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 $n$ vertices and $m$ edges contains a path with at least $\frac{2m}{n}$ 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 $G$ contains a path with at least $\frac{(s+1)N_{s+1}(G)}{N_{s}(G)}+s-1$ edges, where $N_j(G)$ denotes the number of $j$-cliques in $G$ for $1\leq j\leq\omega(G)$. 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 $s$-cliques in an $n$-vertex graph without a path with $l$ vertices (and without cycles of length at least $c$), 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 $s$-cliques in an $n$-vertex 2-connected graph with circumference less than $c$. We prove a similar result for an $n$-vertex 2-connected graph with circumference less than $c$ 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 $G$ is a variation of the well-known Wiener index $W(G)$ defined as the sum of distances between all unordered pairs of vertices of $G$. It is defined as $\Lambda(G)=\frac{1}{2}n(n-1)d-W(G)$, where $n$ is the number of vertices, and $d$ is the diameter of $G$. We now determine the second and the third smallest reverse Wiener indices of $n$-vertex trees and characterize the trees whose reverse Wiener indices attain these values for $n\ge 6$ (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 $-0.5858$

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 $f_0$ 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 $\Phi_{Xk}$ is more efficient than the voltage $V_{Xi}$ 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 $G$ with $n$ vertices satisfies $$e(G)>\max\left\{\binom{n-k}{2}+k^2,\binom{\lceil(n+1)/2\rceil}{2}+\left\lfloor \frac{n-1}{2}\right\rfloor^2\right\},$$ where the minimum degree $\delta(G)\geq k$ and $1\leq k\leq(n-1)/2$, then it is Hamiltonian. For $n \geq 2k+1$, let $E^k_n=K_{k}\vee (kK_1+K_{n-2k})$, where "$\vee$" is the "join" operation. One can observe $e(E^k_n)=\binom{n-k}{2}+k^2$ and $E^k_n$ is not Hamiltonian. As $E^k_n$ contains induced claws for $k\geq 2$, 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