Modular Irregular Labeling on Double-Star and Friendship Graphs

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

Irregularity and Modular Irregularity Strength of Wheels

It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means that no simple graph of the order of at least two is, in this way, irregular. However, a multigraph can be irregular. Chartrand et al., in 1988, posed the following problem: in a loopless multigraph, how can one determine the fewest parallel edges required to ensure that all vertices have distinct degrees? This problem is known as the graph labeling problem and, for its solution, Chartrand et al. introduced irregular assignments. The irregularity strength of a graph G is known as the maximal edge label used in an irregular assignment, minimized over all irregular assignments. Thus, the irregularity strength of a simple graph G is equal to the smallest maximum multiplicity of an edge of G in order to create an irregular multigraph from G. In the present paper, we show the existence of a required irregular labeling scheme that proves the exact value of the irregularity strength of wheels. Then, we modify this irregular mapping in six cases and obtain labelings that determine the exact value of the modular irregularity strength of wheels as a natural modification of the irregularity strength.

Input-Dynamic Distributed Algorithms for Communication Networks

Consider a distributed task where the communication network is fixed but the local inputs given to the nodes of the distributed system may change over time. In this work, we explore the following question: if some of the local inputs change, can an existing solution be updated efficiently, in a dynamic and distributed manner? To address this question, we define the batch dynamic \congest model in which we are given a bandwidth-limited communication network and a dynamic edge labelling defines the problem input. The task is to maintain a solution to a graph problem on the labeled graph under batch changes. We investigate, when a batch of α edge label changes arrive, \beginitemize \item how much time as a function of α we need to update an existing solution, and \item how much information the nodes have to keep in local memory between batches in order to update the solution quickly. \enditemize Our work lays the foundations for the theory of input-dynamic distributed network algorithms. We give a general picture of the complexity landscape in this model, design both universal algorithms and algorithms for concrete problems, and present a general framework for lower bounds. In particular, we derive non-trivial upper bounds for two selected, contrasting problems: maintaining a minimum spanning tree and detecting cliques.

Implicit Landmark Path Indexing for Bounded Label Constrained Reachable Paths

In today’s Big Data era, a graph is an essential tool that models the semi-structured or unstructured data. Graph reachability with vertex or edge constraints is one of the basic queries to extract useful information from the graph data. From the graph reachability with constraints, we obtained the information about the existence of a path between the given two vertices satisfying the vertex or edge constraints. The problem of Label Constraint Reachability (LCR) found the existence of a path between the two given vertices such that the edge-labels along the path are the subset of the given edge-label constraint. We extended the LCR queries by considering weighted directed graphs and proposed a novel technique of finding paths for LCR queries bounded by path weight. We termed these paths as bounded label constrained reachable paths (BLCRP). We extended the landmark path indexing technique [1] by incorporating the implicit paths which satisfy the user constraints but need not satisfy the minimality of edge label sets. We solved the BLCRP by using the extended landmark path indexing and BFS based query processing. We addressed the following challenges through our proposed technique of implicit landmark path indexing in the problem of BLCRP that included (1) the need to handle exponential number of edge label combinations with an additional total path weight constraint, and (2) the need to discover a technique that finds exact reachable paths between the given vertices. This problem could be applied to real network scenarios like road networks, social networks, and proteinprotein interaction networks. Our experiments and statistical analysis revealed the accuracy and efficiency of the proposed approach tested on synthetic and real datasets.

mSphere of Influence: the Rise of Artificial Intelligence in Infection Biology

ABSTRACT Artur Yakimovich works in the field of computational virology and applies machine learning algorithms to study host-pathogen interactions. In this mSphere of Influence article, he reflects on two papers “Holographic Deep Learning for Rapid Optical Screening of Anthrax Spores” by Jo et al. (Y. Jo, S. Park, J. Jung, J. Yoon, et al., Sci Adv 3:e1700606, 2017, https://doi.org/10.1126/sciadv.1700606) and “Bacterial Colony Counting with Convolutional Neural Networks in Digital Microbiology Imaging” by Ferrari and colleagues (A. Ferrari, S. Lombardi, and A. Signoroni, Pattern Recognition 61:629–640, 2017, https://doi.org/10.1016/j.patcog.2016.07.016). Here he discusses how these papers made an impact on him by showcasing that artificial intelligence algorithms can be equally applicable to both classical infection biology techniques and cutting-edge label-free imaging of pathogens.

An Adiabatic Quantum Algorithm for Determining Gracefulness of a Graph

Graph labelling is one of the noticed contexts in combinatorics and graph theory. Graceful labelling for a graph G with e edges, is to label the vertices of G with 0, 1, ℒ, e such that, if we specify to each edge the difference value between its two ends, then any of 1, 2, ℒ, e appears exactly once as an edge label. For a given graph, there are still few efficient classical algorithms that determine either it is graceful or not, even for trees – as a well-known class of graphs. In this paper, we introduce an adiabatic quantum algorithm, which for a graceful graph G finds a graceful labelling. Also, this algorithm can determine if G is not graceful. Numerical simulations of the algorithm reveal that its time complexity has a polynomial behaviour with the problem size up to the range of 15 qubits. A general sufficient condition for a combinatorial optimization problem to have a satisfying adiabatic solution is also derived.