4,521 research outputs found
Extensions of Erd\H{o}s-Gallai Theorem and Luo's Theorem with Applications
The famous Erd\H{o}s-Gallai Theorem on the Tur\'an number of paths states
that every graph with vertices and edges contains a path with at least
edges. In this note, we first establish a simple but novel
extension of the Erd\H{o}s-Gallai Theorem by proving that every graph
contains a path with at least edges,
where denotes the number of -cliques in for . We also construct a family of graphs which shows our extension
improves the estimate given by Erd\H{o}s-Gallai Theorem. Among applications, we
show, for example, that the main results of \cite{L17}, which are on the
maximum possible number of -cliques in an -vertex graph without a path
with vertices (and without cycles of length at least ), can be easily
deduced from this extension. Indeed, to prove these results, Luo \cite{L17}
generalized a classical theorem of Kopylov and established a tight upper bound
on the number of -cliques in an -vertex 2-connected graph with
circumference less than . We prove a similar result for an -vertex
2-connected graph with circumference less than and large minimum degree. We
conclude this paper with an application of our results to a problem from
spectral extremal graph theory on consecutive lengths of cycles in graphs.Comment: 6 page
Bounds for the spectral radius of nonnegative matrices
We give upper and lower bounds for the spectral radius of a nonnegative
matrix by using its average 2-row sums, and characterize the equality cases if
the matrix is irreducible. We also apply these bounds to various nonnegative
matrices associated with a graph, including the adjacency matrix, the signless
Laplacian matrix, the distance matrix, the distance signless Laplacian matrix,
and the reciprocal distance matrix
Creativity and Artificial Intelligence: A Digital Art Perspective
This paper describes the application of artificial intelligence to the
creation of digital art. AI is a computational paradigm that codifies
intelligence into machines. There are generally three types of artificial
intelligence and these are machine learning, evolutionary programming and soft
computing. Machine learning is the statistical approach to building intelligent
systems. Evolutionary programming is the use of natural evolutionary systems to
design intelligent machines. Some of the evolutionary programming systems
include genetic algorithm which is inspired by the principles of evolution and
swarm optimization which is inspired by the swarming of birds, fish, ants etc.
Soft computing includes techniques such as agent based modelling and fuzzy
logic. Opportunities on the applications of these to digital art are explored.Comment: 5 page
Blockchain and Artificial Intelligence
It is undeniable that artificial intelligence (AI) and blockchain concepts
are spreading at a phenomenal rate. Both technologies have distinct degree of
technological complexity and multi-dimensional business implications. However,
a common misunderstanding about blockchain concept, in particular, is that
blockchain is decentralized and is not controlled by anyone. But the underlying
development of a blockchain system is still attributed to a cluster of core
developers. Take smart contract as an example, it is essentially a collection
of codes (or functions) and data (or states) that are programmed and deployed
on a blockchain (say, Ethereum) by different human programmers. It is thus,
unfortunately, less likely to be free of loopholes and flaws. In this article,
through a brief overview about how artificial intelligence could be used to
deliver bug-free smart contract so as to achieve the goal of blockchain 2.0, we
to emphasize that the blockchain implementation can be assisted or enhanced via
various AI techniques. The alliance of AI and blockchain is expected to create
numerous possibilities
Laplacian and signless Laplacian spectral radii of graphs with fixed domination number
In this paper, we determine the maximal Laplacian and signless Laplacian
spectral radii for graphs with fixed number of vertices and domination number,
and characterize the extremal graphs respectively
Ordering trees having small reverse Wiener indices
The reverse Wiener index of a connected graph is a variation of the
well-known Wiener index defined as the sum of distances between all
unordered pairs of vertices of . It is defined as
, where is the number of vertices, and
is the diameter of . We now determine the second and the third smallest
reverse Wiener indices of -vertex trees and characterize the trees whose
reverse Wiener indices attain these values for (it has been known that
the star is the unique tree with the smallest reverse Wiener index)
Graphs characterized by the second distance eigenvalue
We characterize all connected graphs with second distance eigenvalue less
than
Both necessary and sufficient conditions for Bayesian exponential consistency
The last decade has seen a remarkable development in the theory of
asymptotics of Bayesian nonparametric procedures. Exponential consistency has
played an important role in this area. It is known that the condition of
being in the Kullback-Leibler support of the prior cannot ensure exponential
consistency of posteriors. Many authors have obtained additional sufficient
conditions for exponential consistency of posteriors, see, for instance,
Schwartz (1965), Barron, Schervish and Wasserman (1999), Ghosal, Ghosh and
Ramamoorthi (1999), Walker (2004), Xing and Ranneby (2008). However, given the
Kullback-Leibler support condition, less is known about both necessary and
sufficient conditions. In this paper we give one type of both necessary and
sufficient conditions. As a consequence we derive a simple sufficient condition
on Bayesian exponential consistency, which is weaker than the previous
sufficient conditions
Tuning quantum discord in Josephson charge qubits system
A type of two qubits Josephson charge system is constructed in this paper,
and properties of the quantum discord (QD) as well as the differences between
thermal QD and thermal entanglement were investigated. A detailed calculation
shows that the magnetic flux is more efficient than the voltage
in tuning QD. By choosing proper system parameters, one can realize
the maximum QD in our two qubits Josephson charge system.Comment: 5 pages, 5 figures. arXiv admin note: text overlap with
arXiv:cond-mat/0306209 by other author
Extremal problems on the Hamiltonicity of claw-free graphs
In 1962, Erd\H{o}s proved that if a graph with vertices satisfies where the minimum degree
and , then it is Hamiltonian. For , let , where "" is the "join"
operation. One can observe and is not
Hamiltonian. As contains induced claws for , a natural
question is to characterize all 2-connected claw-free non-Hamiltonian graphs
with the largest possible number of edges. We answer this question completely
by proving a claw-free analog of Erd\H{o}s' theorem. Moreover, as byproducts,
we establish several tight spectral conditions for a 2-connected claw-free
graph to be Hamiltonian. Similar results for the traceability of connected
claw-free graphs are also obtained. Our tools include Ryj\'a\v{c}ek's claw-free
closure theory and Brousek's characterization of minimal 2-connected claw-free
non-Hamiltonian graphs.Comment: 22 pages, 8 figures, to appear in Discrete Mathematic
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