab of a lattice array of coupled oscillators modelling a solid is
another example. Assuming that such an open system is all one wants to observe,
we ask how big a part of the original conservative system (possibly very
complex) is relevant to the observations, or, in other words, how big a part of
it is coupled to the open system? We study here the structure of the system
coupling and its coupled and decoupled components, showing, in particular, that
it is only the system's unique minimal extension that is relevant to its
dynamics, and this extension often is tiny part of the original conservative
system. We also give a scenario explaining why certain degrees of freedom of a
solid do not contribute to its specific heat.Comment: 51 page
Lossless Representation of Graphs using Distributions
We consider complete graphs with edge weights and/or node weights taking
values in some set. In the first part of this paper, we show that a large
number of graphs are completely determined, up to isomorphism, by the
distribution of their sub-triangles. In the second part, we propose graph
representations in terms of one-dimensional distributions (e.g., distribution
of the node weights, sum of adjacent weights, etc.). For the case when the
weights of the graph are real-valued vectors, we show that all graphs, except
for a set of measure zero, are uniquely determined, up to isomorphism, from
these distributions. The motivating application for this paper is the problem
of browsing through large sets of graphs.Comment: 19 page
Controllable graphs with least eigenvalue at least -2
Connected graphs whose eigenvalues are distinct and main are called controllable graphs in view of certain applications in control theory. We give some general characterizations of the controllable graphs whose least eigenvalue is bounded from below by - 2; in particular, we determine all the controllable exceptional graphs. We also investigate the controllable graphs whose second largest eigenvalue does not exceed 1
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