12,222 research outputs found
Distance Based Topological Indices of Double graphs and Strong Double graphs
Topological index is a numerical representation of structure of graph. They are mainly classified as Distance and Degree based topological indices. In this article Distance based topological indices of Double graphs and Strong Double graphs are calculated. Let be a graph of order with the vertex set containing vertices . Double graph of graph is constructed by taking two copies of G in which a vertex in one copy is adjacent to a vertex in the another copy if and are adjacent in G. Strong Double graph is a double graph in which a vertex in one copy is adjacent to a vertex in the another copy if .
Distance Based Topological Indices of Double graphs and Strong Double graphs
Topological index is a numerical representation of structure of graph. They are mainly classified as Distance and Degree based topological indices. In this article Distance based topological indices of Double graphs and Strong Double graphs are calculated. Let be a graph of order with the vertex set containing vertices . Double graph of graph is constructed by taking two copies of G in which a vertex in one copy is adjacent to a vertex in the another copy if and are adjacent in G. Strong Double graph is a double graph in which a vertex in one copy is adjacent to a vertex in the another copy if
Accounting for the Role of Long Walks on Networks via a New Matrix Function
We introduce a new matrix function for studying graphs and real-world
networks based on a double-factorial penalization of walks between nodes in a
graph. This new matrix function is based on the matrix error function. We find
a very good approximation of this function using a matrix hyperbolic tangent
function. We derive a communicability function, a subgraph centrality and a
double-factorial Estrada index based on this new matrix function. We obtain
upper and lower bounds for the double-factorial Estrada index of graphs,
showing that they are similar to those of the single-factorial Estrada index.
We then compare these indices with the single-factorial one for simple graphs
and real-world networks. We conclude that for networks containing chordless
cycles---holes---the two penalization schemes produce significantly different
results. In particular, we study two series of real-world networks representing
urban street networks, and protein residue networks. We observe that the
subgraph centrality based on both indices produce significantly different
ranking of the nodes. The use of the double factorial penalization of walks
opens new possibilities for studying important structural properties of
real-world networks where long-walks play a fundamental role, such as the cases
of networks containing chordless cycles
Band Connectivity for Topological Quantum Chemistry: Band Structures As A Graph Theory Problem
The conventional theory of solids is well suited to describing band
structures locally near isolated points in momentum space, but struggles to
capture the full, global picture necessary for understanding topological
phenomena. In part of a recent paper [B. Bradlyn et al., Nature 547, 298
(2017)], we have introduced the way to overcome this difficulty by formulating
the problem of sewing together many disconnected local "k-dot-p" band
structures across the Brillouin zone in terms of graph theory. In the current
manuscript we give the details of our full theoretical construction. We show
that crystal symmetries strongly constrain the allowed connectivities of energy
bands, and we employ graph-theoretic techniques such as graph connectivity to
enumerate all the solutions to these constraints. The tools of graph theory
allow us to identify disconnected groups of bands in these solutions, and so
identify topologically distinct insulating phases.Comment: 19 pages. Companion paper to arXiv:1703.02050 and arXiv:1706.08529
v2: Accepted version, minor typos corrected and references added. Now
19+epsilon page
Fermionic Matrix Models
We review a class of matrix models whose degrees of freedom are matrices with
anticommuting elements. We discuss the properties of the adjoint fermion one-,
two- and gauge invariant D-dimensional matrix models at large-N and compare
them with their bosonic counterparts which are the more familiar Hermitian
matrix models. We derive and solve the complete sets of loop equations for the
correlators of these models and use these equations to examine critical
behaviour. The topological large-N expansions are also constructed and their
relation to other aspects of modern string theory such as integrable
hierarchies is discussed. We use these connections to discuss the applications
of these matrix models to string theory and induced gauge theories. We argue
that as such the fermionic matrix models may provide a novel generalization of
the discretized random surface representation of quantum gravity in which the
genus sum alternates and the sums over genera for correlators have better
convergence properties than their Hermitian counterparts. We discuss the use of
adjoint fermions instead of adjoint scalars to study induced gauge theories. We
also discuss two classes of dimensionally reduced models, a fermionic vector
model and a supersymmetric matrix model, and discuss their applications to the
branched polymer phase of string theories in target space dimensions D>1 and
also to the meander problem.Comment: 139 pages Latex (99 pages in landscape, two-column option); Section
on Supersymmetric Matrix Models expanded, additional references include
The Tensor Track, III
We provide an informal up-to-date review of the tensor track approach to
quantum gravity. In a long introduction we describe in simple terms the
motivations for this approach. Then the many recent advances are summarized,
with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion,
Osterwalder-Schrader positivity...) which, while important for the tensor track
program, are not detailed in the usual quantum gravity literature. We list open
questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
A Givental-like Formula and Bilinear Identities for Tensor Models
In this paper we express some simple random tensor models in a Givental-like
fashion i.e. as differential operators acting on a product of generic
1-Hermitian matrix models. Finally we derive Hirota's equations for these
tensor models. Our decomposition is a first step towards integrability of such
models.Comment: 18 pages, 1 figur
Random tensor models in the large N limit: Uncoloring the colored tensor models
Tensor models generalize random matrix models in yielding a theory of
dynamical triangulations in arbitrary dimensions. Colored tensor models have
been shown to admit a 1/N expansion and a continuum limit accessible
analytically. In this paper we prove that these results extend to the most
general tensor model for a single generic, i.e. non-symmetric, complex tensor.
Colors appear in this setting as a canonical book-keeping device and not as a
fundamental feature. In the large N limit, we exhibit a set of Virasoro
constraints satisfied by the free energy and an infinite family of
multicritical behaviors with entropy exponents \gamma_m=1-1/m.Comment: 15 page
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