2,576 research outputs found
A remark on zeta functions of finite graphs via quantum walks
From the viewpoint of quantum walks, the Ihara zeta function of a finite
graph can be said to be closely related to its evolution matrix. In this note
we introduce another kind of zeta function of a graph, which is closely related
to, as to say, the square of the evolution matrix of a quantum walk. Then we
give to such a function two types of determinant expressions and derive from it
some geometric properties of a finite graph. As an application, we illustrate
the distribution of poles of this function comparing with those of the usual
Ihara zeta function.Comment: 14 pages, 1 figur
A Statistical Model of Current Loops and Magnetic Monopoles
We formulate a natural model of current loops and magnetic monopoles for
arbitrary planar graphs, which we call the monopole-dimer model, and express
the partition function of this model as a determinant. We then extend the
method of Kasteleyn and Temperley-Fisher to calculate the partition function
exactly in the case of rectangular grids. This partition function turns out to
be a square of the partition function of an emergent monomer-dimer model when
the grid sizes are even. We use this formula to calculate the local monopole
density, free energy and entropy exactly. Our technique is a novel
determinantal formula for the partition function of a model of vertices and
loops for arbitrary graphs.Comment: 17 pages, 5 figures, significant stylistic revisions. In particular,
rewritten with a mathematical audience in mind. Numerous errors fixed. This
is the final published version. Maple program file can be downloaded from the
link on the right of this pag
Topological atomic displacements, Kirchhoff and Wiener indices of molecules
We provide a physical interpretation of the Kirchhoff index of any molecules as well as of the Wiener index of acyclic ones. For the purpose, we use a local vertex invariant that is obtained from first principles and describes the atomic displacements due to small vibrations/oscillations of atoms from their equilibrium positions. In addition, we show that the topological atomic displacements correlate with the temperature factors (B-factors) of atoms obtained by X-ray crystallography for both organic molecules and biological macromolecules
Spectral Clustering of Graphs with the Bethe Hessian
Spectral clustering is a standard approach to label nodes on a graph by
studying the (largest or lowest) eigenvalues of a symmetric real matrix such as
e.g. the adjacency or the Laplacian. Recently, it has been argued that using
instead a more complicated, non-symmetric and higher dimensional operator,
related to the non-backtracking walk on the graph, leads to improved
performance in detecting clusters, and even to optimal performance for the
stochastic block model. Here, we propose to use instead a simpler object, a
symmetric real matrix known as the Bethe Hessian operator, or deformed
Laplacian. We show that this approach combines the performances of the
non-backtracking operator, thus detecting clusters all the way down to the
theoretical limit in the stochastic block model, with the computational,
theoretical and memory advantages of real symmetric matrices.Comment: 8 pages, 2 figure
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