The Asymptotic value of the Locating-Chromatic number of nn-ary Tree

In 2013, Welyyanti et al. proved that $\chi_L(T(n,k))\leq n+k-1$ for all positive integers $n,k\geq2$. In this paper we prove that for any fixed integer $k\geq2$, almost all $n$-ary Tree $T(n,k)$ satisfy $\chi_L(T(n,k))=n+k-1$, moreover $\lim\limits_{n\to \infty} \chi_L(T(n,k))-n=k-1$.Comment: 4 Pages, 1 figur

On the size of identifying codes in triangle-free graphs

In an undirected graph $G$, a subset $C\subseteq V(G)$ such that $C$ is a dominating set of $G$, and each vertex in $V(G)$ is dominated by a distinct subset of vertices from $C$, is called an identifying code of $G$. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. For a given identifiable graph $G$, let \M(G) be the minimum cardinality of an identifying code in $G$. In this paper, we show that for any connected identifiable triangle-free graph $G$ on $n$ vertices having maximum degree $\Delta\geq 3$, \M(G)\le n-\tfrac{n}{\Delta+o(\Delta)}. This bound is asymptotically tight up to constants due to various classes of graphs including $(\Delta-1)$-ary trees, which are known to have their minimum identifying code of size $n-\tfrac{n}{\Delta-1+o(1)}$. We also provide improved bounds for restricted subfamilies of triangle-free graphs, and conjecture that there exists some constant $c$ such that the bound \M(G)\le n-\tfrac{n}{\Delta}+c holds for any nontrivial connected identifiable graph $G$

Neighbor-locating colorings in graphs

A k -coloring of a graph G is a k -partition ¿ = { S 1 ,...,S k } of V ( G ) into independent sets, called colors . A k -coloring is called neighbor-locating if for every pair of vertices u,v belonging to the same color S i , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v . The neighbor-locating chromatic number ¿ NL ( G ) is the minimum cardinality of a neighbor-locating coloring of G . We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order n = 5 with neighbor-locating chromatic number n or n - 1. We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphsPreprin

PEA265: Perceptual Assessment of Video Compression Artifacts

The most widely used video encoders share a common hybrid coding framework that includes block-based motion estimation/compensation and block-based transform coding. Despite their high coding efficiency, the encoded videos often exhibit visually annoying artifacts, denoted as Perceivable Encoding Artifacts (PEAs), which significantly degrade the visual Qualityof- Experience (QoE) of end users. To monitor and improve visual QoE, it is crucial to develop subjective and objective measures that can identify and quantify various types of PEAs. In this work, we make the first attempt to build a large-scale subjectlabelled database composed of H.265/HEVC compressed videos containing various PEAs. The database, namely the PEA265 database, includes 4 types of spatial PEAs (i.e. blurring, blocking, ringing and color bleeding) and 2 types of temporal PEAs (i.e. flickering and floating). Each containing at least 60,000 image or video patches with positive and negative labels. To objectively identify these PEAs, we train Convolutional Neural Networks (CNNs) using the PEA265 database. It appears that state-of-theart ResNeXt is capable of identifying each type of PEAs with high accuracy. Furthermore, we define PEA pattern and PEA intensity measures to quantify PEA levels of compressed video sequence. We believe that the PEA265 database and our findings will benefit the future development of video quality assessment methods and perceptually motivated video encoders.Comment: 10 pages,15 figures,4 table

Trees and Markov convexity

We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors

Some invariants related to threshold and chain graphs

Let G = (V, E) be a finite simple connected graph. We say a graph G realizes a code of the type 0^s_1 1^t_1 0^s_2 1^t_2 ... 0^s_k1^t_k if and only if G can obtained from the code by some rule. Some classes of graphs such as threshold and chain graphs realizes a code of the above mentioned type. In this paper, we develop some computationally feasible methods to determine some interesting graph theoretical invariants. We present an efficient algorithm to determine the metric dimension of threshold and chain graphs. We compute threshold dimension and restricted threshold dimension of threshold graphs. We discuss L(2, 1)-coloring of threshold and chain graphs. In fact, for every threshold graph G, we establish a formula by which we can obtain the {\lambda}-chromatic number of G. Finally, we provide an algorithm to compute the {\lambda}-chromatic number of chain graphs

Colour Communication Within Different Languages

For computational methods aiming to reproduce colour names that are meaningful to speakers of different languages, the mapping between perceptual and linguistic aspects of colour is a problem of central information processing. This thesis advances the field of computational colour communication within different languages in five main directions. First, we show that web-based experimental methodologies offer considerable advantages in obtaining a large number of colour naming responses in British and American English, Greek, Russian, Thai and Turkish. We continue with the application of machine learning methods to discover criteria in linguistic, behavioural and geometric features of colour names that distinguish classes of colours. We show that primary colour terms do not form a coherent class, whilst achromatic and basic classes do. We then propose and evaluate a computational model trained by human responses in the online experiment to automate the assignment of colour names in different languages across the full three-dimensional colour gamut. Fourth, we determine for the first time the location of colour names within a physiologically-based cone excitation space through an unconstrained colour naming experiment using a calibrated monitor under controlled viewing conditions. We show a good correspondence between online and offline datasets; and confirm the validity of both experimental methodologies for estimating colour naming functions in laboratory and real-world monitor settings. Finally, we present a novel information theoretic measure, called dispensability, for colour categories that predicts a gradual scale of basicness across languages from both web- and laboratory- based unconstrained colour naming datasets. As a result, this thesis contributes experimental and computational methodologies towards the development of multilingual colour communication schemes